diff --git a/.github/workflows/ci.yml b/.github/workflows/ci.yml index 76aa271..2130925 100644 --- a/.github/workflows/ci.yml +++ b/.github/workflows/ci.yml @@ -117,16 +117,17 @@ jobs: - uses: Swatinem/rust-cache@v2 - name: Check all pub fns are annotated + # Every non-const pub fn in src/ops, methods included, sits under its + # no_panic attribute; const fns cannot carry it. run: | - pub_fns=$(grep -r '^pub fn ' src/ops/*.rs | wc -l) - annotations=$(grep -r 'no_panic::no_panic' src/ops/*.rs | wc -l) - echo "pub fn count: $pub_fns" - echo "no_panic annotations: $annotations" - if [ "$pub_fns" -ne "$annotations" ]; then - echo "::error::Not all public functions in src/ops/ have #[no_panic] annotation ($annotations/$pub_fns)" - grep -n '^pub fn ' src/ops/*.rs + awk 'FNR == 1 { last = -10 } + /no_panic::no_panic/ { last = FNR } + /^[[:space:]]*pub fn / && FNR - last > 3 { print FILENAME ":" FNR ": " $0; bad = 1 } + END { exit bad }' src/ops/*.rs || { + echo "::error::These public functions in src/ops/ lack the no_panic annotation" exit 1 - fi + } + echo "annotated: $(grep -c 'no_panic::no_panic' src/ops/*.rs | awk -F: '{ s += $2 } END { print s }')" - name: Build with no-panic verification run: cargo build --profile no-panic-check --features verify-no-panic --bin verify_no_panic @@ -166,13 +167,15 @@ jobs: if [ "${{ github.event_name }}" = "pull_request" ]; then git fetch origin ${{ github.base_ref }} git show origin/${{ github.base_ref }}:tools/accuracy-bench/baseline.json > /tmp/baseline-from-main.json + git show origin/${{ github.base_ref }}:tools/accuracy-bench/worst.txt > /tmp/worst-from-main.txt || true else git show HEAD~1:tools/accuracy-bench/baseline.json > /tmp/baseline-from-main.json || cp tools/accuracy-bench/baseline.json /tmp/baseline-from-main.json + git show HEAD~1:tools/accuracy-bench/worst.txt > /tmp/worst-from-main.txt || true fi - name: Run accuracy benchmark working-directory: tools/accuracy-bench - run: cargo run --release -- --baseline /tmp/baseline-from-main.json + run: cargo run --release -- --quick --baseline /tmp/baseline-from-main.json --worst-baseline /tmp/worst-from-main.txt - name: Upload report if: always() diff --git a/CHANGELOG.md b/CHANGELOG.md new file mode 100644 index 0000000..4f99bc7 --- /dev/null +++ b/CHANGELOG.md @@ -0,0 +1,93 @@ +# Changelog + +A release that changes any output bit of an existing function names that +function under the release, even for a one-ulp change. `tests/golden.txt` +pins a hash of every function's outputs on six types, and the tests fail when +one moves; `GOLDEN_UPDATE=1 cargo test golden` re-pins it and stamps each +changed row with the crate version, whose section here must then name the +function in backticks. + +Releases before 3.1.1 are described on the +[releases page](https://github.com/GeEom/fixed_analytics/releases). + +## 4.0.0 + +Every function now evaluates in binary floating point at twice its type's +width or more and rounds once, so each is within one ulp of the exact result +on every type, and nearest in every case the accuracy tool has met. Odd and +even functions are exactly so, monotone functions stay monotone, and sin, +cos and tan reduce their argument exactly at any magnitude. + +Output changes, all of them corrections: `sin`, `cos`, `tan`, `sin_cos`, +`asin`, `acos`, `atan`, `atan2`, `sinh`, `cosh`, `tanh`, `sinh_cosh`, `coth`, +`asinh`, `acosh`, `atanh`, `acoth`, `exp`, `pow2`, `ln`, `log2`, `log10` and +`pow`. `sqrt` is unchanged. `tan` saturates only where the tangent is out of +range, and an odd function's result past `MIN` saturates to `MIN` rather than +`-MAX`. `acos` reports domain errors under its own name. + +Breaking: `CordicNumber` is sealed, as its implementations now carry the +working arithmetic. + +New, every one generic over the types: + +- Arithmetic from the exact result, rounded once: `mul_add`, `mul_div`, + `div_or_saturate`, `dot` and `sqrt_or_zero`, and the same rounded toward + −∞ or +∞ by a `Rounding`, `mul_rounded`, `div_rounded`, `mul_div_rounded`, + `mul_add_rounded` and `dot_rounded`. `ERROR_ULPS` is the bound every + function meets. +- Scale-aware algebra: `hypot`, `norm` and `normalize`, which form their + squares without underflow; `scale_exponent` and `scale_by` for exact + power-of-two scaling; `sqrt_product`, `rsqrt`, and `asin_from_gap`, which + takes the gap to 1 in place of a value that would round. +- Exponential and logarithm relatives, each to full relative precision near + 0: `expm1`; `exprel` and `exprel2`, `(e^x − 1)/x` and `(e^x − 1 − x)/x²`; + `exp_moments`, `[E₁, …, E₅, e^(−x)]` with `Eₖ = ∫₀¹ s^(k−1)·e^(−xs) ds`, + all from one exponential; `log1p`, `log1p_ratio`, `log1p_residual` and + `log1p_residual_ratio`; `stumpff`, the Stumpff functions `c₀` to `c₃` of + either sign; and `tan_ratio`, `atan_ratio`, `atanh_ratio` and `tanh_ratio`. +- Elliptic integrals and functions of a `Modulus`, which holds `m` or its + complement `k'` exact: `ellip_k`, `ellip_e` and `ellip_pi_complete`; + `ellip_pi`, the incomplete third kind from `(sn, cn, dn)`, and + `ellip_pi_increment`, its change from `u` to `u + step` for `n < 1`, which + keeps its bits as the step shrinks; `jacobi_sn_cn_dn` and `jacobi_am` of + any argument, and `jacobi_inverse`; and Carlson's `carlson_rf`, + `carlson_rj`, `carlson_rc` and `carlson_rd`, with `carlson_rf_roots` and + `carlson_rj_roots` taking square roots for arguments too small to hold. +- Angles in turns: `sin_cos_turns`, whose reduction is its own bits with no + rounded π, `atan2_turns`, `div_by_tau` and `mul_by_tau`, each rounded + once, and `wrap_radians` and `wrap_principal`, which reduce to `[0, 2π)` + and `(−π, π]` by exact multiples of 2π. +- `Complex`, whose arithmetic rounds each part once and whose division + cannot overflow for a non-zero divisor, the operators also by name as + `saturating_add`, `saturating_sub`, `saturating_mul`, `saturating_div` and + `saturating_neg`; its `exp`, `exprel`, `exprel2`, `ln` and real power + `pow`; `exprel_weighted`, `exprel` and `exprel2` under the weight + `((1 − εt)/(1 − εs))^p/(1 − εs)`, precise as `ε → 0`; and the Faddeeva + function `faddeeva`, `erf`, the exponential integral `e1` and the upper + incomplete gamma function `gamma_upper` for `a` in `(0, 128]`, on the + principal branch. `erf`, `erfc`, `fresnel` and `gamma` take a real + argument. +- Solvers: `root_in`, a bracketed root finder by the ITP method with a fixed + budget, and `kepler_universal`, the universal anomaly of a two-body orbit. +- Enclosures that hold the exact value: `enclose`, one ulp each way, and + ranges over an interval rounded outward, `sin_cos_range`, + `sin_cos_range_turns`, `exp_range`, `atan_range` and `tanh_range`, and for + the kernels `exprel_range`, `exprel2_range`, `exp_moments_range`, + `log1p_range`, `log1p_ratio_range`, `log1p_residual_ratio_range`, + `tanh_ratio_range`, `atan_ratio_range`, `atanh_ratio_range`, + `tan_ratio_range`, `stumpff_range`, `sqrt_range`, `hypot_range` and + `atan2_range`. The Stumpff ranges bound any extreme between the ends. + +Speed against 3.1.1, per call: I16F16's elementary functions take 3.4 to 9 +times as long and I32F32's 1.6 to 6 times, while I64F64's run 0.65 to 2.4 +times as long, `ln` 1.5 and `atan2` 1.2 times faster, as `sin_cos`, `exp`, +`ln` and `atan2` work in 126-bit fixed point on 128-bit types with at most 96 +fractional bits; `sqrt` is unchanged. A function works in a float as wide as +its type when its result leaves 24 bits past the ulp, Carlson's duplication +stops once its closing series is exact, and the Jacobi functions take +Bulirsch's descent: on I64F64 `carlson_rj_roots` takes 6.7 µs and +`jacobi_sn_cn_dn` 2.5 µs. + +## 3.1.1 + +No output changes. The package leaves out the toolchain and CI files. diff --git a/Cargo.lock b/Cargo.lock index d09fe43..1e632ed 100644 --- a/Cargo.lock +++ b/Cargo.lock @@ -222,7 +222,7 @@ dependencies = [ [[package]] name = "fixed_analytics" -version = "3.1.1" +version = "4.0.0" dependencies = [ "criterion", "fixed", diff --git a/Cargo.toml b/Cargo.toml index 5a041ad..cc77afb 100644 --- a/Cargo.toml +++ b/Cargo.toml @@ -1,6 +1,6 @@ [package] name = "fixed_analytics" -version = "3.1.1" +version = "4.0.0" edition.workspace = true rust-version.workspace = true authors = ["David Gathercole"] @@ -16,7 +16,7 @@ categories = ["mathematics", "no-std", "algorithms", "embedded"] # and CI configuration, which would pin anyone building the packaged crate to # this repo's toolchain. exclude = [ - "docs/exp.md", "docs/sin_cos.md", "docs/sinh_cosh.md", "docs/sqrt.md", + "docs/engine.md", "docs/sqrt.md", "rust-toolchain.toml", "clippy.toml", "codecov.yml", ".github/", ] diff --git a/README.md b/README.md index 8a6b5e1..bfa8c0b 100644 --- a/README.md +++ b/README.md @@ -30,14 +30,14 @@ Requires Rust 1.95 or later. ```toml [dependencies] -fixed_analytics = "3.1.1" +fixed_analytics = "4.0.0" ``` For `no_std` environments: ```toml [dependencies] -fixed_analytics = { version = "3.1.1", default-features = false } +fixed_analytics = { version = "4.0.0", default-features = false } ``` ## Available Functions @@ -49,12 +49,20 @@ fixed_analytics = { version = "3.1.1", default-features = false } | Category | Total Functions | Fallible Functions | |----------|-----------------|-------------------| -| Trigonometric | `sin`, `cos`, `tan`, `sin_cos`, `atan`, `atan2` | `asin`, `acos` | +| Trigonometric | `sin`, `cos`, `tan`, `sin_cos`, `atan`, `atan2` | `asin`, `acos`, `asin_from_gap` | | Hyperbolic | `sinh`, `cosh`, `tanh`, `sinh_cosh`, `asinh` | `acosh`, `atanh`, `acoth`, `coth` | -| Exponential | `exp`, `pow2` | `ln`, `log2`, `log10`, `pow` | -| Algebraic | — | `sqrt` | - -Functions are calculated via polynomial evaluation, CORDIC, and Newton-Raphson techniques. Complete absence of panic is verified at the linker level via the [`no-panic`](https://github.com/dtolnay/no-panic) crate. +| Exponential | `exp`, `pow2`, `expm1`, `exprel`, `exprel2`, `exp_moments` | `ln`, `log2`, `log10`, `pow`, `log1p`, `log1p_ratio`, `log1p_residual`, `log1p_residual_ratio` | +| Turns | `sin_cos_turns`, `atan2_turns`, `div_by_tau`, `mul_by_tau`, `wrap_radians`, `wrap_principal` | — | +| Ratios | `stumpff`, `tan_ratio`, `atan_ratio`, `tanh_ratio` | `atanh_ratio` | +| Algebraic | `sqrt_or_zero`, `hypot`, `norm`, `scale_exponent`, `scale_by` | `sqrt`, `sqrt_product`, `rsqrt`, `normalize` | +| Special | `erf`, `erfc`, `fresnel` | `gamma` | +| Complex (`Complex`) | `exp`, `exprel`, `exprel2`, `faddeeva`, `erf` | `ln`, `pow`, `exprel_weighted`, `e1`, `gamma_upper` | +| Elliptic | `ellip_e`, `jacobi_sn_cn_dn`, `jacobi_am` | `ellip_k`, `ellip_pi`, `ellip_pi_complete`, `ellip_pi_increment`, `jacobi_inverse`, `carlson_rf`, `carlson_rf_roots`, `carlson_rj`, `carlson_rj_roots`, `carlson_rc`, `carlson_rd` | +| Arithmetic | `mul_add`, `mul_div`, `div_or_saturate`, `dot`, and by `Rounding` `mul_rounded`, `div_rounded`, `mul_div_rounded`, `mul_add_rounded`, `dot_rounded` | — | +| Solvers | — | `root_in`, `kepler_universal` | +| Enclosures | `enclose`, `sin_cos_range`, `sin_cos_range_turns`, `exp_range`, `atan_range`, `tanh_range`, `exprel_range`, `exprel2_range`, `exp_moments_range`, `tanh_ratio_range`, `atan_ratio_range`, `tan_ratio_range`, `stumpff_range`, `sqrt_range`, `hypot_range`, `atan2_range` | `log1p_range`, `log1p_ratio_range`, `log1p_residual_ratio_range`, `atanh_ratio_range` | + +Each function works in binary floating point at twice its type's width or more, from its arguments converted exactly, and rounds once to the type. Every one is within one ulp of the exact result on every type, `ERROR_ULPS`, and the arithmetic functions within half an ulp, or rounded by `Rounding` to the nearest value below or above. Odd and even functions are exactly so and monotone functions stay monotone. Complete absence of panic is verified at the linker level via the [`no-panic`](https://github.com/dtolnay/no-panic) crate. ### Saturation Behavior @@ -68,7 +76,7 @@ The following total functions saturate, clamping to the representable range near | `pow2` | x ≤ -17.0 | x ≤ -33.0 | Zero | | `sinh` | \|x\| ≥ 11.1 | \|x\| ≥ 22.2 | `T::MAX` or `T::MIN` | | `cosh` | \|x\| ≥ 11.1 | \|x\| ≥ 22.2 | `T::MAX` | -| `tan` | \|x - pole\| < 4e-5 | \|x - pole\| < 5e-10 | `T::MAX` or `T::MIN` | +| `tan` | \|x - pole\| < 3.1e-5 | \|x - pole\| < 4.7e-10 | `T::MAX` or `T::MIN` | Where for `tan`, "pole" refers to ±π/2, ±3π/2, ±5π/2, ... @@ -79,25 +87,153 @@ Relative error statistics measured against MPFR reference implementations. Accur | Function | I16F16 Mean | I16F16 Median | I16F16 P95 | I32F32 Mean | I32F32 Median | I32F32 P95 | |----------|-------------|---------------|------------|-------------|---------------|------------| -| sin | 6.06e-4 | 8.78e-5 | 1.28e-3 | 1.16e-8 | 1.68e-9 | 2.43e-8 | -| cos | 6.45e-4 | 9.03e-5 | 1.38e-3 | 1.22e-8 | 1.72e-9 | 2.64e-8 | -| tan | 7.20e-5 | 3.57e-5 | 2.20e-4 | 1.28e-9 | 3.98e-10 | 3.03e-9 | -| asin | 1.13e-4 | 2.80e-5 | 3.83e-4 | 3.68e-9 | 5.72e-10 | 6.40e-9 | -| acos | 1.79e-5 | 1.17e-5 | 4.81e-5 | 3.50e-10 | 2.11e-10 | 1.04e-9 | -| atan | 7.79e-6 | 6.49e-6 | 1.82e-5 | 1.88e-10 | 1.51e-10 | 4.38e-10 | -| sinh | 9.80e-5 | 6.23e-5 | 2.79e-4 | 1.52e-9 | 9.64e-10 | 4.29e-9 | -| cosh | 9.40e-5 | 5.75e-5 | 2.77e-4 | 1.44e-9 | 8.90e-10 | 4.25e-9 | -| tanh | 1.60e-5 | 1.32e-5 | 2.56e-5 | 2.25e-10 | 1.22e-10 | 3.90e-10 | -| coth | 6.68e-6 | 3.54e-6 | 1.80e-5 | 1.41e-10 | 1.16e-10 | 2.74e-10 | -| asinh | 2.42e-5 | 1.61e-5 | 5.08e-5 | 6.27e-10 | 5.01e-10 | 1.19e-9 | -| acosh | 1.87e-5 | 1.48e-5 | 4.65e-5 | 5.29e-10 | 4.75e-10 | 1.11e-9 | -| atanh | 2.21e-4 | 3.99e-5 | 3.33e-4 | 3.63e-9 | 7.20e-10 | 6.24e-9 | -| acoth | 1.21e-3 | 6.85e-4 | 4.09e-3 | 1.94e-8 | 1.23e-8 | 6.28e-8 | -| exp | 5.98e-3 | 1.47e-5 | 4.13e-2 | 9.49e-8 | 1.50e-9 | 6.50e-7 | -| ln | 1.18e-5 | 7.61e-6 | 2.09e-5 | 4.35e-10 | 3.78e-10 | 6.47e-10 | -| log2 | 9.98e-6 | 6.72e-6 | 2.01e-5 | 1.92e-10 | 1.31e-10 | 3.90e-10 | -| log10 | 1.25e-5 | 8.96e-6 | 2.36e-5 | 4.06e-10 | 3.57e-10 | 6.39e-10 | -| pow2 | 3.62e-4 | 2.24e-5 | 2.37e-3 | 5.64e-9 | 4.29e-10 | 3.67e-8 | -| pow | 6.90e-4 | 6.83e-5 | 3.13e-3 | 1.09e-8 | 1.23e-9 | 4.86e-8 | +| sin | 3.11e-5 | 6.12e-6 | 6.18e-5 | 4.65e-10 | 9.35e-11 | 9.59e-10 | +| cos | 3.14e-5 | 6.17e-6 | 6.29e-5 | 4.73e-10 | 9.33e-11 | 9.89e-10 | +| tan | 3.28e-5 | 1.26e-5 | 7.51e-5 | 5.37e-10 | 1.91e-10 | 1.13e-9 | +| asin | 3.89e-5 | 1.12e-5 | 9.52e-5 | 6.56e-10 | 1.72e-10 | 1.50e-9 | +| acos | 8.39e-6 | 3.69e-6 | 2.85e-5 | 1.29e-10 | 5.64e-11 | 4.25e-10 | +| atan | 2.72e-6 | 2.52e-6 | 4.78e-6 | 4.19e-11 | 3.88e-11 | 7.29e-11 | +| sinh | 7.41e-6 | 4.17e-6 | 1.26e-5 | 1.18e-10 | 6.33e-11 | 1.91e-10 | +| cosh | 3.79e-6 | 3.73e-6 | 7.35e-6 | 5.76e-11 | 5.65e-11 | 1.12e-10 | +| tanh | 5.38e-6 | 2.10e-6 | 9.51e-6 | 1.09e-10 | 6.18e-11 | 1.41e-10 | +| coth | 3.53e-6 | 2.24e-6 | 8.26e-6 | 7.05e-11 | 5.99e-11 | 1.28e-10 | +| asinh | 2.85e-6 | 1.37e-6 | 4.96e-6 | 4.43e-11 | 2.08e-11 | 7.56e-11 | +| acosh | 2.29e-6 | 1.36e-6 | 4.80e-6 | 3.59e-11 | 2.09e-11 | 7.45e-11 | +| atanh | 4.22e-5 | 1.40e-5 | 1.01e-4 | 7.11e-10 | 2.13e-10 | 1.56e-9 | +| acoth | 1.92e-4 | 1.40e-4 | 5.49e-4 | 2.94e-9 | 2.16e-9 | 8.40e-9 | +| exp | 5.98e-3 | 9.19e-6 | 4.13e-2 | 9.45e-8 | 1.39e-10 | 6.50e-7 | +| ln | 1.02e-6 | 6.57e-7 | 1.46e-6 | 1.39e-11 | 1.01e-11 | 2.23e-11 | +| log2 | 6.76e-7 | 4.55e-7 | 1.01e-6 | 9.69e-12 | 6.89e-12 | 1.53e-11 | +| log10 | 1.92e-6 | 1.50e-6 | 3.35e-6 | 3.00e-11 | 2.30e-11 | 5.07e-11 | +| pow2 | 3.54e-4 | 5.03e-6 | 2.37e-3 | 5.47e-9 | 7.69e-11 | 3.67e-8 | +| pow | 6.47e-4 | 1.33e-5 | 3.13e-3 | 1.01e-8 | 2.02e-10 | 4.85e-8 | | sqrt | 8.88e-8 | 5.80e-8 | 2.42e-7 | 1.37e-12 | 8.85e-13 | 3.62e-12 | - \ No newline at end of file + + + +### Worst-case error + +The largest error seen, in ulps of the type, against MPFR on exact inputs. The sweep covers 104 layouts from 8 to 128 bits, every value of each 8- and 16-bit one and 40000 seeded samples of each wider one; tools/accuracy-bench/worst.txt holds every layout. The directed forms, `_down` and `_up`, come within a rounding of a whole ulp, and the sweep checks exactly that each is below one and on its side. + +| Function | I16F16 | I32F32 | I64F64 | +|----------|------|------|------| +| sin | 0.50 | 0.50 | 0.50 | +| cos | 0.50 | 0.50 | 0.50 | +| tan | 0.50 | 0.50 | 0.50 | +| asin | 0.50 | 0.50 | 0.50 | +| acos | 0.50 | 0.50 | 0.50 | +| atan | 0.50 | 0.50 | 0.50 | +| atan2 | 0.50 | 0.50 | 0.50 | +| sinh | 0.50 | 0.50 | 0.50 | +| cosh | 0.50 | 0.50 | 0.50 | +| tanh | 0.50 | 0.50 | 0.50 | +| coth | 0.50 | 0.50 | 0.50 | +| asinh | 0.50 | 0.50 | 0.50 | +| acosh | 0.50 | 0.50 | 0.50 | +| atanh | 0.50 | 0.50 | 0.50 | +| acoth | 0.50 | 0.50 | 0.50 | +| exp | 0.50 | 0.50 | 0.50 | +| pow2 | 0.50 | 0.50 | 0.50 | +| ln | 0.50 | 0.50 | 0.50 | +| log2 | 0.50 | 0.50 | 0.50 | +| log10 | 0.50 | 0.50 | 0.50 | +| pow | 0.50 | 0.50 | 0.50 | +| sqrt | 0.50 | 0.50 | 0.50 | +| sqrt_or_zero | 0.50 | 0.50 | 0.50 | +| mul_add | 0.50 | 0.50 | 0.50 | +| mul_div | 0.50 | 0.50 | 0.50 | +| div_or_saturate | 0.50 | 0.50 | 0.50 | +| dot3 | 0.50 | 0.50 | 0.50 | +| hypot | 0.50 | 0.50 | 0.50 | +| norm3 | 0.50 | 0.50 | 0.50 | +| normalize3 | 0.50 | 0.50 | 0.50 | +| scale_by | 0.50 | 0.50 | 0.50 | +| sqrt_product | 0.50 | 0.50 | 0.50 | +| asin_from_gap | 0.50 | 0.50 | 0.50 | +| rsqrt | 0.50 | 0.50 | 0.50 | +| expm1 | 0.50 | 0.50 | 0.50 | +| exprel | 0.50 | 0.50 | 0.50 | +| exprel2 | 0.50 | 0.50 | 0.50 | +| exp_moments_e1 | 0.50 | 0.50 | 0.50 | +| exp_moments_e2 | 0.50 | 0.50 | 0.50 | +| exp_moments_e3 | 0.50 | 0.50 | 0.50 | +| exp_moments_e4 | 0.50 | 0.50 | 0.50 | +| exp_moments_e5 | 0.50 | 0.50 | 0.50 | +| log1p | 0.50 | 0.50 | 0.50 | +| log1p_ratio | 0.50 | 0.50 | 0.50 | +| log1p_residual | 0.50 | 0.50 | 0.50 | +| log1p_residual_ratio | 0.50 | 0.50 | 0.50 | +| stumpff_c0 | 0.50 | 0.50 | 0.50 | +| stumpff_c1 | 0.50 | 0.50 | 0.50 | +| stumpff_c2 | 0.50 | 0.50 | 0.50 | +| stumpff_c3 | 0.50 | 0.50 | 0.50 | +| tan_ratio | 0.50 | 0.50 | 0.50 | +| atan_ratio | 0.50 | 0.50 | 0.50 | +| atanh_ratio | 0.50 | 0.50 | 0.50 | +| tanh_ratio | 0.50 | 0.50 | 0.50 | +| ellip_k | 0.50 | 0.50 | 0.50 | +| ellip_k_from_m | 0.50 | 0.50 | 0.50 | +| ellip_e | 0.50 | 0.50 | 0.50 | +| jacobi_sn | 0.50 | 0.50 | 0.50 | +| jacobi_cn | 0.50 | 0.50 | 0.50 | +| jacobi_dn | 0.50 | 0.50 | 0.50 | +| jacobi_am | 0.50 | 0.50 | 0.50 | +| jacobi_inverse | 0.50 | 0.50 | 0.50 | +| carlson_rf | 0.50 | 0.50 | 0.50 | +| carlson_rf_roots | 0.50 | 0.50 | 0.50 | +| carlson_rj | 0.50 | 0.50 | 0.50 | +| carlson_rc | 0.50 | 0.50 | 0.50 | +| carlson_rd | 0.50 | 0.50 | 0.50 | +| ellip_pi | 0.50 | 0.50 | 0.50 | +| ellip_pi_complete | 0.50 | 0.50 | 0.50 | +| sin_turns | 0.50 | 0.50 | 0.50 | +| cos_turns | 0.50 | 0.50 | 0.50 | +| atan2_turns | 0.50 | 0.50 | 0.50 | +| div_by_tau | 0.50 | 0.50 | 0.50 | +| wrap_radians | 0.50 | 0.50 | 0.50 | +| wrap_principal | 0.50 | 0.50 | 0.50 | +| mul_by_tau | 0.50 | 0.50 | 0.50 | +| c_mul_re | 0.50 | 0.50 | 0.50 | +| c_mul_im | 0.50 | 0.50 | 0.50 | +| c_div_re | 0.50 | 0.50 | 0.50 | +| c_div_im | 0.50 | 0.50 | 0.50 | +| c_exp_re | 0.50 | 0.50 | 0.50 | +| c_exp_im | 0.50 | 0.50 | 0.50 | +| c_exprel_re | 0.50 | 0.50 | 0.50 | +| c_exprel_im | 0.50 | 0.50 | 0.50 | +| c_exprel2_re | 0.50 | 0.50 | 0.50 | +| c_exprel2_im | 0.50 | 0.50 | 0.50 | +| faddeeva_re | 0.50 | 0.50 | 0.50 | +| faddeeva_im | 0.50 | 0.50 | 0.50 | +| c_erf_re | 0.50 | 0.50 | 0.50 | +| c_erf_im | 0.50 | 0.50 | 0.50 | +| e1_re | 0.50 | 0.50 | 0.50 | +| e1_im | 0.50 | 0.50 | 0.50 | +| gamma_upper_re | 0.50 | 0.50 | 0.50 | +| gamma_upper_im | 0.50 | 0.50 | 0.50 | +| c_ln_re | 0.50 | 0.50 | 0.50 | +| c_ln_im | 0.50 | 0.50 | 0.50 | +| c_pow_re | 0.50 | 0.50 | 0.50 | +| c_pow_im | 0.50 | 0.50 | 0.50 | +| exprel_weighted_k1_re | 0.50 | 0.50 | 0.50 | +| exprel_weighted_k1_im | 0.50 | 0.50 | 0.49 | +| exprel_weighted_k2_re | 0.50 | 0.50 | 0.49 | +| exprel_weighted_k2_im | 0.50 | 0.50 | 0.50 | +| kepler_universal | 0.50 | 0.50 | 0.50 | +| erf | 0.50 | 0.50 | 0.50 | +| erfc | 0.50 | 0.50 | 0.50 | +| fresnel_c | 0.50 | 0.50 | 0.50 | +| fresnel_s | 0.50 | 0.50 | 0.50 | +| gamma | 0.50 | 0.50 | 0.50 | +| mul_down | 1.00 | 1.00 | 1.00 | +| mul_up | 1.00 | 1.00 | 1.00 | +| mul_div_down | 1.00 | 1.00 | 1.00 | +| mul_div_up | 1.00 | 1.00 | 1.00 | +| mul_add_down | 1.00 | 1.00 | 1.00 | +| mul_add_up | 1.00 | 1.00 | 1.00 | +| div_down | 1.00 | 1.00 | 1.00 | +| div_up | 1.00 | 1.00 | 1.00 | +| dot3_down | 1.00 | 1.00 | 1.00 | +| dot3_up | 1.00 | 1.00 | 1.00 | +| ellip_pi_increment | 0.50 | 0.50 | 0.50 | + \ No newline at end of file diff --git a/benches/benchmarks.rs b/benches/benchmarks.rs index 587c413..7cde95a 100644 --- a/benches/benchmarks.rs +++ b/benches/benchmarks.rs @@ -5,8 +5,9 @@ use std::hint::black_box; use criterion::{Criterion, criterion_group, criterion_main}; use fixed::types::{I16F16, I64F64}; use fixed_analytics::{ - CordicNumber, acos, acosh, acoth, asin, asinh, atan, atan2, atanh, cos, cosh, coth, exp, ln, - log2, log10, pow, pow2, sin, sin_cos, sinh, sinh_cosh, sqrt, tan, tanh, + CordicNumber, acos, acosh, acoth, asin, asinh, atan, atan2, atanh, cos, cosh, coth, + div_or_saturate, dot, exp, ln, log2, log10, mul_add, mul_div, pow, pow2, sin, sin_cos, sinh, + sinh_cosh, sqrt, tan, tanh, }; fn bench_type(c: &mut Criterion, name: &str) { @@ -68,6 +69,22 @@ fn bench_type(c: &mut Criterion, name: &str) { g.bench_function("sqrt_small", |b| b.iter(|| sqrt(black_box(small)))); g.finish(); } + { + let mut g = c.benchmark_group(format!("{name}/arith")); + g.bench_function("mul_add", |b| { + b.iter(|| mul_add(black_box(big), black_box(x), black_box(small))); + }); + g.bench_function("mul_div", |b| { + b.iter(|| mul_div(black_box(big), black_box(big), black_box(pos_x))); + }); + g.bench_function("div_or_saturate", |b| { + b.iter(|| div_or_saturate(black_box(x), black_box(large_x))); + }); + g.bench_function("dot3", |b| { + b.iter(|| dot(black_box(&[x, big, small]), black_box(&[pos_x, x, big]))); + }); + g.finish(); + } } fn bench_i16f16(c: &mut Criterion) { diff --git a/docs/crate.md b/docs/crate.md index 3be52f7..23248c1 100644 --- a/docs/crate.md +++ b/docs/crate.md @@ -25,24 +25,42 @@ assert!((log.to_num::() - 1.0).abs() < 0.01); | Category | Total | Fallible | |--------------|-------|----------| -| Trigonometric | [`sin`], [`cos`], [`tan`], [`sin_cos`], [`atan`], [`atan2`] | [`asin`], [`acos`] | +| Trigonometric | [`sin`], [`cos`], [`tan`], [`sin_cos`], [`atan`], [`atan2`] | [`asin`], [`acos`], [`asin_from_gap`] | | Hyperbolic | [`sinh`], [`cosh`], [`tanh`], [`sinh_cosh`], [`asinh`] | [`acosh`], [`atanh`], [`acoth`], [`coth`] | -| Exponential | [`exp`], [`pow2`] | [`ln`], [`log2`], [`log10`], [`pow`] | -| Algebraic | — | [`sqrt`] | - -Functions use polynomial evaluation, CORDIC, and Newton-Raphson techniques. -Complete absence of panic is verified at the linker level via the +| Exponential | [`exp`], [`pow2`], [`expm1`], [`exprel`], [`exprel2`], [`exp_moments`] | [`ln`], [`log2`], [`log10`], [`pow`], [`log1p`], [`log1p_ratio`], [`log1p_residual`], [`log1p_residual_ratio`] | +| Turns | [`sin_cos_turns`], [`atan2_turns`], [`div_by_tau`], [`mul_by_tau`], [`wrap_radians`], [`wrap_principal`] | — | +| Ratios | [`stumpff`], [`tan_ratio`], [`atan_ratio`], [`tanh_ratio`] | [`atanh_ratio`] | +| Algebraic | [`sqrt_or_zero`], [`hypot`], [`norm`], [`scale_exponent`], [`scale_by`] | [`sqrt`], [`sqrt_product`], [`rsqrt`], [`normalize`] | +| Special | [`erf`], [`erfc`], [`fresnel`] | [`gamma`] | +| Complex ([`Complex`]) | [`exp`](Complex::exp), [`exprel`](Complex::exprel), [`exprel2`](Complex::exprel2), [`faddeeva`](Complex::faddeeva), [`erf`](Complex::erf) | [`ln`](Complex::ln), [`pow`](Complex::pow), [`exprel_weighted`](Complex::exprel_weighted), [`e1`](Complex::e1), [`gamma_upper`](Complex::gamma_upper) | +| Elliptic | [`ellip_e`], [`jacobi_sn_cn_dn`], [`jacobi_am`] | [`ellip_k`], [`ellip_pi`], [`ellip_pi_complete`], [`ellip_pi_increment`], [`jacobi_inverse`], [`carlson_rf`], [`carlson_rf_roots`], [`carlson_rj`], [`carlson_rj_roots`], [`carlson_rc`], [`carlson_rd`] | +| Arithmetic | [`mul_add`], [`mul_div`], [`div_or_saturate`], [`dot`], and by [`Rounding`] [`mul_rounded`], [`div_rounded`], [`mul_div_rounded`], [`mul_add_rounded`], [`dot_rounded`] | — | +| Solvers | — | [`root_in`], [`kepler_universal`] | +| Enclosures | [`enclose`], [`sin_cos_range`], [`sin_cos_range_turns`], [`exp_range`], [`atan_range`], [`tanh_range`], [`exprel_range`], [`exprel2_range`], [`exp_moments_range`], [`tanh_ratio_range`], [`atan_ratio_range`], [`tan_ratio_range`], [`stumpff_range`], [`sqrt_range`], [`hypot_range`], [`atan2_range`] | [`log1p_range`], [`log1p_ratio_range`], [`log1p_residual_ratio_range`], [`atanh_ratio_range`] | + +Each function works in binary floating point at twice its type's width or +more, from its arguments converted exactly, and rounds once. Complete absence +of panic is verified at the linker level via the [`no-panic`](https://github.com/dtolnay/no-panic) crate. # Accuracy -| Type | Typical Accuracy | -|------|------------------| -| `I16F16` | ~4 decimal digits | -| `I32F32` | ~8 decimal digits | - -All functions are benchmarked against MPFR reference implementations. -Accuracy regressions are not permitted across releases. +Every function is within [`ERROR_ULPS`], one ulp, of the exact result on +every type, from `I8F0` to `I3F125`, and the arithmetic functions within half +an ulp, or rounded by [`Rounding`] to the nearest value below or above; results +out of range saturate to the nearer end. Odd and even +functions are exactly odd and even, except that an odd one saturating at +`MAX` meets `MIN` on the other side, and functions that are monotone stay +monotone. + +The accuracy tool checks all of this against MPFR on 104 layouts: every +value of the 8- and 16-bit ones, and seeded samples of the wider ones. +Accuracy regressions are not permitted across releases, and a release that +changes any output bit says so in the changelog. The enclosure functions +turn the bound into intervals certain to hold the exact value. + +The elliptic functions take a [`Modulus`], built from the parameter `m` or +from its complement `k' = √(1 − m)`, each kept exact as given. # Features diff --git a/docs/engine.md b/docs/engine.md new file mode 100644 index 0000000..408696f --- /dev/null +++ b/docs/engine.md @@ -0,0 +1,53 @@ +# How the functions evaluate + +This page records the design behind `src/real.rs` and `src/eval.rs`. + +## The working arithmetic + +Every function converts its arguments exactly into a binary floating-point +value, evaluates there, and rounds once to the fixed-point type. The mantissa +has 64 bits for 8-, 16- and 32-bit types, 128 for 64-bit ones and 192 for +128-bit ones, so each type works with at least 64 bits to spare above its +own. Each operation rounds to nearest even. Division by more than one limb +uses a Newton reciprocal, good to a few units in the last place. + +Sign and magnitude are kept apart, so `f(−x)` repeats the work of `f(x)` on +the same magnitude and odd and even functions are exactly so. The final +rounding is monotone, and the working error, below 2^-20 ulp of the type, is +far smaller than any gap between neighbouring results except where the +function is flat, so monotone functions give monotone results. The accuracy +tool checks both on every input of the 8- and 16-bit types. + +## Reductions + +`sin` and `cos` reduce exactly. The magnitude of the argument times +`1/(2π)` to 384 bits gives the fraction of a turn with at least 257 bits +past the point, enough for any 128-bit integer argument. Its top three bits +pick the octant; odd octants count back from their top, so the remainder +`w` lies in `[0, 1]` and `(π/4)·w` in `[0, π/4]`. + +The remaining work is short series around tabled points, the tables holding +384 bits: + +| Function | Table | Remainder | +|----------|-------|-----------| +| `sin`, `cos` | `sin(jπ/128)`, `cos(jπ/128)` | `|d| ≤ π/256` | +| `exp` | `2^(j/32)` | `|r| ≤ ln 2/64` | +| `ln` | `ln(1 + j/16)` on `[√½, √2)` | `|s| ≤ 0.023` in `2·atanh(s)` | +| `atan` | `atan(j/16)` on `[0, 1]` | `|t| ≤ 1/32` | + +`expm1` and `log1p` sum their series directly near 0, so they keep relative +precision, and the hyperbolic functions and their inverses are written in +them so that none cancels: `sinh = (u + u/(1 + u))/2` with `u = expm1(x)`, +`tanh = u/(u + 2)` with `u = expm1(2x)`, `asinh(z) = log1p(z + z²/(1 + +√(1 + z²)))`, `acosh(1 + t) = log1p(t + √(t(t + 2)))`, and `atanh(a) = +log1p(2a/(1 − a))/2`. `asin` and `acos` take `√((1 − a)(1 + a))`, whose +factors are exact. + +## Series + +Each series is a polynomial in the square of the remainder, its coefficients +`1/n!` or `1/(2k + 1)`, summed by Horner's rule in fixed point with 62, 126 +or 190 fractional bits for the three widths. Its length is fixed per width, +long enough for the largest remainder with two bits to spare, so every +function does a bounded amount of work. diff --git a/docs/exp.md b/docs/exp.md deleted file mode 100644 index c77c862..0000000 --- a/docs/exp.md +++ /dev/null @@ -1,40 +0,0 @@ -# How `exp` is computed - -This page records the derivations behind the comments in -`src/ops/exponential.rs`. - -## Argument reduction - -`exp(x) = 2^k · exp(r)` with `r ∈ [0, ln 2)`, so `exp(r) ∈ [1, 2)` carries a -full significand into the final shift. - -`k = ⌊x / ln 2⌋` is estimated to within one and then corrected once in either -direction. `to_i32` floors the quotient. When the integer bits do not -outnumber the fractional bits, the quotient comes from a multiply by -`1/ln 2 − 1` (as I1F63, since `1/ln 2` itself exceeds 1) plus `x`, which is -within `|x|·1.5·2^-frac + 2^-frac < 1` of the true value. Other types divide. - -`2^k · exp(r)` exceeds `MAX` once `k ≥ int_bits − 1` and rounds to zero once -`k ≤ −(frac_bits + 2)`. The early exits allow one of slack, and repeat the test -exactly after the correction. - -## Polynomial - -`exp(r) = 1 + r·(1 + r/2·(1 + r/3·(1 + … r/n)))`, evaluated innermost first -with integer division for each `r/K`. The omitted term `(ln 2)^(n+1)/(n+1)!` -is: - -| Degree | Omitted term | Below half an ulp up to | -|--------|--------------|-------------------------| -| 7 | 1.3e-6 | 18 fractional bits | -| 12 | 1.4e-12 | 38 fractional bits | -| 19 | 2.7e-22 | 70 fractional bits | - -The code uses degree 7 below 24 fractional bits, 12 from 24 and 19 from 40. - -## Final scaling - -A positive `k` shifts left after checking `exp(r)` against `MAX >> k`, since -the shift would wrap. A non-positive `k` shifts right with rounding to -nearest: truncation would bias results that are only a few ulps low by up to -a full ulp. diff --git a/docs/hyperbolic.md b/docs/hyperbolic.md index e159845..96f6b96 100644 --- a/docs/hyperbolic.md +++ b/docs/hyperbolic.md @@ -13,5 +13,6 @@ for extreme inputs: `sinh` to `T::MAX` or `T::MIN` past `±asinh(T::MAX)`, and | I16F16 | \|x\| > ~11.1 | | I32F32 | \|x\| > ~22.2 | -Below that, both come from a Taylor polynomial on `|x| ≤ 1.118`, extended by -halving the argument and doubling back. +Below that, both come from `u = expm1(|x|)`, as `sinh = (u + u/(1 + u))/2` +and `cosh = 1 + u²/(2(1 + u))`, which keep their relative precision at any +size. diff --git a/docs/sin_cos.md b/docs/sin_cos.md deleted file mode 100644 index fc7c9c9..0000000 --- a/docs/sin_cos.md +++ /dev/null @@ -1,20 +0,0 @@ -# How `sin_cos` reduces its argument - -This page records the derivation behind the comments in -`src/ops/circular.rs`. - -An angle outside `[-π, π]` is reduced by `n = round(angle / 2π)` to -`angle − n·2π`, at any magnitude and without iteration limits. - -When the integer bits do not outnumber the fractional bits, the quotient -comes from a multiply by `1/(2π)` as I1F63. That is within -`|angle|·1.5·2^-frac + 2^-frac < 1` of the true quotient, so `n` is off by at -most one, and the clamp to `[-π, π]` that follows absorbs it bit for bit. -Other types divide by 2π. - -`n·2π` is exact and the difference is small, so wrapping arithmetic recovers -it even when `n·2π` exceeds `MAX`, for angles within π of `MAX`. - -The reduced angle then folds to `[-π/2, π/2]`, negating both results. Past -`π/4` the polynomials take `π/2 − |x|` and swap roles, sin from the cosine -polynomial and cos from the sine one, which avoids cancellation near π/2. diff --git a/docs/sinh_cosh.md b/docs/sinh_cosh.md deleted file mode 100644 index 382b8fc..0000000 --- a/docs/sinh_cosh.md +++ /dev/null @@ -1,29 +0,0 @@ -# How `sinh_cosh` is computed - -This page records the derivation behind the comments in -`src/ops/hyperbolic.rs`. - -The argument is halved until `|x| ≤ 1.1182`, the sum of the atanh table, and -both functions come from factored Taylor series in `u = x²`: - -```text -sinh(x) = x·(1 + u/6·(1 + u/20·(1 + u/42·…))) -cosh(x) = 1 + u/2·(1 + u/12·(1 + u/30·…)) -``` - -The divisors are `(2k)(2k+1)` for sinh and `(2k−1)(2k)` for cosh. Integer -division computes each `u/K` in one step without pre-rounding a reciprocal, -and the factored form keeps each multiplicand `u/K` below 1 for `K ≥ 2`. - -At `|x| ≤ 1.1182` the omitted relative term is: - -| Degrees (sinh / cosh) | Omitted term | Below half an ulp up to | -|-----------------------|--------------|-------------------------| -| 9 / 10 | 7.7e-8 / 8.0e-9 | 22 fractional bits | -| 13 / 14 | 3.7e-12 / 2.9e-13 | 37 fractional bits | -| 21 / 22 | 4.5e-22 / 2.4e-23 | 70 fractional bits | - -The code uses the first row below 24 fractional bits, the second from 24 and -the third from 40. The results are rebuilt by doubling, -`sinh(2x) = 2·sinh(x)·cosh(x)` and `cosh(2x) = cosh²(x) + sinh²(x)`, once -per halving. diff --git a/src/bin/verify_no_panic.rs b/src/bin/verify_no_panic.rs index 9c2daaa..8fe78c5 100644 --- a/src/bin/verify_no_panic.rs +++ b/src/bin/verify_no_panic.rs @@ -10,9 +10,30 @@ use fixed_analytics::CordicNumber; use fixed_analytics::bounded::{NonNegative, OpenUnitInterval}; use fixed_analytics::ops::algebraic::sqrt_nonneg; use fixed_analytics::ops::hyperbolic::atanh_open; +use fixed_analytics::{Complex, erf, erfc, fresnel, gamma, kepler_universal, root_in}; use fixed_analytics::{ - acos, acosh, acoth, asin, asinh, atan, atan2, atanh, cos, cosh, coth, exp, ln, log2, log10, - pow, pow2, sin, sin_cos, sinh, sinh_cosh, sqrt, tan, tanh, + Modulus, carlson_rc, carlson_rd, carlson_rf, carlson_rf_roots, carlson_rj, carlson_rj_roots, + ellip_e, ellip_k, ellip_pi, ellip_pi_complete, jacobi_am, jacobi_inverse, jacobi_sn_cn_dn, +}; +use fixed_analytics::{ + Rounding, atan_ratio_range, atan2_range, atanh_ratio_range, div_rounded, dot_rounded, + ellip_pi_increment, exp_moments_range, exprel_range, exprel2_range, hypot_range, log1p_range, + log1p_ratio_range, log1p_residual_ratio_range, mul_add_rounded, mul_div_rounded, mul_rounded, + sqrt_range, stumpff_range, tan_ratio_range, tanh_ratio_range, +}; +use fixed_analytics::{ + acos, acosh, acoth, asin, asin_from_gap, asinh, atan, atan_range, atan2, atanh, cos, cosh, + coth, div_or_saturate, dot, enclose, exp, exp_range, hypot, ln, log2, log10, mul_add, mul_div, + norm, normalize, pow, pow2, rsqrt, scale_by, scale_exponent, sin, sin_cos, sin_cos_range, sinh, + sinh_cosh, sqrt, sqrt_or_zero, sqrt_product, tan, tanh, tanh_range, +}; +use fixed_analytics::{ + atan_ratio, atanh_ratio, exp_moments, expm1, exprel, exprel2, log1p, log1p_ratio, + log1p_residual, log1p_residual_ratio, stumpff, tan_ratio, tanh_ratio, +}; +use fixed_analytics::{ + atan2_turns, div_by_tau, mul_by_tau, sin_cos_range_turns, sin_cos_turns, wrap_principal, + wrap_radians, }; /// Hands `v` to `black_box`, so the optimizer must compute it in full. @@ -47,6 +68,127 @@ fn exercise(x: T, y: T, two: T) { keep(coth(x)); keep(acoth(two)); keep(pow(two, x)); + keep(sqrt_or_zero(x)); + keep(mul_add(x, y, two)); + keep(mul_div(x, y, two)); + keep(div_or_saturate(x, y)); + keep(dot(&[x, y], &[two, x])); + keep(hypot(x, y)); + keep(norm(&[x, y, two])); + keep(normalize([x, y, two])); + keep(scale_exponent(x)); + keep(scale_by(x, 3)); + keep(sqrt_product(x, y)); + keep(rsqrt(x)); + keep(asin_from_gap(x)); + keep(enclose(x)); + keep(sin_cos_range(x, two)); + keep(exp_range(x, two)); + keep(atan_range(x, two)); + keep(tanh_range(x, two)); + keep(expm1(x)); + keep(exprel(x)); + keep(exprel2(x)); + keep(exp_moments(x)); + keep(log1p(x)); + keep(log1p_ratio(x)); + keep(log1p_residual(x)); + keep(log1p_residual_ratio(x)); + keep(stumpff(x)); + keep(tan_ratio(x)); + keep(atan_ratio(x)); + keep(atanh_ratio(x)); + keep(tanh_ratio(x)); + if let Some(m) = Modulus::from_complement(y) { + keep(ellip_k(m)); + keep(ellip_e(m)); + keep(ellip_pi_complete(x, m)); + keep(ellip_pi(x, y, x, y, m)); + keep(jacobi_sn_cn_dn(x, m)); + keep(jacobi_am(x, m)); + keep(jacobi_inverse(x, y, x, m)); + keep(m.parameter()); + keep(m.complement()); + } + keep(Modulus::from_parameter(y)); + keep(carlson_rf(x, y, two)); + keep(carlson_rf_roots(x, y, two)); + keep(carlson_rj(x, y, two, x)); + keep(carlson_rj_roots(x, y, two, x)); + keep(carlson_rc(x, y)); + keep(carlson_rd(x, y, two)); + exercise_more(x, y, two); + exercise_bounds(x, y, two); +} + +/// The range forms and directed arithmetic, which bound what the rest compute. +fn exercise_bounds(x: T, y: T, two: T) { + for mode in [Rounding::Down, Rounding::Nearest, Rounding::Up] { + keep(mul_rounded(x, y, mode)); + keep(div_rounded(x, y, mode)); + keep(mul_div_rounded(x, y, two, mode)); + keep(mul_add_rounded(x, y, two, mode)); + keep(dot_rounded(&[x, y], &[two, x], mode)); + } + keep(exprel_range(x, two)); + keep(exprel2_range(x, two)); + keep(exp_moments_range(x, two)); + keep(log1p_range(x, two)); + keep(log1p_ratio_range(x, two)); + keep(log1p_residual_ratio_range(x, two)); + keep(tanh_ratio_range(x, two)); + keep(atan_ratio_range(x, two)); + keep(atanh_ratio_range(y, x)); + keep(tan_ratio_range(x, two)); + keep(sqrt_range(x, two)); + keep(stumpff_range(x, two)); + keep(hypot_range((x, y), (y, two))); + keep(atan2_range((x, y), (y, two))); + if let Some(m) = Modulus::from_complement(y) { + keep(ellip_pi_increment(x, y, two, m)); + } +} + +/// The turns and complex functions, kept apart so neither list grows too long. +fn exercise_more(x: T, y: T, two: T) { + keep(sin_cos_turns(x)); + keep(atan2_turns(y, x)); + keep(div_by_tau(x)); + keep(mul_by_tau(x)); + keep(sin_cos_range_turns(x, y)); + keep(wrap_radians(x)); + keep(wrap_principal(x)); + let z = Complex::new(x, y); + let w = Complex::new(two, x); + keep(z + w); + keep(z - w); + keep(z * w); + keep(z / w); + keep(-z); + keep(z.saturating_add(w)); + keep(z.saturating_sub(w)); + keep(z.saturating_mul(w)); + keep(z.saturating_div(w)); + keep(z.saturating_neg()); + keep(z.ln()); + keep(z.pow(two)); + keep(z.exprel_weighted(y, two)); + keep(z.conj()); + keep(z.norm()); + keep(z.arg()); + keep(z.exp()); + keep(z.exprel()); + keep(z.exprel2()); + keep(z.faddeeva()); + keep(z.erf()); + keep(z.e1()); + keep(z.gamma_upper(two)); + keep(erf(x)); + keep(erfc(x)); + keep(fresnel(x)); + keep(gamma(x)); + keep(root_in(|t: T| t.saturating_sub(y), x, two, 40)); + keep(kepler_universal(two, y, x, two)); if let Some(nn) = NonNegative::new(x) { keep(sqrt_nonneg(nn)); diff --git a/src/bounded.rs b/src/bounded.rs index 17cfa50..ab54d5f 100644 --- a/src/bounded.rs +++ b/src/bounded.rs @@ -107,15 +107,6 @@ impl OpenUnitInterval { Self(sqrt_x_sq_minus_one.div(x.0)) } - /// Builds `(x − 1) / (x + 1)`, which lies in `[-1/3, 1/3]` for `x` in `[0.5, 2]`. - #[inline] - #[must_use] - pub(crate) fn from_normalized_ln_arg(x: NormalizedLnArg) -> Self { - let x_minus_1 = x.0 - T::one(); - let x_plus_1 = x.0 + T::one(); - Self(x_minus_1.div(x_plus_1)) - } - /// Returns the inner value. #[inline] #[must_use] @@ -144,19 +135,6 @@ impl AtLeastOne { } } -/// The ln argument after normalization, in `[0.5, 2]`. -#[derive(Clone, Copy, Debug)] -pub(crate) struct NormalizedLnArg(T); - -impl NormalizedLnArg { - /// Wraps the normalized value unchecked; `ln_positive` guarantees the range. - #[inline] - #[must_use] - pub(crate) const fn from_normalized(value: T) -> Self { - Self(value) - } -} - #[cfg(test)] mod tests { use super::*; @@ -267,12 +245,4 @@ mod tests { assert_eq!(at_least.get(), I16F16::from_num(2)); Ok(()) } - - #[test] - fn open_unit_interval_from_normalized_ln_arg() { - let norm = NormalizedLnArg::from_normalized(I16F16::from_num(1.5)); - let open = OpenUnitInterval::from_normalized_ln_arg(norm); - let val: f32 = open.get().to_num(); - assert!((val - 0.2).abs() < 0.01); - } } diff --git a/src/consts.rs b/src/consts.rs new file mode 100644 index 0000000..d7aa09d --- /dev/null +++ b/src/consts.rs @@ -0,0 +1,276 @@ +//! Constants to 384 bits, from MPFR (tools/accuracy-bench/src/bin/consts.rs). + +/// Sign, exponent and mantissa, most significant limb last. +pub type C = (bool, i32, [u64; 6]); + +#[rustfmt::skip] +pub const PI: C = (false, 2, [0xEF95_19B3_CD3A_431B, 0x514A_0879_8E34_04DD, 0x020B_BEA6_3B13_9B22, 0x2902_4E08_8A67_CC74, 0xC4C6_628B_80DC_1CD1, 0xC90F_DAA2_2168_C234]); +#[rustfmt::skip] +pub const LN_2: C = (false, 0, [0x5595_52FB_4AFA_1B11, 0xE7B8_7620_6DEB_AC98, 0x8A0D_175B_8BAA_FA2B, 0x40F3_4326_7298_B62D, 0xC9E3_B398_03F2_F6AF, 0xB172_17F7_D1CF_79AB]); +#[rustfmt::skip] +pub const FRAC_1_LN_2: C = (false, 1, [0x24D9_2F75_C16B_E0B4, 0xDE1C_43F7_5517_6CD6, 0x8B25_166C_D1A1_3247, 0xEB57_7AA8_DD69_5A58, 0xBE87_FED0_691D_3E88, 0xB8AA_3B29_5C17_F0BB]); +#[rustfmt::skip] +pub const FRAC_1_LN_10: C = (false, -1, [0x9223_5592_C646_4A15, 0x7542_4EFA_1402_F3F2, 0x3A3F_2D44_F78E_A53C, 0x3EE3_4602_45C9_A202, 0x355B_AAAF_AD33_DC32, 0xDE5B_D8A9_3728_7195]); +#[rustfmt::skip] +pub const FRAC_1_2PI: C = (false, -2, [0x0649_2EEA_09D1_921D, 0xB724_6E3A_424D_D2E0, 0xFE51_63AB_DEBB_C561, 0xDB62_9599_3C43_9041, 0xFC27_57D1_F534_DDC0, 0xA2F9_836E_4E44_1529]); +/// `FRAC_1_2PI` to 576 bits, as an integer times 2^−578, for exact products. +#[rustfmt::skip] +pub const FRAC_1_2PI_LONG: [u64; 9] = [0xE99C_7026_B45F_7E41, 0xE882_35F5_2EBB_4484, 0xFE1D_EB1C_B129_A73E, 0x0649_2EEA_09D1_921C, 0xB724_6E3A_424D_D2E0, 0xFE51_63AB_DEBB_C561, 0xDB62_9599_3C43_9041, 0xFC27_57D1_F534_DDC0, 0xA2F9_836E_4E44_1529]; +#[rustfmt::skip] +pub const EULER_GAMMA: C = (false, 0, [0xD064_9CCB_6210_57D1, 0x8E4B_59FA_03A9_F0EE, 0x0C03_DF34_709A_FFBD, 0xA1CE_CC3A_F65C_C019, 0xD1BE_3F81_0152_CB56, 0x93C4_67E3_7DB0_C7A4]); +#[rustfmt::skip] +pub const FRAC_2_SQRT_PI: C = (false, 1, [0x40C0_3609_6CC7_9AEC, 0xC075_9CF8_5927_0F11, 0x39A1_5830_CCE6_20B0, 0x1409_A0EB_AC3E_7517, 0x71D4_8A7F_6BFE_C344, 0x906E_BA82_14DB_688D]); +/// `1/n!` for n from 0. +#[rustfmt::skip] +pub const INV_FACT: [C; 49] = [ + (false, 1, [0x0000_0000_0000_0000, 0x0000_0000_0000_0000, 0x0000_0000_0000_0000, 0x0000_0000_0000_0000, 0x0000_0000_0000_0000, 0x8000_0000_0000_0000]), + (false, 1, [0x0000_0000_0000_0000, 0x0000_0000_0000_0000, 0x0000_0000_0000_0000, 0x0000_0000_0000_0000, 0x0000_0000_0000_0000, 0x8000_0000_0000_0000]), + (false, 0, [0x0000_0000_0000_0000, 0x0000_0000_0000_0000, 0x0000_0000_0000_0000, 0x0000_0000_0000_0000, 0x0000_0000_0000_0000, 0x8000_0000_0000_0000]), + (false, -2, [0xAAAA_AAAA_AAAA_AAAB, 0xAAAA_AAAA_AAAA_AAAA, 0xAAAA_AAAA_AAAA_AAAA, 0xAAAA_AAAA_AAAA_AAAA, 0xAAAA_AAAA_AAAA_AAAA, 0xAAAA_AAAA_AAAA_AAAA]), + (false, -4, [0xAAAA_AAAA_AAAA_AAAB, 0xAAAA_AAAA_AAAA_AAAA, 0xAAAA_AAAA_AAAA_AAAA, 0xAAAA_AAAA_AAAA_AAAA, 0xAAAA_AAAA_AAAA_AAAA, 0xAAAA_AAAA_AAAA_AAAA]), + (false, -6, [0x8888_8888_8888_8889, 0x8888_8888_8888_8888, 0x8888_8888_8888_8888, 0x8888_8888_8888_8888, 0x8888_8888_8888_8888, 0x8888_8888_8888_8888]), + (false, -9, [0x0B60_B60B_60B6_0B61, 0x60B6_0B60_B60B_60B6, 0xB60B_60B6_0B60_B60B, 0x0B60_B60B_60B6_0B60, 0x60B6_0B60_B60B_60B6, 0xB60B_60B6_0B60_B60B]), + (false, -12, [0x0D00_D00D_00D0_0D01, 0x00D0_0D00_D00D_00D0, 0xD00D_00D0_0D00_D00D, 0x0D00_D00D_00D0_0D00, 0x00D0_0D00_D00D_00D0, 0xD00D_00D0_0D00_D00D]), + (false, -15, [0x0D00_D00D_00D0_0D01, 0x00D0_0D00_D00D_00D0, 0xD00D_00D0_0D00_D00D, 0x0D00_D00D_00D0_0D00, 0x00D0_0D00_D00D_00D0, 0xD00D_00D0_0D00_D00D]), + (false, -18, [0x0B8E_F1D2_AB63_99C8, 0x399C_7D56_0E44_7280, 0x4728_00B8_EF1D_2AB6, 0xD2AB_6399_C7D5_60E4, 0x560E_4472_800B_8EF1, 0xB8EF_1D2A_B639_9C7D]), + (false, -21, [0xA2D8_C175_55E9_47D3, 0x6149_FDDE_71D0_5B99, 0xD286_66FA_58E4_222B, 0x0EEF_82E1_6CAA_B3E9, 0x780B_69F5_333C_725B, 0x93F2_7DBB_C4FA_E397]), + (false, -25, [0xECDE_30AA_AB81_DCD6, 0x47B1_7143_8E46_56B0, 0x7809_4FF7_C71D_48F9, 0x15B9_788D_B555_62C8, 0x3A3F_25C1_BEE3_8F10, 0xD732_2B3F_AA27_1C7F]), + (false, -28, [0x9DE9_75C7_1D01_3DE4, 0xDA76_4B82_5ED9_8F20, 0xA55B_8AA5_2F68_DB50, 0x0E7B_A5B3_CE38_EC85, 0x26D4_C3D6_7F42_5F60, 0x8F76_C77F_C6C4_BDAA]), + (false, -32, [0x4C33_071C_7277_AEA2, 0xD1CC_ABB4_25F8_129E, 0x0698_0D1A_12F7_354F, 0x9BAB_DFA2_38E3_9942, 0x1C19_8E91_D7B4_269D, 0xB092_309D_4368_4BE5]), + (false, -36, [0x3283_75D7_5E3F_A303, 0x38E9_E8CD_E240_1547, 0x7540_0EF9_3A3F_185B, 0x68C4_6D4B_AEBA_F84B, 0xD6F8_A2EF_D1F2_7546, 0xC9CB_A546_03E4_E905]), + (false, -40, [0x24D0_7DB2_86AA_477A, 0x4DC6_4DA8_68CC_E37F, 0x38CC_DCC5_9376_8061, 0x5EAF_6383_ED94_3C0C, 0xC32B_5877_4657_F48F, 0xD73F_9F39_9DC0_F88E]), + (false, -44, [0x24D0_7DB2_86AA_477A, 0x4DC6_4DA8_68CC_E37F, 0x38CC_DCC5_9376_8061, 0x5EAF_6383_ED94_3C0C, 0xC32B_5877_4657_F48F, 0xD73F_9F39_9DC0_F88E]), + (false, -48, [0x0487_FDD5_3373_1619, 0xB29C_8553_3575_8AD2, 0x4484_938C_C706_1E79, 0xA468_D621_D08B_83CF, 0x3028_CBBB_8D7F_F53B, 0xCA96_3B81_856A_5359]), + (false, -52, [0x595C_6FD9_F4D8_13A4, 0x496E_AF66_6868_7B65, 0xE792_4A44_3F21_E233, 0x3CCE_F73A_807C_0362, 0x8024_4351_6155_4BC3, 0xB413_C31D_CBEC_BBDD]), + (false, -56, [0xDF76_433E_4772_9747, 0x9C27_5056_3CF9_AB47, 0xB588_AA54_6B0F_0F53, 0x5BA0_D031_43C6_BF7B, 0x50F6_1DBD_CB3A_5ABF, 0x97A4_DA34_0A0A_B926]), + (false, -61, [0xCBF0_6B96_D8B7_5872, 0x2D0B_B3BD_2E5C_453F, 0xBC0D_DD53_DE7E_7EEC, 0xF901_4D1B_9FA4_6592, 0x4E56_95FC_785D_5DFE, 0xF2A1_5D20_1011_283D]), + (false, -65, [0x523D_45C8_4397_E1DD, 0x1621_4BFD_DA2D_EB9E, 0xF0CD_9C70_A984_F2FD, 0x43D0_3ABF_B695_A2B8, 0x78A3_7E77_3722_90C2, 0xB8DC_77B6_E7AB_8C5F]), + (false, -69, [0xDEB8_32C0_3128_A443, 0xB300_F172_CD38_AB5B, 0x694F_B797_C11A_DF43, 0xD43A_5945_9C0F_BC29, 0x8648_5BF9_9C76_3ABB, 0x8671_CB6D_BFC2_94A2]), + (false, -74, [0x35DE_E26F_924E_CE47, 0xD7A8_44CC_3EEA_ABA1, 0x7C42_639B_7BF8_DD95, 0xB7F8_2329_322C_272E, 0xDC38_26EB_FB13_CC26, 0xBB0D_A098_B1C0_CECB]), + (false, -79, [0x47D3_D894_C313_BDB5, 0x74E0_5BBA_FE8E_3A2C, 0xFB03_2F79_FAA1_2772, 0x9FF5_8436_ED90_343D, 0x25A0_33E5_4EC5_1033, 0xF967_80CB_97AB_BE65]), + (false, -83, [0x612B_6BE4_53E3_AC9D, 0x1799_D44E_B765_43F3, 0xFCCE_D6B4_7771_60ED, 0xFFF9_4A60_980A_5EDF, 0x2252_0CBB_B788_5C49, 0x9F9E_66E8_B2FD_46A7]), + (false, -88, [0xB2AB_987B_7AF0_D472, 0x1D0C_18FE_6B90_53A1, 0xFC12_4354_443C_C610, 0xB132_D1B1_F634_25FF, 0x7902_85D3_580A_4A33, 0xC474_2FE3_5272_CD1C]), + (false, -93, [0x2917_3975_E70A_770C, 0x813D_BECE_B85F_33B6, 0x04D3_4650_E893_E142, 0x3A4F_316A_9F0E_65ED, 0x4D0C_78AE_A13B_9A50, 0xE8D5_8E16_E675_1905]), + (false, -97, [0x3C0D_4567_F1BC_D650, 0xDC23_4876_2036_66B1, 0xB99D_4CC0_84E6_C9DC, 0x6A76_6561_7F75_F119, 0xE2E2_8E1A_A546_A152, 0x850C_5131_A842_E9B9]), + (false, -102, [0x5CBF_321A_6BD9_331A, 0xB51E_1AFD_EE94_4E01, 0xB255_4BDD_4333_A0F3, 0x37AE_C824_F693_CC3F, 0xF187_3BB3_7894_8EB3, 0x92CF_CC5A_1AC5_6BD5]), + (false, -107, [0x9621_4682_952B_F23E, 0x49B9_B664_31AF_4223, 0x46C1_61FD_147B_5659, 0xC3ED_A249_8F8C_9599, 0x56F6_A614_C4E2_BA58, 0x9C99_6282_3EB0_7306]), + (false, -112, [0xCC85_7213_2EA0_FA0F, 0xBFB7_71F3_D030_C866, 0x598D_D081_1D66_8AAE, 0xB179_6DB7_49DB_7122, 0x0F72_37D3_5FE1_C89D, 0xA1A6_973C_1FAD_E217]), + (false, -117, [0xCC85_7213_2EA0_FA0F, 0xBFB7_71F3_D030_C866, 0x598D_D081_1D66_8AAE, 0xB179_6DB7_49DB_7122, 0x0F72_37D3_5FE1_C89D, 0xA1A6_973C_1FAD_E217]), + (false, -122, [0xCE14_CBB5_828C_9564, 0xB226_3FF4_2EBA_F0DF, 0x4753_36CA_C72D_21A1, 0x6E09_2492_B439_A402, 0xC166_FFD4_BA11_3EA8, 0x9CC0_92A6_E86A_8DA9]), + (false, -127, [0x94C8_4741_6BCF_9BA9, 0x2023_FFF4_E0AF_F1C3, 0x9D7B_7EDC_F7B2_0189, 0x2B53_E62F_B8AE_B87A, 0xC515_A57C_EB5B_8644, 0x9388_118E_07EB_D09F]), + (false, -132, [0xB3EA_4FC6_C8F8_53CE, 0xF17F_FFF5_D4BE_1EDE, 0x2995_7B4D_AF43_A9A1, 0xED19_81FF_BECB_A15A, 0x9E3F_AD3F_0311_D9D7, 0x86E2_CE38_B6C8_F941]), + (false, -138, [0x5C4B_387D_D70E_CDDF, 0x57FF_FFED_EBFC_A8A8, 0xBBB4_696D_A95B_D83C, 0x1749_CAAA_36BF_57BC, 0x35C6_8953_93AD_F50E, 0xEFCC_1948_6165_4958]), + (false, -143, [0x5DA8_D6E9_600C_CDBA, 0xCF91_4C0C_0A5D_F9A6, 0xD9B0_C9E2_4D48_83A9, 0x5954_939A_2182_E418, 0x3C58_AE1E_C4E9_79FE, 0xCF64_68E4_A742_D7A6]), + (false, -148, [0x4EDF_05D2_000A_C831, 0xF229_8368_7485_0820, 0xDFBD_4BB1_18A8_DAA9, 0x1554_B230_F3CC_8A2F, 0x9E9B_8527_6273_C50C, 0xAEA5_65CE_061D_5748]), + (false, -153, [0x6EA9_DD64_1A4A_7CE0, 0x0856_9334_E2E3_4EDF, 0x272B_BAD2_F369_B9FB, 0x8117_8BA4_E24B_EE1A, 0x8223_B575_A61D_5979, 0x8F4C_A24D_25D6_6F00]), + (false, -159, [0xE442_FBD3_5D43_FB00, 0x0D57_51EE_37D2_17CB, 0x0B79_2AEB_1F0F_8FF8, 0x9B58_DF6E_36DF_E35D, 0xD039_2255_D695_5BF5, 0xE547_6A15_0957_1800]), + (false, -164, [0x8CB1_2873_041C_1513, 0x9A05_B694_767E_82F6, 0x08F4_6C6C_951E_E0C1, 0xD0A9_443D_0B9C_02A0, 0x12E7_E8D8_D96E_5442, 0xB2F3_0E1C_E812_063F]), + (false, -169, [0xF14A_0670_0321_9627, 0x07A2_D440_5A48_0241, 0x80BA_3A3A_593C_18F5, 0x0CB1_BA16_2139_E99E, 0x26C9_12EE_5C84_D27C, 0x8857_A93A_986F_41B6]), + (false, -175, [0x19BB_9282_FEB4_F74C, 0x0569_6B89_7482_FD68, 0x0CFD_5C9E_4947_9643, 0x0CF0_B5B5_C64A_49DA, 0x5D72_B6F7_9B90_1B83, 0xCAED_A292_BF28_916E]), + (false, -180, [0x6FCE_3C02_2D9A_E266, 0x6106_D9D8_54BC_5B34, 0xDAE6_CEFE_C0EE_3EBC, 0x7DC6_559B_78F0_35B5, 0xCF99_3F3F_B6F4_7119, 0x9395_8D81_FF63_527E]), + (false, -186, [0x6BCF_FA52_BE04_1A2A, 0x78ED_4C94_61C1_EDCD, 0xFE70_1553_8F8B_B9EF, 0xA1D0_1EBA_FBA5_46B2, 0x162F_48E3_2655_F62F, 0xD1E5_C391_1032_3C70]), + (false, -191, [0x8DC8_5518_2B24_3EB9, 0xD9B0_3545_D4B3_6DC6, 0x8F9B_E250_63DB_A2BC, 0xD4BD_4D07_A3ED_68D5, 0x5237_2792_E304_25B1, 0x9204_027B_2CA8_8316]), + (false, -197, [0xAB47_42DA_1A10_AC8F, 0x7F93_58E1_CFEE_E1BC, 0x40D4_4483_412B_1411, 0x16CB_4DA8_5C7F_3228, 0x5A2A_6C5B_1472_955F, 0xC6D4_7050_93F5_CDBC]), + (false, -202, [0xC784_D73C_1160_730A, 0x550C_E5EB_DFF4_967D, 0xD5E2_D857_80C7_62B6, 0x6487_891A_E854_CC1A, 0xE6C6_F2E7_62F7_0E3F, 0x848D_A035_B7F9_33D2]), +]; +/// `1/(2k + 1)` for k from 0. +#[rustfmt::skip] +pub const INV_ODD: [C; 49] = [ + (false, 1, [0x0000_0000_0000_0000, 0x0000_0000_0000_0000, 0x0000_0000_0000_0000, 0x0000_0000_0000_0000, 0x0000_0000_0000_0000, 0x8000_0000_0000_0000]), + (false, -1, [0xAAAA_AAAA_AAAA_AAAB, 0xAAAA_AAAA_AAAA_AAAA, 0xAAAA_AAAA_AAAA_AAAA, 0xAAAA_AAAA_AAAA_AAAA, 0xAAAA_AAAA_AAAA_AAAA, 0xAAAA_AAAA_AAAA_AAAA]), + (false, -2, [0xCCCC_CCCC_CCCC_CCCD, 0xCCCC_CCCC_CCCC_CCCC, 0xCCCC_CCCC_CCCC_CCCC, 0xCCCC_CCCC_CCCC_CCCC, 0xCCCC_CCCC_CCCC_CCCC, 0xCCCC_CCCC_CCCC_CCCC]), + (false, -2, [0x4924_9249_2492_4925, 0x2492_4924_9249_2492, 0x9249_2492_4924_9249, 0x4924_9249_2492_4924, 0x2492_4924_9249_2492, 0x9249_2492_4924_9249]), + (false, -3, [0x8E38_E38E_38E3_8E39, 0x38E3_8E38_E38E_38E3, 0xE38E_38E3_8E38_E38E, 0x8E38_E38E_38E3_8E38, 0x38E3_8E38_E38E_38E3, 0xE38E_38E3_8E38_E38E]), + (false, -3, [0xBA2E_8BA2_E8BA_2E8C, 0x8BA2_E8BA_2E8B_A2E8, 0xE8BA_2E8B_A2E8_BA2E, 0x2E8B_A2E8_BA2E_8BA2, 0xA2E8_BA2E_8BA2_E8BA, 0xBA2E_8BA2_E8BA_2E8B]), + (false, -3, [0x89D8_9D89_D89D_89D9, 0xD89D_89D8_9D89_D89D, 0x9D89_D89D_89D8_9D89, 0x89D8_9D89_D89D_89D8, 0xD89D_89D8_9D89_D89D, 0x9D89_D89D_89D8_9D89]), + (false, -3, [0x8888_8888_8888_8889, 0x8888_8888_8888_8888, 0x8888_8888_8888_8888, 0x8888_8888_8888_8888, 0x8888_8888_8888_8888, 0x8888_8888_8888_8888]), + (false, -4, [0xF0F0_F0F0_F0F0_F0F1, 0xF0F0_F0F0_F0F0_F0F0, 0xF0F0_F0F0_F0F0_F0F0, 0xF0F0_F0F0_F0F0_F0F0, 0xF0F0_F0F0_F0F0_F0F0, 0xF0F0_F0F0_F0F0_F0F0]), + (false, -4, [0x0D79_435E_50D7_9436, 0x7943_5E50_D794_35E5, 0x435E_50D7_9435_E50D, 0x5E50_D794_35E5_0D79, 0x50D7_9435_E50D_7943, 0xD794_35E5_0D79_435E]), + (false, -4, [0x0C30_C30C_30C3_0C31, 0x30C3_0C30_C30C_30C3, 0xC30C_30C3_0C30_C30C, 0x0C30_C30C_30C3_0C30, 0x30C3_0C30_C30C_30C3, 0xC30C_30C3_0C30_C30C]), + (false, -4, [0x642C_8590_B216_42C8, 0x90B2_1642_C859_0B21, 0x42C8_590B_2164_2C85, 0x0B21_642C_8590_B216, 0x2C85_90B2_1642_C859, 0xB216_42C8_590B_2164]), + (false, -4, [0xA3D7_0A3D_70A3_D70A, 0x0A3D_70A3_D70A_3D70, 0x70A3_D70A_3D70_A3D7, 0xD70A_3D70_A3D7_0A3D, 0x3D70_A3D7_0A3D_70A3, 0xA3D7_0A3D_70A3_D70A]), + (false, -4, [0x097B_425E_D097_B426, 0x7B42_5ED0_97B4_25ED, 0x425E_D097_B425_ED09, 0x5ED0_97B4_25ED_097B, 0xD097_B425_ED09_7B42, 0x97B4_25ED_097B_425E]), + (false, -4, [0xDCB0_8D3D_CB08_D3DD, 0xD3DC_B08D_3DCB_08D3, 0x08D3_DCB0_8D3D_CB08, 0xCB08_D3DC_B08D_3DCB, 0x3DCB_08D3_DCB0_8D3D, 0x8D3D_CB08_D3DC_B08D]), + (false, -4, [0x8421_0842_1084_2108, 0x0842_1084_2108_4210, 0x1084_2108_4210_8421, 0x2108_4210_8421_0842, 0x4210_8421_0842_1084, 0x8421_0842_1084_2108]), + (false, -5, [0xF83E_0F83_E0F8_3E10, 0x0F83_E0F8_3E0F_83E0, 0xE0F8_3E0F_83E0_F83E, 0x3E0F_83E0_F83E_0F83, 0x83E0_F83E_0F83_E0F8, 0xF83E_0F83_E0F8_3E0F]), + (false, -5, [0x0EA0_EA0E_A0EA_0EA1, 0xA0EA_0EA0_EA0E_A0EA, 0xEA0E_A0EA_0EA0_EA0E, 0x0EA0_EA0E_A0EA_0EA0, 0xA0EA_0EA0_EA0E_A0EA, 0xEA0E_A0EA_0EA0_EA0E]), + (false, -5, [0x0DD6_7C8A_60DD_67C9, 0xD67C_8A60_DD67_C8A6, 0x7C8A_60DD_67C8_A60D, 0x8A60_DD67_C8A6_0DD6, 0x60DD_67C8_A60D_D67C, 0xDD67_C8A6_0DD6_7C8A]), + (false, -5, [0x0D20_D20D_20D2_0D21, 0x20D2_0D20_D20D_20D2, 0xD20D_20D2_0D20_D20D, 0x0D20_D20D_20D2_0D20, 0x20D2_0D20_D20D_20D2, 0xD20D_20D2_0D20_D20D]), + (false, -5, [0xC7CE_0C7C_E0C7_CE0C, 0x0C7C_E0C7_CE0C_7CE0, 0xE0C7_CE0C_7CE0_C7CE, 0xCE0C_7CE0_C7CE_0C7C, 0x7CE0_C7CE_0C7C_E0C7, 0xC7CE_0C7C_E0C7_CE0C]), + (false, -5, [0x2FA0_BE82_FA0B_E830, 0xE82F_A0BE_82FA_0BE8, 0x0BE8_2FA0_BE82_FA0B, 0xFA0B_E82F_A0BE_82FA, 0x82FA_0BE8_2FA0_BE82, 0xBE82_FA0B_E82F_A0BE]), + (false, -5, [0x0B60_B60B_60B6_0B61, 0x60B6_0B60_B60B_60B6, 0xB60B_60B6_0B60_B60B, 0x0B60_B60B_60B6_0B60, 0x60B6_0B60_B60B_60B6, 0xB60B_60B6_0B60_B60B]), + (false, -5, [0x2B93_1057_2620_AE4C, 0x7262_0AE4_C415_C988, 0x4C41_5C98_82B9_3105, 0x882B_9310_5726_20AE, 0x0572_620A_E4C4_15C9, 0xAE4C_415C_9882_B931]), + (false, -5, [0xE5E0_A72F_0539_782A, 0x72F0_5397_829C_BC14, 0x3978_29CB_C14E_5E0A, 0x9CBC_14E5_E0A7_2F05, 0x4E5E_0A72_F053_9782, 0xA72F_0539_7829_CBC1]), + (false, -5, [0xA0A0_A0A0_A0A0_A0A1, 0xA0A0_A0A0_A0A0_A0A0, 0xA0A0_A0A0_A0A0_A0A0, 0xA0A0_A0A0_A0A0_A0A0, 0xA0A0_A0A0_A0A0_A0A0, 0xA0A0_A0A0_A0A0_A0A0]), + (false, -5, [0x90E7_D95B_C609_A90E, 0x09A9_0E7D_95BC_609A, 0xBC60_9A90_E7D9_5BC6, 0xD95B_C609_A90E_7D95, 0x0E7D_95BC_609A_90E7, 0x9A90_E7D9_5BC6_09A9]), + (false, -5, [0x94F2_094F_2094_F209, 0x094F_2094_F209_4F20, 0x2094_F209_4F20_94F2, 0xF209_4F20_94F2_094F, 0x4F20_94F2_094F_2094, 0x94F2_094F_2094_F209]), + (false, -5, [0x08FB_823E_E08F_B824, 0xFB82_3EE0_8FB8_23EE, 0x823E_E08F_B823_EE08, 0x3EE0_8FB8_23EE_08FB, 0xE08F_B823_EE08_FB82, 0x8FB8_23EE_08FB_823E]), + (false, -5, [0xEA4E_1A08_AD8F_2FBB, 0xFBA9_3868_22B6_3CBE, 0xCBEE_A4E1_A08A_D8F2, 0x8F2F_BA93_8682_2B63, 0xB63C_BEEA_4E1A_08AD, 0x8AD8_F2FB_A938_6822]), + (false, -5, [0xA7DE_6D1D_6086_4B8A, 0x8A7D_E6D1_D608_64B8, 0xB8A7_DE6D_1D60_864B, 0x4B8A_7DE6_D1D6_0864, 0x64B8_A7DE_6D1D_6086, 0x864B_8A7D_E6D1_D608]), + (false, -5, [0x0820_8208_2082_0821, 0x2082_0820_8208_2082, 0x8208_2082_0820_8208, 0x0820_8208_2082_0820, 0x2082_0820_8208_2082, 0x8208_2082_0820_8208]), + (false, -6, [0x0FC0_FC0F_C0FC_0FC1, 0xC0FC_0FC0_FC0F_C0FC, 0xFC0F_C0FC_0FC0_FC0F, 0x0FC0_FC0F_C0FC_0FC0, 0xC0FC_0FC0_FC0F_C0FC, 0xFC0F_C0FC_0FC0_FC0F]), + (false, -6, [0x503D_2263_57E1_6ECE, 0x40F4_898D_5F85_BB39, 0x03D2_2635_7E16_ECE5, 0x0F48_98D5_F85B_B395, 0x3D22_6357_E16E_CE54, 0xF489_8D5F_85BB_3950]), + (false, -6, [0x303B_5CC0_ED73_03B6, 0xC0ED_7303_B5CC_0ED7, 0x03B5_CC0E_D730_3B5C, 0x0ED7_303B_5CC0_ED73, 0x3B5C_C0ED_7303_B5CC, 0xED73_03B5_CC0E_D730]), + (false, -6, [0xD856_8903_9B0A_D120, 0x15A2_40E6_C2B4_481C, 0x6890_39B0_AD12_0736, 0x240E_6C2B_4481_CD85, 0x039B_0AD1_2073_615A, 0xE6C2_B448_1CD8_5689]), + (false, -6, [0x0E07_0381_C0E0_7038, 0x0703_81C0_E070_381C, 0x0381_C0E0_7038_1C0E, 0x81C0_E070_381C_0E07, 0xC0E0_7038_1C0E_0703, 0xE070_381C_0E07_0381]), + (false, -6, [0xDA74_0DA7_40DA_740E, 0x0DA7_40DA_740D_A740, 0x40DA_740D_A740_DA74, 0x740D_A740_DA74_0DA7, 0xA740_DA74_0DA7_40DA, 0xDA74_0DA7_40DA_740D]), + (false, -6, [0xB035_31DE_C0D4_C77B, 0x7B03_531D_EC0D_4C77, 0x77B0_3531_DEC0_D4C7, 0xC77B_0353_1DEC_0D4C, 0x4C77_B035_31DE_C0D4, 0xD4C7_7B03_531D_EC0D]), + (false, -6, [0x6474_A881_9EC8_E951, 0x2A20_67B2_3A54_40CF, 0x19EC_8E95_1033_D91D, 0x23A5_440C_F647_4A88, 0x5103_3D91_D2A2_067B, 0xCF64_74A8_819E_C8E9]), + (false, -6, [0x0CA4_587E_6B74_F033, 0x4F03_2916_1F9A_DD3C, 0xADD3_C0CA_4587_E6B7, 0x7E6B_74F0_3291_61F9, 0x161F_9ADD_3C0C_A458, 0xCA45_87E6_B74F_0329]), + (false, -6, [0x40C5_65C8_7B5F_9D4D, 0x9721_ED7E_7534_6F09, 0xB5F9_D4D1_BC25_0315, 0x5346_F094_0C56_5C87, 0xC250_3159_721E_D7E7, 0xC565_C87B_5F9D_4D1B]), + (false, -6, [0xC0C0_C0C0_C0C0_C0C1, 0xC0C0_C0C0_C0C0_C0C0, 0xC0C0_C0C0_C0C0_C0C0, 0xC0C0_C0C0_C0C0_C0C0, 0xC0C0_C0C0_C0C0_C0C0, 0xC0C0_C0C0_C0C0_C0C0]), + (false, -6, [0x2640_BC52_640B_C526, 0xC526_40BC_5264_0BC5, 0x0BC5_2640_BC52_640B, 0x640B_C526_40BC_5264, 0x5264_0BC5_2640_BC52, 0xBC52_640B_C526_40BC]), + (false, -6, [0x702E_05C0_B817_02E0, 0xC0B8_1702_E05C_0B81, 0x02E0_5C0B_8170_2E05, 0x0B81_702E_05C0_B817, 0x2E05_C0B8_1702_E05C, 0xB817_02E0_5C0B_8170]), + (false, -6, [0x0B40_B40B_40B4_0B41, 0x40B4_0B40_B40B_40B4, 0xB40B_40B4_0B40_B40B, 0x0B40_B40B_40B4_0B40, 0x40B4_0B40_B40B_40B4, 0xB40B_40B4_0B40_B40B]), + (false, -6, [0xB02C_0B02_C0B0_2C0B, 0x0B02_C0B0_2C0B_02C0, 0xC0B0_2C0B_02C0_B02C, 0x2C0B_02C0_B02C_0B02, 0x02C0_B02C_0B02_C0B0, 0xB02C_0B02_C0B0_2C0B]), + (false, -6, [0x0AC7_6918_40AC_7692, 0xC769_1840_AC76_9184, 0x6918_40AC_7691_840A, 0x1840_AC76_9184_0AC7, 0x40AC_7691_840A_C769, 0xAC76_9184_0AC7_6918]), + (false, -6, [0x17C0_A8E8_3F57_17C1, 0x3F57_17C0_A8E8_3F57, 0xA8E8_3F57_17C0_A8E8, 0x17C0_A8E8_3F57_17C0, 0x3F57_17C0_A8E8_3F57, 0xA8E8_3F57_17C0_A8E8]), +]; +/// `atan(j/16)` for j from 0 to 16. +#[rustfmt::skip] +pub const ATAN_SIXTEENTHS: [C; 17] = [ + (false, 0, [0; 6]), + (false, -4, [0x93FE_2926_AC74_803B, 0x0249_0094_73E9_B7D4, 0x387A_9F80_3C4B_8AED, 0x319C_12CF_59D4_B2DC, 0xCB27_92DC_0E2E_0D51, 0xFFAA_DDB9_67EF_4E36]), + (false, -3, [0x3FFA_5D51_E234_D71D, 0xD4D0_0802_01D9_C265, 0xEC93_577D_FB6C_1768, 0x8B39_57D9_5D9A_D922, 0xC897_989F_3E88_8EF7, 0xFEAD_D4D5_617B_6E32]), + (false, -2, [0x8B83_8DF4_FD70_0BD0, 0x9710_460E_84C5_1D02, 0xF05A_6A4F_1E7D_72D1, 0x93AD_00D6_838A_2B2A, 0x7B0B_4F88_1C9C_7487, 0xBDCB_DA5E_72D8_1134]), + (false, -2, [0x05BE_8C5B_A583_1A3E, 0x92B9_2516_68E5_7653, 0x7A3A_767C_9449_A765, 0xAA7F_A903_88B3_836B, 0x6DC7_9EF5_F7A2_17E5, 0xFADB_AFC9_6406_EB15]), + (false, -1, [0x4406_7EF4_68C8_7459, 0xD338_D324_A9DD_F7E6, 0xF5A5_900C_1304_C7AE, 0x08A4_8F85_0B2F_55FF, 0xC5AB_B498_D27A_F328, 0x9B13_B9B8_3F5E_5E69]), + (false, -1, [0xDCCC_E4A8_00CF_73D3, 0x64CC_5820_39FD_C0D6, 0x4733_E5CA_3349_08E0, 0x3D84_3977_AE54_55CC, 0x8AA3_2122_DCFE_4483, 0xB7B0_CA0F_26F7_8473]), + (false, -1, [0xF796_8592_537F_6C5C, 0x2857_E840_BC44_83B4, 0x2667_7F8F_13DD_7E28, 0x6E00_40DD_2161_A73C, 0x427C_95E9_001E_7136, 0xD327_761E_611F_E5B6]), + (false, -1, [0x87C8_12F8_C6A4_6183, 0xFA9C_5917_892B_516C, 0xE2CF_16E6_E962_4709, 0x6E14_1587_261C_DF00, 0x6FE4_45EC_BC3A_8D03, 0xED63_382B_0DDA_7B45]), + (false, 0, [0x82C3_BD5B_D4BD_DC25, 0x43D7_D60C_D4F1_3F4E, 0x55C6_BDCF_1E5B_65D0, 0xF115_5CD8_774D_DFBC, 0x4B6A_09CB_61A5_15C0, 0x832B_F4A6_D986_7E2A]), + (false, 0, [0x66CD_7DD3_073D_6A58, 0x3669_DCD3_918D_712B, 0xF53D_A46D_1338_9EB2, 0x918A_67E0_652B_375C, 0x5C83_5E16_65C4_3747, 0x8F00_5D5E_F7F5_9F9B]), + (false, 0, [0x1162_0AE2_966B_DA9C, 0x4CCD_BFE6_B74A_D88F, 0x7787_4D1E_753A_FEE4, 0xDAE4_6F06_1748_9D5C, 0x4226_F8E2_204F_F3BC, 0x9A2F_80E6_71BD_DA20]), + (false, 0, [0x5762_206E_D3D0_24B4, 0xA7F7_B7DB_933C_B84F, 0x2148_6665_8CC4_EF3A, 0xE3F0_8689_EEB2_B9E7, 0x19A8_7F2A_457D_AC9E, 0xA4BC_7D19_34F7_0924]), + (false, 0, [0xC658_6FE1_DD43_9AF3, 0x95EE_0FD1_EEF1_F3D7, 0x0440_7947_C44F_DD30, 0x0374_2B46_43EF_FE26, 0x1472_5E2F_3E52_070A, 0xAEAC_4C38_B4D8_C080]), + (false, 0, [0x49C4_9EC7_2668_272D, 0x4655_F065_CEC9_8182, 0xA012_155F_64CA_E530, 0x3D7A_ECC1_14C7_9A80, 0xCB2D_A552_10A4_443D, 0xB805_3E2B_C231_9E73]), + (false, 0, [0xA34A_C5CF_485B_C5C2, 0x44FF_A53C_51A6_2F57, 0x3708_1467_A10B_2D25, 0xD58E_E867_AEF4_36F6, 0x89DD_62C4_6E92_FA24, 0xC0CE_85B8_AC52_6640]), + (false, 0, [0xEF95_19B3_CD3A_431B, 0x514A_0879_8E34_04DD, 0x020B_BEA6_3B13_9B22, 0x2902_4E08_8A67_CC74, 0xC4C6_628B_80DC_1CD1, 0xC90F_DAA2_2168_C234]), +]; +/// `ln(1 + j/16)` for j from −5 to 7. +#[rustfmt::skip] +pub const LN_SIXTEENTHS: [C; 13] = [ + (true, -1, [0x4A05_F8F9_AFF5_41C2, 0x714C_D163_6486_9A35, 0xA83C_C3FD_C46D_FB26, 0x9C4C_3CCC_CEA8_F9E1, 0x37ED_A996_244B_CCAF, 0xBFD7_D1DE_C0A8_DF6F]), + (true, -1, [0xB571_5D87_24B3_4E99, 0x49B2_D2DD_9E21_81C0, 0xAD04_7F99_8C19_7D96, 0x87A4_86E6_5AA1_BCD5, 0xDF5B_B3B6_0554_E151, 0x934B_1089_A6DC_93C1]), + (true, -2, [0xE306_1D01_03B9_CA85, 0x9CE3_D1B4_1D3D_BA7A, 0x7EC0_D7BF_B6B6_5F0C, 0x7188_AF8F_4F45_B9EE, 0x5F53_BD2E_406E_66E7, 0xD49F_69E4_56CF_1B79]), + (true, -2, [0xBC72_3671_B137_6A00, 0xE51F_93FB_0A4B_5A21, 0xBF07_ED54_C028_B125, 0xDA8F_EC3C_24B7_8A37, 0x9C5A_0FE3_96F4_0F1D, 0x88BC_7411_3F23_DEF1]), + (true, -3, [0x4DAB_C5A6_0E5E_16F4, 0x1A36_038A_919C_29B1, 0x2C88_5EBB_0A67_BC24, 0xE362_C7F8_DD18_E5CB, 0x1FEC_DFA8_19B9_6097, 0x842C_C5AC_F1D0_3445]), + (false, 0, [0; 6]), + (false, -4, [0x0872_2D98_5BAD_CCAA, 0x172D_A6BD_E861_0069, 0x1851_F0A9_6F69_8496, 0x3474_D337_5B52_5967, 0xE64B_8B77_5997_898D, 0xF851_8600_8B15_330B]), + (false, -3, [0x011F_ABA1_3236_63C0, 0xF02D_1A16_7E51_56BD, 0xE844_BE0F_FC8B_E4AC, 0xCA75_E200_BFB7_CABE, 0x543F_FF0F_F4F0_AAED, 0xF138_3B71_5797_2F4F]), + (false, -2, [0x5F51_3316_D504_9A3D, 0xB6EB_F7F4_08CC_47E6, 0x02B3_FB5B_64A0_A477, 0x5571_CD57_F7CF_98C7, 0x9F10_5039_091D_D7F3, 0xAFF9_8385_3C9E_9E43]), + (false, -2, [0xC40C_D83B_4237_91B8, 0x864A_A3F5_F374_EEA8, 0xC3C4_CFD5_92FF_1D1A, 0x1D97_A9D0_46B7_06C5, 0x2EC0_F797_FDCD_1257, 0xE47F_BE3C_D4D1_0D61]), + (false, -1, [0x1780_2D36_98A5_3289, 0x932E_4F65_B890_2A5F, 0x8791_B873_2B28_1E2E, 0x0CFA_0948_7833_EA69, 0xE63E_AB88_3717_047E, 0x8B3A_E55D_5D30_701C]), + (false, -1, [0x6124_ACFC_E5FE_F460, 0x5E24_1ADD_7750_BEFB, 0x6BDD_6AB9_52E7_F931, 0xE59A_4980_1688_7279, 0x5BD9_BD99_E39A_20AE, 0xA30C_5E10_E2F6_13E8]), + (false, -1, [0x8936_F8F7_3C29_44D2, 0xA0BA_5B82_958C_BA55, 0x7924_F09A_4688_DE04, 0xBBC2_1B9D_4B2E_AF77, 0x1262_9FD6_9F7B_CB55, 0xB9CE_BFB5_DE80_34E7]), +]; +/// `sin(jπ/128)` for j from 0 to 32. +#[rustfmt::skip] +pub const SIN_PI_128THS: [C; 33] = [ + (false, 0, [0; 6]), + (false, -5, [0x9E06_47AF_97E1_D695, 0x0721_7866_CB16_FAC9, 0x9E63_5806_DBD0_38D5, 0x4CA5_F3C3_54A1_1581, 0xC539_EDCB_FDA0_CF2C, 0xC90A_AFBD_1B33_EFC9]), + (false, -4, [0x2E72_BF5A_F601_0C46, 0x8902_B24F_DD1F_B467, 0x36FC_97B1_FEF1_FDEA, 0x2167_69FC_DA0D_0EBF, 0x6A17_954B_2B7C_5171, 0xC8FB_2F88_6EC0_9F37]), + (false, -3, [0xE9B3_A30A_4CF5_4FF2, 0x1EC8_2521_1F1A_06FF, 0xDC9C_4943_BA9B_D204, 0x1293_0F6D_BA22_F08A, 0xF775_7409_4D3C_35C4, 0x96A9_0496_70CF_AE65]), + (false, -3, [0x400C_BA03_CF01_C89D, 0x0D79_045D_EB7A_0095, 0xCD5F_082F_F07D_F518, 0x0818_4085_46EE_849E, 0xC739_6C89_4BBF_7389, 0xC8BD_35E1_4DA1_5F0E]), + (false, -3, [0xC58F_76A0_B1BB_D883, 0xD7E2_9DED_879D_6AF4, 0x8645_35F8_9D09_1486, 0x1213_A3BD_3767_98F7, 0x7047_29AE_56D7_8A37, 0xFAB2_72B5_4B98_71A2]), + (false, -2, [0xDF75_D2AA_BAD5_B167, 0x3A1E_679A_44E0_0F80, 0xF1BD_C9C1_AC65_2E22, 0x7F85_0122_F9E1_B840, 0x000A_89A1_1E07_C1FE, 0x9640_8374_7309_D113]), + (false, -2, [0x6D46_8F36_B596_8643, 0xC4AF_AC07_4C89_06C4, 0x70CB_EE14_1E9D_4ED5, 0x8F56_8AE3_7078_0D07, 0x3FEE_F3BB_58B1_F10C, 0xAF10_A224_59FE_32A6]), + (false, -2, [0x9216_5E4C_6070_DC60, 0x8A3E_F6F9_FAC6_61F8, 0xE50E_1D9E_2550_5306, 0xCE53_9634_4A5C_BC74, 0x5CC8_C00E_4FCC_D84F, 0xC7C5_C1E3_4D30_55B2]), + (false, -2, [0x988E_8AA0_3132_C4C5, 0x9D88_D3DB_CF87_87AE, 0x04ED_C5B9_C415_799C, 0xB26E_0071_A479_C6EE, 0x0EBC_61AD_E6CA_83CC, 0xE05C_1353_F27B_17E5]), + (false, -2, [0x3FEB_AC8B_5D23_5B51, 0xA6FC_7EB9_57A5_CA25, 0xDB3B_B977_B9B4_B552, 0x5600_217E_7089_1ECD, 0x4221_DC4B_A772_598D, 0xF8CF_CBD9_0AF8_D57A]), + (false, -1, [0xFA32_F5F4_0783_089C, 0x4131_767E_D0A5_A563, 0xD4DC_74C1_FB5C_711C, 0xEE6A_6256_8EEE_784C, 0x9841_56F5_5334_4305, 0x888E_9315_8FB3_BB04]), + (false, -1, [0xED8E_8C12_9794_23BC, 0x2DD1_526E_8427_3C8E, 0x38AC_3D47_CA10_9385, 0x8E0A_682E_37ED_1502, 0xAE4B_A773_DA6B_F753, 0x94A0_3176_ACF8_2D45]), + (false, -1, [0x106E_2513_96F3_B9CE, 0x2D5E_B80A_F442_D518, 0xC912_A50B_E56F_D2A0, 0xF614_1485_E8B8_837A, 0x2F89_5F44_A303_CC0A, 0xA09A_E4A0_BB30_0A19]), + (false, -1, [0x764F_BFF7_F805_565F, 0xFB3A_15ED_1F80_DCFE, 0xC3CB_6EAB_7052_2AB0, 0x342A_F9DB_36A9_C8A9, 0x811F_9539_84EF_F83E, 0xAC7C_D3AD_58FE_E7F0]), + (false, -1, [0x737E_260F_BAC9_03EC, 0xD078_3FFB_47D8_8EDE, 0x56FA_1D40_0A43_67B3, 0xDEEA_8EFE_BB77_0256, 0x9CC3_EF36_746D_E3B7, 0xB844_2987_D22C_F576]), + (false, -1, [0xF714_2064_C9DB_5E4C, 0xDA25_AB37_7538_9102, 0x7478_2B20_C4E3_1EE2, 0x7BE3_EB69_567B_C498, 0x3122_C2A5_9EFD_DC37, 0xC3EF_1535_754B_168D]), + (false, -1, [0x2A96_E94A_CD1A_0B1A, 0x2519_FDB3_8D4A_4B58, 0xED91_83A4_39F7_1A88, 0x2DFD_8B0D_C011_C592, 0xA812_90BD_BAAD_62E4, 0xCF7B_CA1D_476C_516D]), + (false, -1, [0x5152_4F06_DF01_4687, 0xBCF4_2402_8E4E_639A, 0x3655_FF4D_38D2_7A26, 0x083C_E50F_3E59_8515, 0x362C_B974_182E_3030, 0xDAE8_804F_0AE6_015B]), + (false, -1, [0x4955_D87C_EA7D_0632, 0xDF06_150F_1DCE_2A4B, 0x97ED_C1CA_ECD3_AC70, 0xB4B9_28A7_F449_4A68, 0x2872_CE1B_FC7A_D1CC, 0xE633_74C9_8E22_F0B4]), + (false, -1, [0x2E8C_49F7_6FBA_B289, 0x3592_C3E1_60F0_D91D, 0x9503_39DA_70B9_7314, 0xAD15_5B05_3B59_41A1, 0x6C48_E9E3_420B_0F1D, 0xF15A_E9C0_37B1_D8F0]), + (false, -1, [0x7A5A_3DE7_6E83_00A3, 0x5171_5874_355A_6E64, 0xA6F5_0FF4_8B9D_0717, 0x6AB5_C209_AE2F_4097, 0x27C0_7C91_1290_B8D1, 0xFC5D_26DF_C4D5_CFDA]), + (false, 0, [0xC4E8_DAC7_4071_0C18, 0x718F_B60D_736D_CE7C, 0x2D3B_FE70_D9AB_409A, 0xD6BC_7FE5_7653_AEC3, 0xBFD7_9717_F288_0ABE, 0x839C_3CC9_17FF_6CB4]), + (false, 0, [0x275C_1D52_1F91_CA92, 0xDD54_9BB3_52A2_47B8, 0x5C00_D180_C765_5A29, 0x1DDC_902C_4A28_D838, 0xB892_CA83_61D8_C84C, 0x88F5_9AA0_DA59_1421]), + (false, 0, [0xDF93_EAC8_EC56_1D41, 0x7F91_52CB_A72A_E50C, 0xCA99_6068_C296_FD79, 0x7CA7_D749_ADFB_A33E, 0xBBA4_CFEC_BFF5_4867, 0x8E39_D9CD_7346_4364]), + (false, 0, [0x085D_7226_8351_3DF8, 0x4F95_EC57_47E0_9033, 0x7AD1_253F_5EC1_4FF0, 0x44B5_62FE_29A5_3A66, 0xB178_2191_1E71_C16E, 0x9368_2A66_E896_F544]), + (false, 0, [0x5169_4C24_22E6_A3FE, 0x01AC_D947_A164_7870, 0x1D9C_E3A2_743A_1804, 0x6B71_4DF5_A5E7_2D4F, 0x19CE_C845_AC87_A5C6, 0x987F_BFE7_0B81_A708]), + (false, 0, [0x81CB_56DE_4833_3065, 0xC54B_364F_1553_B6CC, 0x21F5_F9A3_921B_151E, 0x8742_4855_DAF5_D9F3, 0xE25E_3954_9638_AE67, 0x9D7F_D149_0285_C9E3]), + (false, 0, [0x1537_0B1D_FF51_5E33, 0xA7DE_0615_A750_1006, 0x04C7_B378_B215_B01D, 0xC381_700B_DF75_7600, 0x3B51_67EE_359A_234D, 0xA267_9928_48EE_B0C0]), + (false, 0, [0xA8AA_22AB_31E0_36A8, 0xE237_EEAA_1B32_DAC2, 0x3D51_7247_A54A_9EA8, 0x9AAB_00F4_88EE_A6AD, 0x149F_6E75_9934_68A2, 0xA736_55DF_1F2F_489E]), + (false, 0, [0x307F_343E_B797_3BB4, 0x3C2E_A3BE_7E7A_3226, 0x6474_2CA4_28C8_0C16, 0x2FD0_F385_9ABE_CFF4, 0x1BEC_DA80_89C1_A94C, 0xABEB_49A4_6764_FD15]), + (false, 0, [0xDC1D_A803_0635_A8B3, 0xFEBC_57D4_74CF_8BA5, 0xC057_0800_FB96_7A6B, 0xA642_438D_CE55_E433, 0xE4CA_D00D_5C94_BCD1, 0xB085_BAA8_E966_F6DA]), + (false, 0, [0xA8B1_FE6F_DC83_DB39, 0x4AFC_8304_3AB8_A2C3, 0xED17_AC85_8333_9915, 0x1D6F_60BA_893B_A84C, 0x597D_89B3_754A_BE9F, 0xB504_F333_F9DE_6484]), +]; +/// `cos(jπ/128)` for j from 0 to 32. +#[rustfmt::skip] +pub const COS_PI_128THS: [C; 33] = [ + (false, 1, [0x0000_0000_0000_0000, 0x0000_0000_0000_0000, 0x0000_0000_0000_0000, 0x0000_0000_0000_0000, 0x0000_0000_0000_0000, 0x8000_0000_0000_0000]), + (false, 0, [0xB707_F211_99FE_9209, 0xAB5D_C1A4_1978_5764, 0xEF6F_30DA_8FE3_07E8, 0x568F_B692_8218_2F32, 0x5657_5523_6696_1732, 0xFFEC_4304_2668_65D9]), + (false, 0, [0xA94F_F904_6C34_D4E4, 0x1BF5_8F6A_90D5_20D5, 0x5750_700A_6DF2_6524, 0xCF53_E4BA_A403_250B, 0x421E_8EDA_AF59_453D, 0xFFB1_0F1B_CB6B_EF1D]), + (false, 0, [0xF141_9BCC_0FF7_192E, 0x3B7C_57BC_5C73_5193, 0x964A_AD96_D95A_E9FB, 0x91CF_862E_7FAD_71EE, 0x0F66_8633_F1AB_8589, 0xFF4E_6D68_0C41_D0A9]), + (false, 0, [0x4852_24A9_82A7_C85A, 0xA0E9_EE7A_8525_8E45, 0xB268_1B46_8EFB_A361, 0x5A2C_976D_196A_3567, 0x4139_0EFD_C726_E9EF, 0xFEC4_6D1E_8929_2CF0]), + (false, 0, [0xF65D_64CE_2EA9_40EE, 0x300E_707E_9A9A_66E5, 0x92AE_1E5D_881F_153B, 0xCC9D_5D78_7E81_F867, 0x90CD_1D95_9DB6_74EE, 0xFE13_2387_0CFE_9A3D]), + (false, 0, [0xB393_BE74_C838_1589, 0x6731_71E6_A655_4F35, 0x62B8_1762_1FF0_F760, 0xA197_02D6_D140_D729, 0xEAE6_BD95_1C1D_ABBD, 0xFD3A_ABF8_4528_B50B]), + (false, 0, [0x9C19_91B6_86A0_0C7D, 0x184B_1D6F_CCFC_856E, 0xB859_505A_0B30_1225, 0x246D_DC4E_63F5_9390, 0x2567_78FF_CB5C_1769, 0xFC3B_27D3_8A5D_49AB]), + (false, 0, [0x8D5B_15EA_A583_68EF, 0x947A_5373_353C_45A4, 0xB199_7321_B48E_8B1B, 0xC5F4_0E3F_D8F1_8AE1, 0x2172_A361_FD2A_722E, 0xFB14_BE7F_BAE5_8156]), + (false, 0, [0x5AF9_10DE_D9DE_EB17, 0xA87C_F32C_E084_AF6A, 0x61E6_3B20_928C_47AF, 0x79B5_67A6_7CDE_8B61, 0x4504_AE08_D19B_2980, 0xF9C7_9D63_272C_4628]), + (false, 0, [0x9958_F29A_37EF_D515, 0x18DA_934F_3FC6_3793, 0x6D19_3284_5F38_13DD, 0x95A1_ACB3_4336_3FB2, 0xC7AD_C6B4_9888_91BA, 0xF853_F7DC_9186_B952]), + (false, 0, [0x0F46_E91A_CB4E_EAB8, 0xAFDC_4073_2A1B_03F6, 0xF2F6_8677_A56E_FB23, 0xB068_54A7_D313_9B87, 0x2C79_1F59_CC1F_FC22, 0xF6BA_073B_424B_19E8]), + (false, 0, [0x2A53_096F_0C12_78A8, 0xD5FD_8A1C_AFB2_CCE5, 0x8E77_EE63_670F_A090, 0x5EE4_B099_89A1_1728, 0x163C_5C7F_03B7_18C5, 0xF4FA_0AB6_316E_D2EC]), + (false, 0, [0xE285_567B_61BD_57D7, 0xC3D4_240D_5D9A_890A, 0x511A_ACDC_D3B8_EB99, 0x6CC9_0B7D_06C5_06D8, 0xA548_6BDC_455D_56A2, 0xF314_4762_4708_8F74]), + (false, 0, [0xB39A_CCC3_025A_1B44, 0xD2CF_ECDD_D6B7_BA8A, 0x894C_0E9A_AC77_BFE4, 0x89DA_B2E0_FE03_A229, 0x6712_7DB3_5B28_7315, 0xF109_0827_B437_25FD]), + (false, 0, [0xA8AC_C772_534B_BB89, 0x0E0E_A289_950A_DD57, 0xB6A4_DEC4_98E3_FBC3, 0xFC4F_ACDF_D3D6_4E3F, 0x86F8_C20F_B664_B01A, 0xEED8_9DB6_6611_E307]), + (false, 0, [0xB1EC_03E9_7399_98C9, 0xA634_0CB6_5C4B_11D5, 0xDB89_7C23_8408_3746, 0xF0A8_3D3C_D0DA_E9B5, 0x7E61_0231_AC1D_6180, 0xEC83_5E79_946A_3145]), + (false, 0, [0xE4F7_620C_FF6D_E0C8, 0x23CB_86A3_C7B2_9248, 0x2397_EA24_D504_9427, 0x3D57_05F1_0A9A_7F6F, 0x15AD_45B4_A1B5_E823, 0xEA09_A68A_6E49_CD62]), + (false, 0, [0xC0A7_6C66_9F14_305B, 0xC924_4112_52C0_9E83, 0x09DE_14EF_EA57_5873, 0x2F1B_B328_2663_E451, 0x1251_2952_9D48_A92F, 0xE76B_D7A1_E63B_9786]), + (false, 0, [0x4995_E362_E423_E0B2, 0xB858_FEC8_2022_732F, 0x2790_2AA3_6084_A51B, 0x8743_F3FE_219A_6A8F, 0x1227_85AE_67F5_515C, 0xE4AA_5909_A08F_A7B4]), + (false, 0, [0xC80F_46BE_AF1A_70B6, 0xF775_E41E_C001_5D69, 0x82C4_FE35_F8F2_9CEA, 0xFDEA_3FF0_DC05_3327, 0xF4E8_A837_2F8C_580F, 0xE1C5_978C_05ED_8691]), + (false, 0, [0xACD6_4CCE_066F_8326, 0xA01D_809B_EB9C_BEF8, 0x86A3_422B_1823_0D41, 0xD4A2_5796_6E11_6379, 0x4B19_AA71_FEC3_AE6C, 0xDEBE_0563_7CA9_4CFB]), + (false, 0, [0xCBA1_E4CA_9B20_EF6B, 0xA718_57FD_6A65_3DB4, 0xF8AE_9C38_D38A_7789, 0xB672_92CB_DF4C_B882, 0x2C19_B632_53DA_43FB, 0xDB94_1A28_CB71_EC87]), + (false, 0, [0x2220_CF60_3079_04FB, 0xF209_94C8_B2E8_F9F4, 0x17B1_5CB0_A701_425B, 0xDD2C_2B52_901F_E500, 0x24B9_FE00_6635_74A3, 0xD848_52C0_A80F_FCDB]), + (false, 0, [0xC63D_F891_EFB8_7D5C, 0x3F46_DE4F_BAFD_C08B, 0xAC08_E58A_7CD3_9544, 0xB469_1D2F_99EC_9EAA, 0xF630_E8B6_DAC8_3E68, 0xD4DB_3148_750D_1819]), + (false, 0, [0x5B31_1EE0_C5A1_E4F3, 0x18E8_58D2_2518_7BD4, 0x39D9_56BC_FE9A_5D9A, 0xE386_CBDA_4F76_D559, 0x744F_EA20_E8AB_EF91, 0xD14D_3D02_313C_0EED]), + (false, 0, [0xA9AA_C44F_B448_6325, 0x6C21_62F3_7C43_1F7F, 0xC881_A550_2747_6AF7, 0x9D02_F0E6_CAA1_7D3D, 0x23AF_31DB_7179_A4A9, 0xCD9F_023F_9C3A_059E]), + (false, 0, [0x7C86_3E3C_5FB2_4298, 0xA0FA_0954_D3D0_40BB, 0x3E33_F423_0B84_8C8A, 0x4BFC_4D9A_BFEB_5F9F, 0x8335_241B_E169_3225, 0xC9D1_124C_931F_DA7A]), + (false, 0, [0x0E2F_6071_F0D4_1621, 0x810C_85A6_661D_3BAF, 0x9BCD_AEF0_2B67_3F05, 0xD474_37D3_710B_81D7, 0x43DA_25D9_9267_326A, 0xC5E4_0358_A8BA_05A7]), + (false, 0, [0x6C34_A47E_DDD9_D480, 0x4203_B9BF_F493_5AFD, 0x164C_59C4_FC98_9F35, 0xD785_C0CF_C7C2_E7A2, 0xBDF0_715C_B8B2_0BD6, 0xC1D8_705F_FCBB_6E90]), + (false, 0, [0xE9FD_235A_467A_6348, 0x6A8E_7399_0FF5_1F79, 0xE0BD_3568_6BED_05F0, 0x9C67_7149_ABC2_CAAF, 0xAC85_320F_528D_6D5C, 0xBDAE_F913_557D_76F0]), + (false, 0, [0xC7B2_2E56_47B3_09C5, 0xF052_B359_06EA_C2EE, 0xD1AB_4F85_1F61_04B5, 0x43B1_297E_FF84_2A9F, 0x9DE1_E3B2_2B8B_F4DB, 0xB968_41BF_7FFC_B21A]), + (false, 0, [0xA8B1_FE6F_DC83_DB39, 0x4AFC_8304_3AB8_A2C3, 0xED17_AC85_8333_9915, 0x1D6F_60BA_893B_A84C, 0x597D_89B3_754A_BE9F, 0xB504_F333_F9DE_6484]), +]; +/// `2^(j/32)` for j from 0 to 31. +#[rustfmt::skip] +pub const EXP2_32NDS: [C; 32] = [ + (false, 1, [0x0000_0000_0000_0000, 0x0000_0000_0000_0000, 0x0000_0000_0000_0000, 0x0000_0000_0000_0000, 0x0000_0000_0000_0000, 0x8000_0000_0000_0000]), + (false, 1, [0xD31A_9B21_2C31_DA8A, 0x1A0A_663A_A009_92EA, 0x2502_F150_6737_8A17, 0x29F1_A4AF_BEFA_5D7C, 0x3E2A_475B_4652_0BFF, 0x82CD_8698_AC2B_A1D7]), + (false, 1, [0x82DC_9BF3_4218_40D2, 0x7835_AF9A_B4E6_355C, 0x5D42_B362_AF1E_E859, 0x148A_0459_E758_5151, 0xC5C9_5B8C_2154_C1B2, 0x85AA_C367_CC48_7B14]), + (false, 1, [0x372E_BD45_721C_7672, 0x6768_735A_D779_487D, 0x8BC3_587F_B118_C94D, 0xE623_D58B_3772_BA13, 0x5DF8_D76C_98C6_7562, 0x8898_0E80_92DA_8527]), + (false, 1, [0x9994_FAC7_F4EA_654C, 0xB4F2_8C87_9A52_4BD4, 0x91E1_35EE_84A3_F733, 0x1AA8_4FFB_EBAC_349F, 0xFBE4_6287_58A5_3C90, 0x8B95_C1E3_EA8B_D6E6]), + (false, 1, [0x9ADA_7463_A678_3E85, 0x7D3E_DC47_B8A2_A6E3, 0xEB90_CE37_00BF_59B6, 0xA110_3723_0B36_7828, 0x2DC0_144C_8783_D4C5, 0x8EA4_398B_45CD_53C0]), + (false, 1, [0x23B5_04FB_5DAD_BB25, 0x4E0D_990C_27C4_3643, 0xF120_3CAF_65BF_B9B9, 0x1942_B348_16FB_4F26, 0x0FD6_D8E0_AE5A_C9D8, 0x91C3_D373_AB11_C336]), + (false, 1, [0x2484_B2CB_2958_AD46, 0xA2C0_832E_C246_2B1A, 0x035F_B634_C2E6_3A0E, 0x4856_0469_01FF_6C05, 0x2E8A_FAD1_2551_DE54, 0x94F4_EFA8_FEF7_0961]), + (false, 1, [0x4E9F_94EE_0C90_DBBD, 0xBE47_C34D_3802_50A8, 0xD78B_65CB_EFA7_BB6F, 0x5E13_9A1B_14FA_8178, 0x46AD_2318_2E42_F6F6, 0x9837_F051_8DB8_A96F]), + (false, 1, [0x3C9F_D243_0EC8_5332, 0x6452_9A5A_58FE_31DB, 0x9769_D9B0_A908_A786, 0x0928_B5FC_E34C_DF21, 0xA2A8_17A2_A3CC_3F1F, 0x9B8D_39B9_D54E_5538]), + (false, 1, [0xA264_696B_6F6E_2B1C, 0xCD12_E5AD_A91E_EF7B, 0x21F9_77FE_7C7F_A117, 0x65C1_5C12_2133_E2A2, 0xA091_1F09_EBB9_FDD1, 0x9EF5_3260_91A1_11AD]), + (false, 1, [0x6CE3_DDC6_8357_C2BD, 0x4BB9_8F72_CE6E_577F, 0x5A7A_7992_2180_8DE9, 0x9DA4_384D_BC2C_8EAE, 0x9B7A_04EF_80CF_DEA7, 0xA270_4303_0C49_6818]), + (false, 1, [0xAD35_E8E5_8C5C_E4F5, 0x5343_B4A0_330A_8052, 0x2589_C98A_8290_D3F0, 0x1DD1_70AC_E2BC_FC17, 0x1CBD_7F62_1710_701B, 0xA5FE_D6A9_B151_38EA]), + (false, 1, [0x28EE_6DF7_03CF_75F1, 0x8906_EC5F_A145_A886, 0x325C_9E22_0350_4516, 0x3951_F214_C02D_824A, 0x541E_24EC_3531_FA73, 0xA9A1_5AB4_EA7C_0EF8]), + (false, 1, [0x739A_3D06_1FEA_7593, 0x458F_D5F4_4E26_A7A7, 0xB165_F141_833A_67DA, 0x6BE4_0940_7034_FDED, 0x4980_A8C8_F59A_2EC4, 0xAD58_3EEA_42A1_4AC6]), + (false, 1, [0x7E93_B6B5_EF54_B640, 0x4E1C_E1E5_9FFC_7C71, 0x97CE_D890_D5B0_B0C0, 0x757C_FB99_13AD_C577, 0x87D0_37E9_6D21_5D8E, 0xB123_F581_D2AC_258F]), + (false, 1, [0xA8B1_FE6F_DC83_DB39, 0x4AFC_8304_3AB8_A2C3, 0xED17_AC85_8333_9915, 0x1D6F_60BA_893B_A84C, 0x597D_89B3_754A_BE9F, 0xB504_F333_F9DE_6484]), + (false, 1, [0xD978_A131_52B2_718B, 0x9D85_7B40_8C48_F4FD, 0x322D_7893_ED4D_A9A7, 0xA5AB_16CF_4510_56ED, 0x1B87_9778_566B_65A1, 0xB8FB_AF47_62FB_9EE9]), + (false, 1, [0x283C_66DB_56DF_97F8, 0x9B98_5F3A_0EAE_1C1D, 0x0D9A_4BE0_23EC_E031, 0x15B3_4BBC_B029_8F41, 0xA881_1FB6_6D0F_AF7A, 0xBD08_A39F_580C_36BE]), + (false, 1, [0xFBD6_529A_59CF_B8C8, 0x470A_F538_3ECD_3ADC, 0xDEFE_FEE7_2AE7_A33D, 0x6B2E_5DD6_07A9_969C, 0x7C45_7D59_A500_87B5, 0xC12C_4CCA_6670_9456]), + (false, 1, [0xEFBA_190E_D1A5_8A52, 0xBB20_68BE_2375_12DF, 0xC768_6006_E4E6_C092, 0x6B0F_9399_9825_1A36, 0x3E2A_D0C9_64DD_9F37, 0xC567_2A11_5506_DADD]), + (false, 1, [0xC133_A6AC_D534_6749, 0x610D_A545_A82A_EB32, 0xF4DD_023F_F93C_7FFB, 0x257A_C0DB_1F41_9377, 0x80E1_F92A_0511_697E, 0xC9B9_BD86_6E2F_27A2]), + (false, 1, [0xCD8C_AEBC_5D89_4C26, 0x1B83_4308_8BBD_ADD4, 0x2BBD_398A_F35C_079F, 0x6F28_610B_8C36_485A, 0xE235_838F_95F2_C6ED, 0xCE24_8C15_1F84_80E3]), + (false, 1, [0xA05A_7730_9F43_6B62, 0xE699_75A7_684D_759B, 0xFA76_6303_3F05_357A, 0x5202_9C0B_81F7_BE57, 0x1224_8E57_C3DE_4028, 0xD2A8_1D91_F12A_E45A]), + (false, 1, [0x8124_7458_FB61_E576, 0xEFB0_1FDA_334B_CA9A, 0xB5C1_3ADA_0E77_8299, 0x1D73_3AF5_2205_8B16, 0x39A6_8BB9_902D_3FDE, 0xD744_FCCA_D69D_6AF4]), + (false, 1, [0xC12D_F1FE_DBCD_8B32, 0xC635_0F57_37A1_16BA, 0x613B_0D1D_BFA0_D716, 0x481E_1AB7_25B1_2D56, 0x3D84_0D5A_9E29_AA64, 0xDBFB_B797_DAF2_3755]), + (false, 1, [0xF44E_1726_2DA4_9B5D, 0x1880_81FE_7062_F61E, 0x1CB9_9D3F_1FF2_98A2, 0x224B_251B_3309_2002, 0x0658_9504_8DD3_33CA, 0xE0CC_DEEC_2A94_E111]), + (false, 1, [0x3F7A_1E03_0A0A_43BA, 0x5CD7_D504_25C6_ACE6, 0xCEFC_D5B6_2A14_B818, 0xAACD_6065_B6E9_F6AC, 0x1E5E_8F4A_4EDB_B0EC, 0xE5B9_06E7_7C83_48A8]), + (false, 1, [0x0914_8285_4D81_9E44, 0xB338_FCD2_AC2F_FBC8, 0x17D8_D1E8_CA31_880A, 0xC4FA_ACE0_43B7_F91C, 0xD02D_75B3_706E_54FA, 0xEAC0_C6E7_DD24_392E]), + (false, 1, [0x4281_03C6_BABF_7CCC, 0x22B9_8A27_35B2_A131, 0x9392_8708_34F2_1A53, 0xD4A2_77EA_DDAA_925C, 0x4656_1CF6_948D_B912, 0xEFE4_B99B_DCDA_F5CB]), + (false, 1, [0x29F0_75F2_3730_E918, 0xD22D_D036_F190_6094, 0xBDD8_0329_364A_A29F, 0x6F51_0308_6777_09F5, 0x7B9D_0C7A_ED98_0FC3, 0xF525_7D15_2486_CC2C]), + (false, 1, [0x42E7_4BD5_B359_AF22, 0x4B02_0716_6EE1_375E, 0x4844_B29B_F4AF_18E8, 0x8006_FE21_A95D_14DC, 0x7C25_BB14_315D_7FCC, 0xFA83_B2DB_722A_033A]), +]; diff --git a/src/cx.rs b/src/cx.rs new file mode 100644 index 0000000..090dd33 --- /dev/null +++ b/src/cx.rs @@ -0,0 +1,786 @@ +//! Complex arithmetic in the working type, and the functions built on it. + +use crate::consts::{EULER_GAMMA, FRAC_2_SQRT_PI, INV_FACT}; +use crate::eval::{ + HUGE, atan_ratio, beyond, by_width, exp, exp_wide, expm1, half_pi, ln, log1p, log1p_ratio, + phi as phi_real, pi, sin_cos, sin_cos_octant, +}; +use crate::real::Work; +use crate::stirling::{LN_GAMMA_1P, LN_SQRT_2PI, STIRLING}; +use core::ops::{Add, Div, Mul, Neg, Sub}; + +#[derive(Clone, Copy, Debug)] +pub struct Cx { + pub re: V, + pub im: V, +} + +pub fn atan2(y: V, x: V) -> V { + let angle = if x.is_zero() { + if y.is_zero() { V::zero() } else { half_pi() } + } else { + let base = atan_ratio(y.abs(), x.abs()); + if x.is_neg() { pi::() - base } else { base } + }; + angle.with_sign(y.is_neg()) +} + +impl Cx { + pub const fn new(re: V, im: V) -> Self { + Self { re, im } + } + + pub fn real(re: V) -> Self { + Self::new(re, V::zero()) + } + + pub fn zero() -> Self { + Self::real(V::zero()) + } + + pub const fn parts(self) -> (V, V) { + (self.re, self.im) + } + + #[must_use] + pub fn scale(self, k: V) -> Self { + Self::new(self.re * k, self.im * k) + } + + #[must_use] + pub fn ldexp(self, k: i32) -> Self { + Self::new(self.re.ldexp(k), self.im.ldexp(k)) + } + + #[must_use] + pub fn div_int(self, k: u64) -> Self { + Self::new(self.re.div_int(k), self.im.div_int(k)) + } + + #[must_use] + pub fn times_i(self) -> Self { + Self::new(-self.im, self.re) + } + + #[must_use] + pub fn conj(self) -> Self { + Self::new(self.re, -self.im) + } + + pub fn norm2(self) -> V { + self.re.square() + self.im.square() + } + + pub fn abs(self) -> V { + self.norm2().sqrt() + } + + pub fn is_zero(&self) -> bool { + self.re.is_zero() && self.im.is_zero() + } + + #[must_use] + pub fn square(self) -> Self { + Self::new( + self.re.square() - self.im.square(), + (self.re * self.im).ldexp(1), + ) + } + + #[must_use] + pub fn recip(self) -> Self { + let n = self.norm2(); + Self::new(self.re / n, -self.im / n) + } + + /// `e^self`, saturating or vanishing past [`HUGE`]. + #[must_use] + pub fn exp(self) -> Self { + let modulus = exp_wide(self.re); + let (s, c) = sin_cos(self.im); + Self::new(modulus * c, modulus * s) + } + + /// `e^self − 1`, its real part `expm1(x)·cos y − 2sin²(y/2)` to hold near 0. + #[must_use] + pub fn exp_m1(self) -> Self { + let (s, c) = sin_cos(self.im); + if self.re > V::from_int(HUGE) { + return Self::new(beyond::() * c, beyond::() * s); + } + let (half, _) = sin_cos(self.im.ldexp(-1)); + let em1 = if self.re < V::from_int(-HUGE) { + -V::one() + } else { + expm1(self.re) + }; + Self::new(em1 * c - half.square().ldexp(1), (em1 + V::one()) * s) + } + + /// `e^(−self²)` from exact `y² − x²` and `2xy`, its phase right at any size. + #[must_use] + pub fn exp_neg_square(self) -> Self { + let modulus = exp_wide(V::dot2(self.im, self.im, -self.re, self.re)); + let (oct, frac) = V::octant_of_product(self.re.ldexp(1), self.im); + let (s, c) = sin_cos_octant(oct, frac); + let s = if self.re.is_neg() == self.im.is_neg() { + -s + } else { + s + }; + Self::new(modulus * c, modulus * s) + } + + /// The principal logarithm, `ln|z|` kept exact in form near `|z| = 1`. + #[must_use] + pub fn ln(self) -> Self { + let one = V::one(); + let near = (self.re - one) * (self.re + one) + self.im.square(); + let modulus = if near.abs() <= one.ldexp(-2) { + log1p(near).ldexp(-1) + } else { + ln(self.norm2()).ldexp(-1) + }; + Self::new(modulus, atan2(self.im, self.re)) + } +} + +impl Add for Cx { + type Output = Self; + fn add(self, o: Self) -> Self { + Self::new(self.re + o.re, self.im + o.im) + } +} + +impl Sub for Cx { + type Output = Self; + fn sub(self, o: Self) -> Self { + Self::new(self.re - o.re, self.im - o.im) + } +} + +impl Neg for Cx { + type Output = Self; + fn neg(self) -> Self { + Self::new(-self.re, -self.im) + } +} + +impl Mul for Cx { + type Output = Self; + fn mul(self, o: Self) -> Self { + Self::new( + self.re * o.re - self.im * o.im, + self.re * o.im + self.im * o.re, + ) + } +} + +impl Div for Cx { + type Output = Self; + fn div(self, o: Self) -> Self { + let n = o.norm2(); + let t = self * o.conj(); + Self::new(t.re / n, t.im / n) + } +} + +fn settled(t: Cx, sum: Cx) -> bool { + V::negligible(&t.norm2().sqrt(), &sum.norm2().sqrt()) +} + +fn two_over_root_pi() -> V { + V::constant(&FRAC_2_SQRT_PI) +} + +/// Terms past which every series and fraction here stops. +const CAP: u32 = 600; + +/// The radius, 4 or 5.5, within which a series cancelling like `e^|z|²` keeps +/// 9 bits past a 32-bit ulp with one limb, or 20 past a 128-bit one. +fn kummer_reach() -> V { + V::from_int(i64::from(crate::eval::by_width::([8, 11, 11]))).ldexp(-1) +} + +/// The same for `e^(2|z|)`, 6 or 8. Twice it bounds `r + Re z`, the power of +/// `e` by which the terms for `E₁` and `γ(a, z)` outgrow their sum; past it +/// Lentz's fraction converges like `e^(−2√(2k(r + Re z)))`. +fn linear_reach() -> V { + V::from_int(i64::from(crate::eval::by_width::([6, 8, 8]))) +} + +/// `D(z) = z·Σ (−2z²)ⁿ/(2n + 1)!!`, where `w(z) = e^(−z²) + (2i/√π)·D(z)`. +pub fn dawson_series(z: Cx) -> Cx { + let step = -(z.square().ldexp(1)); + let mut term = z; + let mut sum = z; + let past = V::from_int(2) * z.norm2(); + for n in 1..CAP { + term = (term * step).div_int(u64::from(2 * n + 1)); + sum = sum + term; + if V::from_int(i64::from(n)) > past && settled(term, sum) { + break; + } + } + sum +} + +/// `w(z) ~ (i/√π)/z·Σ (2n − 1)!!/(2z²)ⁿ`, for `|z|` where the smallest term, +/// about `e^(−|z|²)`, is past the working precision. +fn asymptotic(z: Cx) -> Cx { + let inv = (z.square().ldexp(1)).recip(); + let mut term = Cx::real(V::one()); + let mut sum = term; + let mut last = term.norm2(); + for n in 1..CAP { + let next = (term * inv).scale(V::from_int(i64::from(2 * n - 1))); + let size = next.norm2(); + if size > last || settled(next, sum) { + break; + } + sum = sum + next; + term = next; + last = size; + } + (sum / z).scale(two_over_root_pi::().ldexp(-1)).times_i() +} + +/// `w(z)` for `Im z ≥ 1`, `(i/√π)/(z − ½/(z − 1/(z − (3/2)/(z − …))))`. +fn continued(z: Cx) -> Cx { + let g = lentz(z, |k| (Cx::real(V::from_int(-i64::from(k)).ldexp(-1)), z)); + g.recip().scale(two_over_root_pi::().ldexp(-1)).times_i() +} + +fn upper(z: Cx) -> Cx { + let r = z.abs(); + if r <= kummer_reach() { + return z.exp_neg_square() + dawson_series(z).scale(two_over_root_pi()).times_i(); + } + let far = V::from_int(i64::from(crate::eval::by_width::([8, 11, 13]))); + if r >= far { + return asymptotic(z); + } + let one = V::one(); + if z.im >= one { + return continued(z); + } + // Near the real axis, Taylor from z₀ = x + i by w' = −2zw + 2i/√π. + let z0 = Cx::new(z.re, one); + let w0 = if z0.abs() >= far { + asymptotic(z0) + } else { + continued(z0) + }; + let delta = Cx::new(V::zero(), z.im - one); + let lead = (-(z0 * w0).ldexp(1) + Cx::real(two_over_root_pi()).times_i()) * delta; + let (mut prev, mut cur) = (w0, lead); + let mut sum = w0 + lead; + let a = -(z0 * delta).ldexp(1); + let b = -(delta.square()).ldexp(1); + for k in 1..CAP { + let next = (a * cur + b * prev).div_int(u64::from(k + 1)); + sum = sum + next; + if settled(next, sum) && settled(cur, sum) { + break; + } + (prev, cur) = (cur, next); + } + sum +} + +/// `w(z) = e^(−z²)·erfc(−iz)`, real on the imaginary axis. +pub fn faddeeva(z: Cx) -> Cx { + let w = if z.im.is_neg() { + // w(z) = 2e^(−z²) − w(−z). + z.exp_neg_square().ldexp(1) - upper(-z) + } else { + upper(z) + }; + if z.re.is_zero() { Cx::real(w.re) } else { w } +} + +/// `Re erf(x + iy)` for `|2xy| ≤ ½`, which `|erf z| ≈ e^(y²)/y` swamps below: +/// `(2/√π)·e^(y²)·∫₀ˣ e^(−s²)·cos 2ys ds` in `x`, by `gₖ₊₁ = (−2iy·gₖ − 2gₖ₋₁)/(k + 1)` +/// for `e^(−2iys − s²)`, `gₖ/i` held for odd `k`. +fn erf_re_near_axis(x: V, y: V) -> V { + let two_y = y.ldexp(1); + let (mut prev, mut cur) = (V::one(), -two_y); + let x2 = x.square(); + let mut power = x; + let mut sum = x; + for k in 1..CAP { + let turned = if k % 2 == 1 { + two_y * cur + } else { + -(two_y * cur) + }; + (prev, cur) = (cur, (turned - prev.ldexp(1)).div_int(u64::from(k + 1))); + if k % 2 == 1 { + power = power * x2; + let term = cur.div_int(u64::from(k + 2)) * power; + sum = sum + term; + if V::negligible(&term, &sum) { + break; + } + } + } + exp_wide(y.square()) * sum * two_over_root_pi() +} + +/// `erf z`, odd and real on the real axis: `(2/√π)·e^(−z²)·Σ (2z²)ⁿz/(2n + 1)!!` +/// near 0, and past it `1 − e^(−z²)·w(iz)`. +pub fn erf(z: Cx) -> Cx { + let flip = z.re.is_neg(); + let u = if flip { -z } else { z }; + let r = u.abs(); + let value = if r <= kummer_reach() { + // −i·D(iu) = u·Σ (2u²)ⁿ/(2n + 1)!!. + let s = -(dawson_series(u.times_i()).times_i()); + (u.exp_neg_square() * s).scale(two_over_root_pi()) + } else { + Cx::real(V::one()) - u.exp_neg_square() * faddeeva(u.times_i()) + }; + let value = if u.im.abs() > V::one() && (u.re * u.im).abs().ldexp(2) <= V::one() { + Cx::new(erf_re_near_axis(u.re, u.im), value.im) + } else { + value + }; + let value = if flip { -value } else { value }; + match (z.re.is_zero(), z.im.is_zero()) { + (true, _) => Cx::new(V::zero(), value.im), + (_, true) => Cx::real(value.re), + _ => value, + } +} + +/// `b₀ + a₁/(b₁ + a₂/(b₂ + …))` by Lentz's method, `(aₖ, bₖ)` from `term(k)`. +fn lentz(b0: Cx, term: impl Fn(u32) -> (Cx, Cx)) -> Cx { + let tiny = Cx::real(V::one().ldexp(-1000)); + let mut value = if b0.is_zero() { tiny } else { b0 }; + let mut upper = value; + let mut lower = Cx::zero(); + let limit = V::one().ldexp(20 - 2 * V::PREC.cast_signed()); + for k in 1..8 * CAP { + let (num, den) = term(k); + lower = den + num * lower; + if lower.is_zero() { + lower = tiny; + } + upper = den + num / upper; + if upper.is_zero() { + upper = tiny; + } + lower = lower.recip(); + let delta = upper * lower; + value = value * delta; + if (delta - Cx::real(V::one())).norm2() < limit { + break; + } + } + value +} + +/// `Γ(a, z) ~ z^(a−1)·e^(−z)·Σ (a − 1)…(a − k)/z^k`, for `|z|` past 256. +fn gamma_asymptotic(a: V, z: Cx) -> Cx { + let inv = z.recip(); + let mut term = Cx::real(V::one()); + let mut sum = term; + let mut last = V::one(); + for k in 1..CAP { + let next = (term * inv).scale(a - V::from_int(i64::from(k))); + let size = next.norm2(); + if size > last || settled(next, sum) { + break; + } + sum = sum + next; + term = next; + last = size; + } + power(z, a - V::one()) * (-z).exp() * sum +} + +/// `E₁(z)`, `z ≠ 0`: `−γ − ln z − Σ (−z)ⁿ/(n·n!)` near 0 or the negative axis, +/// asymptotic far along it, else `e^(−z)/(z + 1 − 1/(z + 3 − 4/(z + 5 − …)))`. +pub fn e1(z: Cx) -> Cx { + let r = z.abs(); + let series = r + z.re <= linear_reach::().ldexp(1); + if series && r > V::from_int(256) { + let value = gamma_asymptotic(V::zero(), z); + // On the axis, the value from above it: −Ei(−z) − iπ. + return if z.im.is_zero() { + Cx::new(value.re, -pi::()) + } else { + value + }; + } + if series { + let mut term = Cx::real(V::one()); + let mut sum = Cx::zero(); + for n in 1..8 * CAP { + term = (-(term * z)).div_int(u64::from(n)); + let part = term.div_int(u64::from(n)); + sum = sum + part; + if V::from_int(i64::from(n)) > r && settled(part, sum) { + break; + } + } + return -(Cx::real(V::constant(&EULER_GAMMA)) + z.ln() + sum); + } + let one = Cx::real(V::one()); + let f = lentz(z + one, |k| { + let i = V::from_int(i64::from(k)); + (Cx::real(-i.square()), z + one + Cx::real(i.ldexp(1))) + }); + (-z).exp() / f +} + +/// `ln Γ(x)` for real `x > 0`: shifted to 40 or past, then Stirling's series. +fn ln_gamma(x: V) -> V { + let mut shift = V::one(); + let mut y = x; + for _ in 0..40 { + if y >= V::from_int(40) { + break; + } + shift = shift * y; + y = y + V::one(); + } + let inv = V::one() / y; + let inv2 = inv.square(); + let mut power = inv; + let mut sum = V::zero(); + for c in STIRLING + .iter() + .take(crate::eval::by_width::([12, 24, 40]) as usize) + { + sum = sum + V::constant(c) * power; + power = power * inv2; + } + (y - V::one().ldexp(-1)) * ln(y) - y + V::constant(&LN_SQRT_2PI) + sum - ln(shift) +} + +pub fn gamma(x: V) -> V { + let lg = ln_gamma(x); + if lg > V::from_int(HUGE) { + beyond() + } else { + exp(lg) + } +} + +/// `(Γ(1 + a) − 1)/a`, below 2^−8 by the Taylor series of `ln Γ(1 + a)`. +fn gamma_1p_ratio(a: V) -> V { + if a >= V::one().ldexp(-8) { + return (gamma(a + V::one()) - V::one()) / a; + } + let mut sum = V::zero(); + for c in LN_GAMMA_1P.iter().rev() { + sum = V::constant(c) + a * sum; + } + expm1(a * sum) / a +} + +/// `z^a` for real `a`; left of the imaginary axis `e^(±iπa)`, reduced exactly, +/// turned by `∓a·atan|y/x|`, so parts near the negative axis keep precision. +pub fn power(z: Cx, a: V) -> Cx { + let log = z.ln(); + if !z.re.is_neg() { + return log.scale(a).exp(); + } + let whole = a.round_int(); + let rest = a - V::from_int(whole); + let (sin, cos) = if rest.abs() == V::one().ldexp(-1) { + (rest.ldexp(1), V::zero()) + } else { + sin_cos(rest * pi::()) + }; + let (st, ct) = sin_cos(a * atan_ratio(z.im.abs(), z.re.abs())); + let turn = Cx::new(cos, sin) * Cx::new(ct, -st); + let unit = if whole % 2 == 0 { turn } else { -turn }; + let unit = if z.im.is_neg() { unit.conj() } else { unit }; + unit.scale(exp_wide(a * log.re)) +} + +/// `Γ(a, z)` for `a` in `(0, 128]`: `Γ(a) − z^a·Σ (−z)ⁿ/(n!(a + n))` near the +/// negative axis or for `a ≤ ½`, asymptotic far along it, `Γ(a) − z^a·e^(−z)·Σ zⁿ/(a)ₙ₊₁` +/// within `a + 1` and more of 0, else Legendre's continued fraction. +pub fn gamma_upper(a: V, z: Cx) -> Cx { + if z.is_zero() { + return Cx::real(gamma(a)); + } + let r = z.abs(); + let one = V::one(); + let small = a <= one.ldexp(-1); + let left = z.re.is_neg() && r > a + one; + if (small || left) && r + z.re <= linear_reach::().ldexp(1) { + if r > V::from_int(256) { + let value = gamma_asymptotic(a, z); + // On the axis Γ(a, z) = Γ(a) − e^(iπa)·G, G real: the series is the latter. + let whole = V::from_int(a.round_int()) == a; + return if z.im.is_zero() && !whole { + Cx::new(value.re + gamma(a), value.im) + } else { + value + }; + } + let mut term = Cx::real(one); + let mut sum = Cx::zero(); + for n in 1..8 * CAP { + term = (-(term * z)).div_int(u64::from(n)); + let part = term.scale(one / (a + V::from_int(i64::from(n)))); + sum = sum + part; + if V::from_int(i64::from(n)) > r && settled(part, sum) { + break; + } + } + if small { + // Γ(a) − z^a/a as (Γ(1 + a) − 1)/a − (z^a − 1)/a. + let head = gamma_1p_ratio(a); + let ratio = z.ln().scale(a).exp_m1().scale(one / a); + return Cx::real(head) - ratio - power(z, a) * sum; + } + return Cx::real(gamma(a)) - power(z, a) * (sum + Cx::real(one / a)); + } + let reach = (a + V::one()).max(linear_reach()); + let lead = power(z, a) * (-z).exp(); + if r <= reach { + let mut term = Cx::real(V::one() / a); + let mut sum = term; + let mut denom = a; + for _ in 1..8 * CAP { + denom = denom + V::one(); + term = (term * z).scale(V::one() / denom); + sum = sum + term; + if denom > r && settled(term, sum) { + break; + } + } + return Cx::real(gamma(a)) - lead * sum; + } + let f = lentz(z + Cx::real(V::one() - a), |k| { + let i = V::from_int(i64::from(k)); + ( + Cx::real(-(i * (i - a))), + z + Cx::real(i.ldexp(1) + V::one() - a), + ) + }); + lead / f +} + +/// `φₖ(z) = (e^z − Σ_{n(k: u32, z: Cx) -> Cx { + if z.abs() <= V::one().ldexp(-1) { + let n = by_width::([18, 30, 39]); + let mut p = Cx::zero(); + for j in (0..=n).rev() { + let coef = INV_FACT + .get((k + j) as usize) + .map_or_else(V::zero, V::constant); + p = Cx::real(coef) + z * p; + } + return p; + } + let mut num = z.exp_m1(); + let mut den = z; + if k >= 2 { + num = num - z; + den = den * z; + } + num / den +} + +const TERMS: usize = 320; + +/// `Σ cₙ·Jₙ(w)`, `Jₙ(w) = ∫₀¹ sⁿ·e^(ws) ds`, `Re w ≤ 0`: forward to `|w|` by `Jₙ = (e^w − nJₙ₋₁)/w`, +/// past it backward from `J_M = e^w·Σ (−w)^k·M!/(M + k + 1)!`, `M ≥ 2|w|`, each way damping. +fn moment_sum(w: Cx, coefs: &[V]) -> Cx { + let n = coefs.len().saturating_sub(1); + let reach = usize::try_from(w.abs().floor_int().max(0)).unwrap_or(usize::MAX); + let ew = w.exp(); + let coef = |k: usize| coefs.get(k).copied().unwrap_or_else(V::zero); + let mut cur = phi(1, w); + let mut sum = cur.scale(coef(0)); + let forward = reach.min(n); + if forward > 0 { + let inv = w.recip(); + for k in 1..=forward { + cur = (ew - cur.scale(V::from_int(i64::try_from(k).unwrap_or(0)))) * inv; + sum = sum + cur.scale(coef(k)); + } + } + if forward == n { + return sum; + } + let top = n.max(reach.saturating_mul(2).saturating_add(1)); + let top_v = V::from_int(i64::try_from(top).unwrap_or(0)); + let mut term = Cx::real(V::one() / (top_v + V::one())); + let mut series = term; + for k in 2..TERMS { + let kv = V::from_int(i64::try_from(k).unwrap_or(0)); + term = -(term * w).scale(V::one() / (top_v + kv)); + series = series + term; + if settled(term, series) { + break; + } + } + let mut down = ew * series; + for k in (forward + 1..=top).rev() { + if k <= n { + sum = sum + down.scale(coef(k)); + } + down = (ew - w * down).div_int(u64::try_from(k).unwrap_or(1)); + } + sum +} + +/// `W(u) = (1 + δu)^(−a)` and `V(u) = (1 − W(u))/(aδ)`, `a = p + 1`, `δ = ε/(1 − ε)`. +struct Weight { + power: V, + delta: V, +} + +impl Weight { + fn log_at(&self, u: V) -> V { + -(self.power * log1p(self.delta * u)) + } + + fn complement_at(&self, u: V) -> V { + u * log1p_ratio(self.delta * u) * phi_real(1, self.log_at(u)) + } +} + +/// `∫₀¹ e^(z(1 − u))·W(u) du`, or of `V` with `second`, from each end toward the least of the +/// log-convex `|e^(−zu)·W(u)|`: each step's `e^(ct)` whole by [`moment_sum`], `W`'s curve as series. +fn weighted_passes(z: Cx, wt: &Weight, second: bool) -> Cx { + let one = V::one(); + let (power, delta) = (wt.power, wt.delta); + let meet = if z.re.is_neg() { + (-(power / z.re) - one / delta).max(V::zero()).min(one) + } else { + one + }; + let tiny = V::one().ldexp(-(V::PREC.cast_signed() + 8)); + let most = wt.complement_at(one); + let curve = one.ldexp(-1) / power.sqrt(); + let (mut ahead, mut behind) = (Cx::zero(), Cx::zero()); + for forward in [true, false] { + let pass = if forward { &mut ahead } else { &mut behind }; + let sign = if forward { one } else { -one }; + let mut u = if forward { V::zero() } else { one }; + for _ in 0..1024 { + let rest = if forward { meet - u } else { u - meet }; + if rest <= V::zero() { + break; + } + let log_w = wt.log_at(u); + // Forward the integrand is e^(−zu)·W, e^z applied at the end. + let from = if forward { -u } else { one - u }; + let lead = Cx::new(z.re * from + log_w, z.im * from).exp(); + let base = Cx::new(z.re * from, z.im * from).exp(); + // Its size falls toward the meeting point, so this bounds what is left. + let left = if second { + base.abs() * most + } else { + lead.abs() + } * rest; + if left < tiny.ldexp(-8) * pass.abs() { + break; + } + let ratio = delta / (one + delta * u); + let h = rest.min(curve / ratio).min(one.ldexp(-1) / ratio); + let (rh, mut coefs) = (ratio * h, [V::zero(); TERMS]); + let used = if second { + // 1 − (1 ± rt)^(∓a) over aδ: rt/(δuⱼ + 1)·…, ratio ∓(a + k)rt/(k + 1). + let mut c = h * h / (one + delta * u); + let mut used = 1; + for (k, slot) in coefs.iter_mut().enumerate().skip(1) { + *slot = c; + used = k; + if c.abs() < tiny * h { + break; + } + let kv = V::from_int(i64::try_from(k).unwrap_or(0)); + c = -(sign * c * (power + kv) * rh).div_int(u64::try_from(k + 1).unwrap_or(1)); + } + used + } else { + // e^(aρ(±rt)): b₀ = 1, b₁ = 0, bₖ₊₁ = (a·r²·bₖ₋₁ ∓ r·k·bₖ)/(k + 1), times h^(k+1). + let curvature = power * rh.square(); + let (mut prev, mut cur) = (h, V::zero()); + let mut used = 1; + if let Some(slot) = coefs.first_mut() { + *slot = h; + } + for (k, slot) in coefs.iter_mut().enumerate().skip(1) { + *slot = cur; + used = k; + if cur.abs() < tiny * h && prev.abs() < tiny * h { + break; + } + let kv = V::from_int(i64::try_from(k).unwrap_or(0)); + let next = (curvature * prev - sign * rh * kv * cur) + .div_int(u64::try_from(k + 1).unwrap_or(1)); + (prev, cur) = (cur, next); + } + used + }; + let series = coefs.get(..=used).unwrap_or(&coefs); + if second { + let rate = (-z).scale(sign).scale(h); + let flat = phi(1, rate).scale(h * wt.complement_at(u)); + let bend = moment_sum(rate, series).scale(sign * exp_wide(log_w)); + *pass = *pass + base * (flat + bend); + } else { + let rate = (-(z + Cx::real(power * ratio))).scale(sign * h); + *pass = *pass + lead * moment_sum(rate, series); + } + u = if forward { u + h } else { u - h }; + } + } + z.exp() * ahead + behind +} + +fn parts_sum(z: Cx, a: V, q: V, first: Cx) -> Cx { + let step = z.recip().scale(q); + let mut term = first; + let mut sum = term; + for k in 0..TERMS { + term = -(term * step).scale(a + V::from_int(i64::try_from(k).unwrap_or(0))); + sum = sum + term; + if settled(term, sum) { + break; + } + } + sum +} + +/// `(K₁, K₂)` of `Complex::exprel_weighted`, `ε > 0`: `e^z/(1 − ε)·∫₀¹ e^(−zu)·{W, V}(u) du`, by +/// parts past `|z| = 4(a + PREC)δ` and 1, else stepped, `K₂ = (φ₁ − (1 − ε)K₁)/(εa)` for `εa ≥ ¼`. +pub fn exprel_weighted(z: Cx, eps: V, p: V) -> (Cx, Cx) { + let one = V::one(); + let a = p + one; + let keep = one - eps; + let delta = eps / keep; + let log_keep = log1p(-eps); + let ez = z.exp(); + let far = V::from_int(4) * (a + V::from_int(i64::from(V::PREC))) * delta; + if z.abs() >= far.max(one) { + let inv2 = z.square().recip(); + let k1 = (ez * parts_sum(z, a, delta, z.recip())).scale(one / keep) + - parts_sum(z, a, eps, z.recip()).scale(exp_wide(p * log_keep)); + // V(1)/(1 − ε) = (1 − (1 − ε)^a)/(aε) = (Λ/ε)·φ₁(−aΛ), Λ = −log1p(−ε). + let tail = log1p_ratio(-eps) * phi_real(1, a * log_keep); + let k2 = -(z.recip().scale(tail)) + + (ez * parts_sum(z, a + one, delta, inv2)).scale(one / keep) + - parts_sum(z, a + one, eps, inv2).scale(exp_wide(a * log_keep)); + return (k1, k2); + } + let wt = Weight { power: a, delta }; + let k1 = weighted_passes(z, &wt, false).scale(one / keep); + let k2 = if eps * a >= one.ldexp(-2) { + (phi(1, z) - k1.scale(keep)).scale(one / (eps * a)) + } else { + weighted_passes(z, &wt, true).scale(one / keep) + }; + (k1, k2) +} diff --git a/src/eval.rs b/src/eval.rs new file mode 100644 index 0000000..61abc14 --- /dev/null +++ b/src/eval.rs @@ -0,0 +1,416 @@ +//! Function cores in the working arithmetic, shared by the public functions, +//! which convert in exactly and round once on the way out. + +use crate::consts::{ + ATAN_SIXTEENTHS, COS_PI_128THS, EXP2_32NDS, FRAC_1_LN_2, INV_FACT, LN_2, LN_SIXTEENTHS, PI, + SIN_PI_128THS, +}; +use crate::real::Work; +use crate::series::{FACT, ODD}; +use crate::traits::CordicNumber; +use crate::traits::sealed::Sealed; + +/// The working type of `T`. +pub type W = ::Work; + +/// The narrow working type of `T`, as wide as `T`. +pub type Narrow = ::Narrow; + +/// Bits a narrow result must keep past `T`'s ulp to be trusted. +const GUARD: i32 = 24; + +pub fn narrow(x: T) -> Narrow { + Narrow::::from_raw(x.raw_i128(), T::frac_bits()) +} + +/// Whether `T` has a working type narrower than `W` that leaves guard bits, +/// and `extra` more for what the function itself may cancel. +pub fn has_narrow(extra: i32) -> bool { + Narrow::::PREC < W::::PREC + && T::frac_bits().cast_signed() + GUARD + extra < Narrow::::PREC.cast_signed() +} + +/// Results in either working type, rounded to `T` all at once. +pub trait Outputs { + type Rounded; + fn fits(&self) -> bool; + fn round(self) -> Self::Rounded; +} + +impl Outputs for V { + type Rounded = T; + fn fits(&self) -> bool { + let room = V::PREC.cast_signed() - T::frac_bits().cast_signed() - GUARD; + self.is_zero() || self.exponent() <= room + } + fn round(self) -> T { + let lo = T::min_value().raw_i128(); + let hi = T::max_value().raw_i128(); + T::from_raw_i128(self.to_raw(T::frac_bits(), lo, hi, false)) + } +} + +impl, B: Outputs> Outputs for (A, B) { + type Rounded = (A::Rounded, B::Rounded); + fn fits(&self) -> bool { + self.0.fits() && self.1.fits() + } + fn round(self) -> Self::Rounded { + (self.0.round(), self.1.round()) + } +} + +impl, const K: usize> Outputs for [A; K] { + type Rounded = [A::Rounded; K]; + fn fits(&self) -> bool { + self.iter().all(Outputs::fits) + } + fn round(self) -> Self::Rounded { + self.map(Outputs::round) + } +} + +/// `$body`, arguments converted by `$v`, in `Narrow` where every result fits, else in `W`. +macro_rules! tiered { + ($t:ty, |$v:ident| $body:expr) => { + $crate::eval::tiered!($t, 0, |$v| $body) + }; + ($t:ty, $extra:expr, |$v:ident| $body:expr) => {{ + let mut done = None; + if $crate::eval::has_narrow::<$t>($extra) { + let $v = $crate::eval::narrow::<$t>; + let out = $body; + if $crate::eval::Outputs::<$t>::fits(&out) { + done = Some($crate::eval::Outputs::<$t>::round(out)); + } + } + match done { + Some(r) => r, + None => { + let $v = $crate::eval::work::<$t>; + $crate::eval::Outputs::<$t>::round($body) + } + } + }}; +} +pub(crate) use tiered; + +/// Arguments past this saturate every function of them that grows. +pub const HUGE: i64 = 4096; + +pub fn work(x: T) -> W { + W::::from_raw(x.raw_i128(), T::frac_bits()) +} + +/// Rounds to the nearest value of `T`, saturating. +pub fn fixed(v: W) -> T { + let lo = T::min_value().raw_i128(); + let hi = T::max_value().raw_i128(); + T::from_raw_i128(v.to_raw(T::frac_bits(), lo, hi, false)) +} + +pub fn pi() -> V { + V::constant(&PI) +} + +pub fn half_pi() -> V { + pi::().ldexp(-1) +} + +/// The entry for this width, from entries for 64-, 128- and 192-bit mantissas. +pub const fn by_width(n: [u32; 3]) -> u32 { + match V::PREC { + 0..=64 => n[0], + 65..=128 => n[1], + _ => n[2], + } +} + +fn entry(table: &[crate::consts::C], i: i64) -> V { + usize::try_from(i) + .ok() + .and_then(|i| table.get(i)) + .map_or_else(V::zero, |c| V::constant(c)) +} + +/// `expm1(r)` for `|r| ≤ ln2/64`, as `r·Σ r^n/(n+1)!`. +fn expm1_series(r: V) -> V { + r * V::horner(r, &FACT, 1, 1, by_width::([8, 14, 20]), false) +} + +/// `e^x` for `|x| < HUGE`, as `2^(k/32)·e^r`, `|r| ≤ ln2/64`, the power from the table. +pub fn exp(x: V) -> V { + let steps = (x * V::constant(&FRAC_1_LN_2).ldexp(5)).round_int(); + let rest = x - V::constant(&LN_2).ldexp(-5).mul_int(steps); + let index = steps.rem_euclid(32); + let shift = i32::try_from(steps.div_euclid(32)).unwrap_or(0); + let power = entry::(&EXP2_32NDS, index); + (power + power * expm1_series(rest)).ldexp(shift) +} + +/// `e^x − 1` for `|x| < HUGE`, to full relative precision near 0. +pub fn expm1(x: V) -> V { + if x.abs() <= V::constant(&LN_2).ldexp(-6) { + expm1_series(x) + } else { + exp(x) - V::one() + } +} + +/// `atanh(s) = s + s³/3 + s⁵/5 + …` for `|s| ≤ 0.023`. +fn atanh_series(s: V) -> V { + s * V::horner(s.square(), &ODD, 0, 1, by_width::([8, 13, 19]), false) +} + +/// `ln x` on `[√½, √2)`: `ln c`, `c = 1 + j/16` nearest, plus `2·atanh((x − c)/(x + c))`. +fn ln_near_one(y: V) -> V { + let one = V::one(); + let j = (y - one).ldexp(4).round_int().clamp(-5, 7); + let c = one + V::from_int(j).ldexp(-4); + entry::(&LN_SIXTEENTHS, j + 5) + atanh_series((y - c) / (y + c)).ldexp(1) +} + +/// `x = y·2^k` with `y` in `[√½, √2)`. +fn split_log(x: V) -> (V, i32) { + let e = x.exponent(); + let y = x.ldexp(-e); + if y.square() < V::one().ldexp(-1) { + (y.ldexp(1), e - 1) + } else { + (y, e) + } +} + +/// Natural logarithm of `x > 0`. +pub fn ln(x: V) -> V { + let (y, k) = split_log(x); + ln_near_one(y) + V::constant(&LN_2).mul_int(i64::from(k)) +} + +/// Base-2 logarithm of `x > 0`, its integer part exact. +pub fn log2(x: V) -> V { + let (y, k) = split_log(x); + ln_near_one(y) * V::constant(&FRAC_1_LN_2) + V::from_int(i64::from(k)) +} + +/// `ln(1 + u)` for `u > −1`, to full relative precision near 0. +pub fn log1p(u: V) -> V { + let one = V::one(); + if u.abs() <= one.ldexp(-5) { + atanh_series(u / (u + V::from_int(2))).ldexp(1) + } else { + ln(one + u) + } +} + +/// `atan(num/den)` for `0 ≤ num ≤ den`: `atan c`, `c = j/16` nearest, plus that of +/// `(16num − j·den)/(16den + j·num)`, one division in all. +fn atan_unit(num: V, den: V) -> V { + let j = V::sixteenths(num, den).clamp(0, 16); + let t = (num.ldexp(4) - den.mul_int(j)) / (den.ldexp(4) + num.mul_int(j)); + let series = V::horner(t.square(), &ODD, 0, 1, by_width::([8, 14, 21]), true); + entry::(&ATAN_SIXTEENTHS, j) + t * series +} + +/// `atan(num/den)` for `0 ≤ num` and `0 < den`, never forming a ratio above 1. +pub fn atan_ratio(num: V, den: V) -> V { + if num <= den { + atan_unit(num, den) + } else { + half_pi::() - atan_unit(den, num) + } +} + +/// Sine and cosine of `(π/4)·(oct + w)`, `w` from [`Work::octant`]: the table's +/// `jπ/128` nearest, then series for the rest, `|d| ≤ π/256`. +pub fn sin_cos_octant(oct: u32, frac: V) -> (V, V) { + let index = frac.ldexp(5).round_int().clamp(0, 32); + let rest = (frac - V::from_int(index).ldexp(-5)) * pi::().ldexp(-2); + let rest2 = rest.square(); + let terms = by_width::([4, 7, 11]); + let (sd, cd) = V::horner2(rest2, &FACT, [1, 0], 2, terms, true); + let sd = rest * sd; + let (sj, cj) = ( + entry::(&SIN_PI_128THS, index), + entry::(&COS_PI_128THS, index), + ); + let s = sj * cd + cj * sd; + let c = cj * cd - sj * sd; + match oct & 7 { + 0 => (s, c), + 1 => (c, s), + 2 => (c, -s), + 3 => (s, -c), + 4 => (-s, -c), + 5 => (-c, -s), + 6 => (-c, s), + _ => (-s, c), + } +} + +/// A value past `e^HUGE`, which every type saturates from even divided by its values. +pub fn beyond() -> V { + V::one().ldexp(8192) +} + +/// `e^v`, past [`HUGE`] a value every type saturates or rounds to zero from. +pub fn exp_wide(v: V) -> V { + if v.abs() > V::from_int(HUGE) { + return if v.is_neg() { V::zero() } else { beyond() }; + } + exp(v) +} + +/// `φₖ(z) = (e^z − Σ_{n(k: u32, z: V) -> V { + if z.abs() <= V::one().ldexp(-1) { + return V::horner(z, &FACT, k, 1, by_width::([18, 30, 39]), false); + } + if z > V::from_int(HUGE) { + return beyond(); + } + let mut num = if z < V::from_int(-HUGE) { + -V::one() + } else { + expm1(z) + }; + if k >= 2 { + num = num - z; + } + if k >= 3 { + num = num - z.square().ldexp(-1); + } + let mut den = z; + for _ in 1..k { + den = den * z; + } + num / den +} + +/// `φ₁` to `φ₅` given `e^(−z)`: near 0 `φ₅`'s series and `φₖ = 1/k! + z·φₖ₊₁`, past it +/// `φ₁` from `e^z − 1` and `φₖ₊₁ = (φₖ − 1/k!)/z`, losing at most 11 bits, at `|z| = ½`. +pub fn phis(z: V, decay: V) -> [V; 5] { + let inv_fact = |k: usize| INV_FACT.get(k).map_or_else(V::zero, V::constant); + let mut out = [V::zero(); 5]; + if z.abs() <= V::one().ldexp(-1) { + let mut next = V::horner(z, &FACT, 5, 1, by_width::([18, 30, 39]), false); + for (k, slot) in out.iter_mut().enumerate().rev() { + *slot = next; + next = inv_fact(k) + z * next; + } + return out; + } + if z > V::from_int(HUGE) { + return [beyond(); 5]; + } + // e^z − 1 = 1/e^(−z) − 1, which past |z| = ½ cancels by at most two bits. + let mut cur = if z < V::from_int(-HUGE) { + -V::one() + } else { + V::one() / decay - V::one() + } / z; + for (k, slot) in out.iter_mut().enumerate() { + *slot = cur; + cur = (cur - inv_fact(k + 1)) / z; + } + out +} + +/// `ln(1 + v) − v`, `v > −1`; near 0 `2s³·Σ s^{2k}/(2k + 3) − v²/(2 + v)`, `s = v/(2 + v)`, +/// two terms of order `v²` that do not cancel. +pub fn log1p_minus(v: V) -> V { + if v.abs() <= V::one().ldexp(-2) { + let den = v + V::from_int(2); + let s = v / den; + let sum = V::horner(s.square(), &ODD, 1, 1, by_width::([13, 24, 36]), false); + return (s * s.square() * sum).ldexp(1) - v.square() / den; + } + log1p(v) - v +} + +/// `(ln(1 + u) − u)/u²`, which is −½ at 0. +pub fn log1p_minus_ratio(u: V) -> V { + if u.abs() <= V::one().ldexp(-2) { + let den = u + V::from_int(2); + let s = u / den; + let sum = V::horner(s.square(), &ODD, 1, 1, by_width::([13, 24, 36]), false); + return (u * sum).ldexp(1) / (den * den.square()) - V::one() / den; + } + log1p_minus(u) / u.square() +} + +/// `ln(1 + z)/z`, which is 1 at 0. +pub fn log1p_ratio(z: V) -> V { + if z.abs() <= V::one().ldexp(-2) { + let den = z + V::from_int(2); + let s = z / den; + let sum = V::horner(s.square(), &ODD, 0, 1, by_width::([13, 24, 36]), false); + return sum.ldexp(1) / den; + } + log1p(z) / z +} + +/// The Stumpff functions `cₖ(z) = Σ (−z)ⁿ/(2n + k)!`, k from 0 to 3: the series near +/// 0, past it closed forms in `√|z|`, `c₂` from the half angle so it does not cancel. +pub fn stumpff(z: V) -> [V; 4] { + if z.abs() <= V::one().ldexp(-2) { + let n = by_width::([10, 17, 21]); + return [0, 1, 2, 3].map(|k| V::horner(-z, &FACT, k, 2, n, false)); + } + let one = V::one(); + let w = z.abs().sqrt(); + if z.is_neg() { + if w > V::from_int(HUGE) { + return [beyond(); 4]; + } + let (half, half_c) = sinh_cosh(w.ldexp(-1)); + let c1 = (half * half_c).ldexp(1) / w; + [ + one + half.square().ldexp(1), + c1, + (half.square() / w.square()).ldexp(1), + (c1 - one) / -z, + ] + } else { + // sin w = 2·s·c and cos w = (c − s)(c + s), exact in form at its zeros. + let (half, half_c) = sin_cos(w.ldexp(-1)); + let c1 = (half * half_c).ldexp(1) / w; + let c0 = (half_c - half) * (half_c + half); + [c0, c1, (half.square() / z).ldexp(1), (one - c1) / z] + } +} + +/// Sine and cosine of a working value, reduced exactly past π/4. +pub fn sin_cos(w: V) -> (V, V) { + let a = w.abs(); + let quarter = pi::().ldexp(-2); + let (oct, frac) = if a <= quarter { + (0, a / quarter) + } else { + a.octant_of() + }; + let (s, c) = sin_cos_octant(oct, frac); + (if w.is_neg() { -s } else { s }, c) +} + +/// `frac(x/2π)` in `[0, 1)`, from the same exact reduction as `sin` and `cos`. +pub fn turn_fraction(x: T) -> W { + let raw = x.raw_i128(); + let (oct, w) = W::::octant(raw.unsigned_abs(), T::frac_bits(), true); + let within = if oct & 1 == 1 { W::::one() - w } else { w }; + let u = (W::::from_int(i64::from(oct)) + within).ldexp(-3); + if raw < 0 && !u.is_zero() { + W::::one() - u + } else { + u + } +} + +/// `(sinh x, cosh x)` for `0 ≤ x < HUGE`, from `u = expm1(x)` so neither +/// cancels: `sinh = (u + u/(1 + u))/2` and `cosh = 1 + u²/(2(1 + u))`. +pub fn sinh_cosh(x: V) -> (V, V) { + let u = expm1(x); + let one = V::one(); + let e = one + u; + ((u + u / e).ldexp(-1), one + (u.square() / e).ldexp(-1)) +} diff --git a/src/fast.rs b/src/fast.rs new file mode 100644 index 0000000..00750de --- /dev/null +++ b/src/fast.rs @@ -0,0 +1,226 @@ +//! Kernels for 128-bit types in Q126, good to 2^−120: 24 guard bits at 96 fractional. + +use crate::consts::{ + ATAN_SIXTEENTHS, C, COS_PI_128THS, EXP2_32NDS, FRAC_1_LN_2, LN_2, LN_SIXTEENTHS, PI, + SIN_PI_128THS, +}; +use crate::real::{Real, mul_q126, recip_seed}; +use crate::series::{FACT, ODD, q126}; +use crate::traits::CordicNumber; + +const fn table(src: &[C; 33]) -> [i128; 33] { + let mut out = [0i128; 33]; + let mut rest: &mut [i128] = &mut out; + let mut from: &[C] = src; + while let ([slot, tail @ ..], [c, more @ ..]) = (rest, from) { + *slot = q126(c); + rest = tail; + from = more; + } + out +} + +const fn table_of(src: &[C; K]) -> [i128; K] { + let mut out = [0i128; K]; + let mut rest: &mut [i128] = &mut out; + let mut from: &[C] = src; + while let ([slot, tail @ ..], [c, more @ ..]) = (rest, from) { + *slot = q126(c); + rest = tail; + from = more; + } + out +} + +const fn top(c: &C) -> u128 { + let (_, _, m) = *c; + (m[5] as u128) << 64 | m[4] as u128 +} + +const SIN: [i128; 33] = table(&SIN_PI_128THS); +const EXP2: [i128; 32] = table_of(&EXP2_32NDS); +const ATAN16: [i128; 17] = table_of(&ATAN_SIXTEENTHS); +const LN16: [i128; 13] = table_of(&LN_SIXTEENTHS); +/// `ln2/32·2^132`, rounded; `ln 2·2^128` is its top 128 bits. +const LN2_32NDS: i128 = ((top(&LN_2) >> 1) + (top(&LN_2) & 1)).cast_signed(); +/// `32/ln 2·2^40`, from `2^127/ln 2`. +const PER_LN2_32NDS: i128 = (top(&FRAC_1_LN_2) >> 82).cast_signed(); +const COS: [i128; 33] = table(&COS_PI_128THS); +/// `π/4·2^126`, from π's top 128 bits, `π·2^126`. +const QUARTER_PI: i128 = { + let (_, _, m) = PI; + (((m[5] as u128) << 64 | m[4] as u128) >> 2).cast_signed() +}; + +/// Whether `T` takes these kernels: 128 bits, at most 96 of them fractional. +pub fn applies() -> bool { + T::total_bits() == 128 && T::frac_bits() <= 96 +} + +/// `|v|·2^(frac − 126)` rounded to nearest even, with `v`'s sign. +pub fn round(v: i128) -> T { + let shift = 126 - T::frac_bits(); + let m = v.unsigned_abs(); + let (q, rem, half) = (m >> shift, m & ((1 << shift) - 1), 1u128 << (shift - 1)); + let q = q + u128::from(rem > half || (rem == half && q & 1 == 1)); + let raw = q.cast_signed(); + T::from_raw_i128(if v < 0 { -raw } else { raw }) +} + +/// `sin` and `cos` of `mag·2^−point` radians, or turns, in Q126. +pub fn sin_cos(mag: u128, point: u32, radians: bool) -> (i128, i128) { + let (oct, w) = Real::<2>::octant_q127(mag, point, radians); + let index = ((w + (1 << 121)) >> 122).min(32); + // (w − index/32)·π/4, from Q127 to Q126; |d| ≤ π/256. + let rest = w.wrapping_sub(index << 122).cast_signed(); + let d = mul_q126(rest, QUARTER_PI) >> 1; + let d2 = mul_q126(d, d); + let coef = |i: u32| FACT.q126.get(i as usize).copied().unwrap_or(0); + let (mut ps, mut pc) = (0i128, 0i128); + for k in (0..=7u32).rev() { + let sign = if k % 2 == 1 { -1 } else { 1 }; + ps = sign * coef(2 * k + 1) + mul_q126(d2, ps); + pc = sign * coef(2 * k) + mul_q126(d2, pc); + } + let sd = mul_q126(d, ps); + let at = |t: &[i128; 33]| { + t.get(usize::try_from(index).unwrap_or(0)) + .copied() + .unwrap_or(0) + }; + let (sj, cj) = (at(&SIN), at(&COS)); + let s = mul_q126(sj, pc) + mul_q126(cj, sd); + let c = mul_q126(cj, pc) - mul_q126(sj, sd); + match oct & 7 { + 0 => (s, c), + 1 => (c, s), + 2 => (c, -s), + 3 => (s, -c), + 4 => (-s, -c), + 5 => (-c, -s), + 6 => (-c, s), + _ => (-s, c), + } +} + +/// `e^x = v·2^(q − 126)`, `v` in `[2^126, 2^127)`, for `x = raw·2^−frac` with +/// `|x| ≤ 64` and `8 ≤ frac ≤ 96`: `x = k·ln2/32 + r` exactly, as `r` fits +/// 128 bits whatever the products wrap to, then `2^(j/32)·(1 + expm1 r)`. +pub fn exp(raw: i128, frac: u32) -> Option<(i128, i32)> { + if raw.unsigned_abs() > 64u128 << frac || !(8..=96).contains(&frac) { + return None; + } + let estimate = (raw << (120 - frac)) >> 80; + let k = (estimate * PER_LN2_32NDS + (1 << 79)) >> 80; + let r = raw + .wrapping_shl(132 - frac) + .wrapping_sub(k.wrapping_mul(LN2_32NDS)) + >> 6; + let coef = |i: usize| FACT.q126.get(i).copied().unwrap_or(0); + let mut p = 0i128; + for n in (0..=14).rev() { + p = coef(n + 1) + mul_q126(r, p); + } + let base = EXP2 + .get(usize::try_from(k.rem_euclid(32)).unwrap_or(0)) + .copied() + .unwrap_or(0); + let value = base + mul_q126(base, mul_q126(r, p)); + Some((value, i32::try_from(k.div_euclid(32)).unwrap_or(0))) +} + +/// `v·2^−s` rounded to nearest even, `v ≥ 0`. +pub fn shift_round(v: i128, s: u32) -> i128 { + if s >= 128 { + return 0; + } + let m = v.unsigned_abs(); + let (q, rem, half) = (m >> s, m & ((1 << s) - 1), 1u128 << (s - 1)); + (q + u128::from(rem > half || (rem == half && q & 1 == 1))).cast_signed() +} + +/// `2^252/d` in Q126 for `d` in `[2^125, 2^127)`: a seed, then two steps of `g + g(1 − δg)`. +fn recip(d: i128) -> i128 { + let z = d.unsigned_abs().leading_zeros(); + let seed = recip_seed(u64::try_from((d.unsigned_abs() << z) >> 64).unwrap_or(u64::MAX)); + let mut g = (seed << (z + 60)).cast_signed(); + for _ in 0..2 { + g += mul_q126(g, (1i128 << 126) - mul_q126(d, g)); + } + g +} + +fn series(u: i128, table: &[i128], n: usize, alternate: bool) -> i128 { + let mut p = 0i128; + for k in (0..=n).rev() { + let c = table.get(k).copied().unwrap_or(0); + p = if alternate && k % 2 == 1 { -c } else { c } + mul_q126(u, p); + } + p +} + +/// `atan2(y, x)` in Q125, neither 0: `atan(j/16)` plus that of `(16a − j·b)/(16b + j·a)`, +/// `a ≤ b` the sizes cut to 122 bits. +pub fn atan2(y: i128, x: i128) -> i128 { + let (ay, ax) = (y.unsigned_abs(), x.unsigned_abs()); + let (small, large) = if ay <= ax { (ay, ax) } else { (ax, ay) }; + let lz = large.leading_zeros(); + let (small, large) = if lz >= 6 { + (small << (lz - 6), large << (lz - 6)) + } else { + (small >> (6 - lz), large >> (6 - lz)) + }; + let j = ((((small >> 57) << 5) / (large >> 57).max(1)) + 1) >> 1; + let j = j.min(16); + let (small, large, ji) = (small.cast_signed(), large.cast_signed(), j.cast_signed()); + let ratio = mul_q126((small << 4) - ji * large, recip((large << 4) + ji * small)); + let rest = mul_q126(ratio, series(mul_q126(ratio, ratio), ODD.q126, 13, true)); + let base = (ATAN16 + .get(usize::try_from(j).unwrap_or(0)) + .copied() + .unwrap_or(0) + + rest) + >> 1; + let pi = (top(&PI) >> 1).cast_signed(); + let angle = if ay > ax { (pi >> 1) - base } else { base }; + let angle = if x < 0 { pi - angle } else { angle }; + if y < 0 { -angle } else { angle } +} + +/// `ln x` in Q119, `x = Y·2^k`, `Y` in `[√½, √2)`: `ln c + 2·atanh((Y − c)/(Y + c)) + k·ln 2`. +pub fn ln(raw: i128, frac: u32) -> i128 { + let lz = raw.unsigned_abs().leading_zeros(); + let mut mant = ((raw.unsigned_abs() << lz) >> 1).cast_signed(); + let mut k = 127 - i64::from(lz) - i64::from(frac); + if mant > 0x5A82_7999_FCEF_3242_2CBE_C4E8_DB3B_9C48 { + mant >>= 1; + k += 1; + } + let one = 1i128 << 126; + let j = ((mant - one + (1 << 121)) >> 122).clamp(-5, 7); + let centre = one + (j << 122); + // Halved, so mant + centre stays below 2 in Q126. + let ratio = mul_q126((mant - centre) >> 1, recip((mant >> 1) + (centre >> 1))); + let tail = mul_q126(ratio, series(mul_q126(ratio, ratio), ODD.q126, 11, false)) << 1; + let head = LN16 + .get(usize::try_from(j + 5).unwrap_or(0)) + .copied() + .unwrap_or(0); + let ln2 = (top(&LN_2) >> 2).cast_signed(); + let (hi, lo) = (ln2 >> 64, ln2 & ((1 << 64) - 1)); + let k = i128::from(k); + ((head + tail) >> 7) + ((k * hi) << 57) + ((k * lo) >> 7) +} + +#[cfg(test)] +mod tests { + use super::{COS, EXP2, SIN, table, table_of}; + use crate::consts::{COS_PI_128THS, EXP2_32NDS, SIN_PI_128THS}; + + #[test] + fn tables_build_the_same_at_run_time() { + assert_eq!(table(&SIN_PI_128THS), SIN); + assert_eq!(table(&COS_PI_128THS), COS); + assert_eq!(table_of(&EXP2_32NDS), EXP2); + } +} diff --git a/src/lib.rs b/src/lib.rs index 86e9cf4..53e0eb9 100644 --- a/src/lib.rs +++ b/src/lib.rs @@ -3,18 +3,56 @@ #![cfg_attr(docsrs, feature(doc_cfg))] pub mod bounded; +mod consts; +mod cx; pub mod error; +mod eval; +mod fast; pub mod kernel; pub mod ops; +mod real; +mod series; +mod stirling; pub mod tables; pub mod traits; pub use fixed; +/// Every function's worst-case error in ulps, strict, wherever the exact +/// value is in range; out of range a result saturates. +pub const ERROR_ULPS: u32 = 1; + pub use error::{Error, Result}; pub use traits::CordicNumber; -pub use ops::algebraic::sqrt; -pub use ops::circular::{acos, asin, atan, atan2, cos, sin, sin_cos, tan}; -pub use ops::exponential::{exp, ln, log2, log10, pow, pow2}; +pub use ops::algebraic::{ + hypot, norm, normalize, rsqrt, scale_by, scale_exponent, sqrt, sqrt_product, +}; +pub use ops::arith::{ + Rounding, div_or_saturate, div_rounded, dot, dot_rounded, mul_add, mul_add_rounded, mul_div, + mul_div_rounded, mul_rounded, sqrt_or_zero, +}; +pub use ops::circular::{acos, asin, asin_from_gap, atan, atan2, cos, sin, sin_cos, tan}; +pub use ops::complex::Complex; +pub use ops::elliptic::{ + Modulus, carlson_rc, carlson_rd, carlson_rf, carlson_rf_roots, carlson_rj, carlson_rj_roots, + ellip_e, ellip_k, ellip_pi, ellip_pi_complete, ellip_pi_increment, jacobi_am, jacobi_inverse, + jacobi_sn_cn_dn, +}; +pub use ops::exponential::{ + exp, exp_moments, expm1, exprel, exprel2, ln, log1p, log1p_ratio, log1p_residual, + log1p_residual_ratio, log2, log10, pow, pow2, +}; pub use ops::hyperbolic::{acosh, acoth, asinh, atanh, cosh, coth, sinh, sinh_cosh, tanh}; +pub use ops::interval::{ + atan_range, atan_ratio_range, atan2_range, atanh_ratio_range, enclose, exp_moments_range, + exp_range, exprel_range, exprel2_range, hypot_range, log1p_range, log1p_ratio_range, + log1p_residual_ratio_range, sin_cos_range, sin_cos_range_turns, sqrt_range, stumpff_range, + tan_ratio_range, tanh_range, tanh_ratio_range, +}; +pub use ops::ratio::{atan_ratio, atanh_ratio, stumpff, tan_ratio, tanh_ratio}; +pub use ops::solve::{kepler_universal, root_in}; +pub use ops::special::{erf, erfc, fresnel, gamma}; +pub use ops::turns::{ + atan2_turns, div_by_tau, mul_by_tau, sin_cos_turns, wrap_principal, wrap_radians, +}; diff --git a/src/ops/algebraic.rs b/src/ops/algebraic.rs index b13e4aa..cdcb1af 100644 --- a/src/ops/algebraic.rs +++ b/src/ops/algebraic.rs @@ -1,7 +1,9 @@ -//! Algebraic functions (sqrt). +//! Roots, norms and power-of-two scaling, formed so no square or product underflows. use crate::bounded::NonNegative; use crate::error::{Error, Result}; +use crate::eval::{W, fixed, tiered, work}; +use crate::real::Work; use crate::traits::CordicNumber; /// Square root, rounded to the nearest representable value. Domain: `x ≥ 0`. @@ -30,3 +32,107 @@ pub fn sqrt(x: T) -> Result { pub fn sqrt_nonneg(x: NonNegative) -> T { x.get().sqrt_round() } + +/// `√(x² + y²)`, rounded once, with no overflow or underflow in between. +#[must_use] +#[cfg_attr( + all(feature = "verify-no-panic", not(debug_assertions)), + no_panic::no_panic +)] +pub fn hypot(x: T, y: T) -> T { + tiered!(T, |v| (v(x).square() + v(y).square()).sqrt()) +} + +/// The Euclidean norm `√(Σ vᵢ²)` of any number of components, rounded once. +#[must_use] +#[cfg_attr( + all(feature = "verify-no-panic", not(debug_assertions)), + no_panic::no_panic +)] +pub fn norm(v: &[T]) -> T { + tiered!(T, |w| norm_work(v, w)) +} + +fn inverse(x: V) -> V { + V::one() / x +} + +fn norm_work(v: &[T], as_work: fn(T) -> V) -> V { + v.iter() + .fold(V::zero(), |acc, &x| acc + as_work(x).square()) + .sqrt() +} + +/// `v / |v|`, each component rounded once, the direction exact even a few ulps long. +/// # Errors +/// Returns `DomainError` for the zero vector. +#[must_use = "returns the unit vector which should be handled"] +#[cfg_attr( + all(feature = "verify-no-panic", not(debug_assertions)), + no_panic::no_panic +)] +pub fn normalize(v: [T; N]) -> Result<[T; N]> { + if v.iter().all(|&x| x == T::zero()) { + return Err(Error::domain("normalize", "non-zero vector")); + } + Ok(tiered!(T, |w| { + let inv = inverse(norm_work(&v, w)); + v.map(|x| w(x) * inv) + })) +} + +/// `⌊log₂|x|⌋`, the power of two of the leading bit, or `None` for zero. +#[must_use] +#[cfg_attr( + all(feature = "verify-no-panic", not(debug_assertions)), + no_panic::no_panic +)] +pub fn scale_exponent(x: T) -> Option { + let mag = x.raw_i128().unsigned_abs(); + if mag == 0 { + return None; + } + let lead = 127 - mag.leading_zeros().cast_signed(); + Some(lead - T::frac_bits().cast_signed()) +} + +/// `x · 2^e`, exact unless it rounds below an ulp or saturates. +#[must_use] +#[cfg_attr( + all(feature = "verify-no-panic", not(debug_assertions)), + no_panic::no_panic +)] +pub fn scale_by(x: T, e: i32) -> T { + fixed(work(x).ldexp(e.clamp(-512, 512))) +} + +/// `√(a·b)`, rounded once, with the product formed without underflow. +/// # Errors +/// Returns `DomainError` if `a` and `b` have opposite signs. +#[must_use = "returns the geometric mean which should be handled"] +#[cfg_attr( + all(feature = "verify-no-panic", not(debug_assertions)), + no_panic::no_panic +)] +pub fn sqrt_product(a: T, b: T) -> Result { + let (wa, wb) = (work(a), work(b)); + if wa.is_neg() != wb.is_neg() && !wa.is_zero() && !wb.is_zero() { + return Err(Error::domain("sqrt_product", "arguments of one sign")); + } + Ok(fixed((wa * wb).sqrt())) +} + +/// Reciprocal square root `1/√x`, rounded once. Domain: `x > 0`. +/// # Errors +/// Returns `DomainError` if `x ≤ 0`. +#[must_use = "returns the reciprocal square root which should be handled"] +#[cfg_attr( + all(feature = "verify-no-panic", not(debug_assertions)), + no_panic::no_panic +)] +pub fn rsqrt(x: T) -> Result { + if x <= T::zero() { + return Err(Error::domain("rsqrt", "positive value")); + } + Ok(fixed(W::::one() / work(x).sqrt())) +} diff --git a/src/ops/arith.rs b/src/ops/arith.rs new file mode 100644 index 0000000..b612835 --- /dev/null +++ b/src/ops/arith.rs @@ -0,0 +1,296 @@ +//! Products, quotients and sums of products formed exactly, then rounded once and saturated. + +use crate::traits::CordicNumber; + +const LOW: u128 = u64::MAX as u128; + +/// The way a result rounds, for bounds that must each round their own way. +#[derive(Clone, Copy, Debug, PartialEq, Eq)] +pub enum Rounding { + /// To the nearest value, ties to even. + Nearest, + /// Toward −∞. + Down, + /// Toward +∞. + Up, +} + +impl Rounding { + /// Whether a magnitude steps from 0, given its sign, its parity and its dropped half and rest. + const fn away(self, neg: bool, odd: bool, (half, rest): (bool, bool)) -> bool { + match self { + Self::Nearest => half && (rest || odd), + Self::Down => neg && (half || rest), + Self::Up => !neg && (half || rest), + } + } +} + +/// `a · b` as `(high, low)` halves of 256 bits. +const fn wide(a: u128, b: u128) -> (u128, u128) { + let (ah, al) = (a >> 64, a & LOW); + let (bh, bl) = (b >> 64, b & LOW); + let (mid, carry) = (al * bh).overflowing_add(ah * bl); + let (lo, c2) = (al * bl).overflowing_add(mid << 64); + let hi = ah * bh + (mid >> 64) + ((carry as u128) << 64) + c2 as u128; + (hi, lo) +} + +/// `(hi·2^128 + lo) / d` and remainder, `hi < d`, by two 64-bit digits (divlu). +const fn div_wide(hi: u128, lo: u128, d: u128) -> (u128, u128) { + let s = d.leading_zeros(); + let d = d << s; + let (hi, lo) = if s == 0 { + (hi, lo) + } else { + ((hi << s) | (lo >> (128 - s)), lo << s) + }; + let (dh, dl) = (d >> 64, d & LOW); + let (lh, ll) = (lo >> 64, lo & LOW); + let (mut q1, mut r) = div_rem(hi, dh); + while q1 >> 64 != 0 || q1 * dl > ((r << 64) | lh) { + q1 -= 1; + r += dh; + if r >> 64 != 0 { + break; + } + } + let mid = ((hi << 64) | lh).wrapping_sub(q1.wrapping_mul(d)); + let (mut q0, mut r0) = div_rem(mid, dh); + while q0 >> 64 != 0 || q0 * dl > ((r0 << 64) | ll) { + q0 -= 1; + r0 += dh; + if r0 >> 64 != 0 { + break; + } + } + let rem = ((mid << 64) | ll).wrapping_sub(q0.wrapping_mul(d)); + ((q1 << 64) | q0, rem >> s) +} + +/// `a / b` and `a % b`, or zeros for a zero `b`, which callers rule out. +const fn div_rem(a: u128, b: u128) -> (u128, u128) { + match (a.checked_div(b), a.checked_rem(b)) { + (Some(q), Some(r)) => (q, r), + _ => (0, 0), + } +} + +/// The raw value of `T` nearest `±mag`, saturating. +fn saturate(neg: bool, mag: u128) -> T { + let raw = if neg { + 0i128.checked_sub_unsigned(mag).unwrap_or(i128::MIN) + } else { + i128::try_from(mag).unwrap_or(i128::MAX) + }; + T::from_raw_i128(raw) +} + +/// `|(hi, lo)| / 2^s` of sign `neg` rounded by `mode`, saturating past 128 bits, `s < 128`. +const fn shift_round(hi: u128, lo: u128, s: u32, neg: bool, mode: Rounding) -> u128 { + if s == 0 { + return if hi == 0 { lo } else { u128::MAX }; + } + if hi >> s != 0 { + return u128::MAX; + } + let q = (hi << (128 - s)) | (lo >> s); + let half = (lo >> (s - 1)) & 1 == 1; + let rest = s > 1 && lo << (129 - s) != 0; + if mode.away(neg, q & 1 == 1, (half, rest)) { + q.saturating_add(1) + } else { + q + } +} + +/// `a · b + c`, rounded once. +#[must_use] +#[cfg_attr( + all(feature = "verify-no-panic", not(debug_assertions)), + no_panic::no_panic +)] +pub fn mul_add(a: T, b: T, c: T) -> T { + mul_add_rounded(a, b, c, Rounding::Nearest) +} + +/// [`mul_add`] rounded by `mode`. +#[must_use] +#[cfg_attr( + all(feature = "verify-no-panic", not(debug_assertions)), + no_panic::no_panic +)] +pub fn mul_add_rounded(a: T, b: T, c: T, mode: Rounding) -> T { + let (ra, rb, rc) = (a.raw_i128(), b.raw_i128(), c.raw_i128()); + let (p_hi, p_lo) = wide(ra.unsigned_abs(), rb.unsigned_abs()); + let p_neg = (ra < 0) != (rb < 0); + let f = T::frac_bits(); + let cm = rc.unsigned_abs(); + let (c_hi, c_lo) = if f == 0 { + (0, cm) + } else { + (cm >> (128 - f), cm << f) + }; + let (neg, hi, lo) = signed_add(p_neg, p_hi, p_lo, rc < 0, c_hi, c_lo); + saturate(neg, shift_round(hi, lo, f, neg, mode)) +} + +/// `a · b` rounded by `mode`. +#[must_use] +#[cfg_attr( + all(feature = "verify-no-panic", not(debug_assertions)), + no_panic::no_panic +)] +pub fn mul_rounded(a: T, b: T, mode: Rounding) -> T { + mul_add_rounded(a, b, T::zero(), mode) +} + +/// `±(ah, al) ± (bh, bl)` as a sign and magnitude. +const fn signed_add( + an: bool, + ah: u128, + al: u128, + bn: bool, + bh: u128, + bl: u128, +) -> (bool, u128, u128) { + if an == bn { + let (lo, c) = al.overflowing_add(bl); + return (an, ah.saturating_add(bh).saturating_add(c as u128), lo); + } + if ah > bh || (ah == bh && al >= bl) { + let (lo, b) = al.overflowing_sub(bl); + (an, ah - bh - b as u128, lo) + } else { + let (lo, b) = bl.overflowing_sub(al); + (bn, bh - ah - b as u128, lo) + } +} + +/// `a · b / c` from the exact product, rounded once; a zero `c` saturates by its sign. +#[must_use] +#[cfg_attr( + all(feature = "verify-no-panic", not(debug_assertions)), + no_panic::no_panic +)] +pub fn mul_div(a: T, b: T, c: T) -> T { + mul_div_rounded(a, b, c, Rounding::Nearest) +} + +/// [`mul_div`] rounded by `mode`. +#[must_use] +#[cfg_attr( + all(feature = "verify-no-panic", not(debug_assertions)), + no_panic::no_panic +)] +pub fn mul_div_rounded(a: T, b: T, c: T, mode: Rounding) -> T { + let (ra, rb, rc) = (a.raw_i128(), b.raw_i128(), c.raw_i128()); + let neg = ((ra < 0) != (rb < 0)) != (rc < 0); + let (hi, lo) = wide(ra.unsigned_abs(), rb.unsigned_abs()); + let divisor = rc.unsigned_abs(); + if hi == 0 && lo == 0 { + return T::zero(); + } + if divisor == 0 || hi >= divisor { + return saturate(neg, u128::MAX); + } + let (quotient, rem) = div_wide(hi, lo, divisor); + let twice = rem << 1; + let dropped = (twice >= divisor, twice != divisor && rem != 0); + let away = mode.away(neg, quotient & 1 == 1, dropped); + saturate(neg, quotient.saturating_add(u128::from(away))) +} + +/// `a / b` rounded to nearest; a zero `b` saturates by sign, 0/0 is 0, and it never panics. +#[must_use] +#[cfg_attr( + all(feature = "verify-no-panic", not(debug_assertions)), + no_panic::no_panic +)] +pub fn div_or_saturate(a: T, b: T) -> T { + mul_div(a, T::one(), b) +} + +/// [`div_or_saturate`] rounded by `mode`. +#[must_use] +#[cfg_attr( + all(feature = "verify-no-panic", not(debug_assertions)), + no_panic::no_panic +)] +pub fn div_rounded(a: T, b: T, mode: Rounding) -> T { + mul_div_rounded(a, T::one(), b, mode) +} + +/// `x += (hi, lo)` for a 384-bit `x`, most significant word first. +const fn accumulate(x: &mut (u128, u128, u128), hi: u128, lo: u128) { + let (l, c0) = x.2.overflowing_add(lo); + let (m1, c1) = x.1.overflowing_add(hi); + let (m, c2) = m1.overflowing_add(c0 as u128); + x.2 = l; + x.1 = m; + x.0 = x.0.wrapping_add(c1 as u128 + c2 as u128); +} + +/// `Σ aᵢ·bᵢ` over the pairs, up to the shorter, summed exactly and rounded once. +#[must_use] +#[cfg_attr( + all(feature = "verify-no-panic", not(debug_assertions)), + no_panic::no_panic +)] +pub fn dot(a: &[T], b: &[T]) -> T { + dot_rounded(a, b, Rounding::Nearest) +} + +/// [`dot`] rounded by `mode`. +#[must_use] +#[cfg_attr( + all(feature = "verify-no-panic", not(debug_assertions)), + no_panic::no_panic +)] +pub fn dot_rounded(a: &[T], b: &[T], mode: Rounding) -> T { + let mut pos = (0, 0, 0); + let mut neg = (0, 0, 0); + for (&x, &y) in a.iter().zip(b) { + let (rx, ry) = (x.raw_i128(), y.raw_i128()); + let (hi, lo) = wide(rx.unsigned_abs(), ry.unsigned_abs()); + accumulate( + if (rx < 0) == (ry < 0) { + &mut pos + } else { + &mut neg + }, + hi, + lo, + ); + } + let negative = neg > pos; + let (big, small) = if negative { (neg, pos) } else { (pos, neg) }; + let (lo, b0) = big.2.overflowing_sub(small.2); + let (m1, b1) = big.1.overflowing_sub(small.1); + let (mid, b2) = m1.overflowing_sub(u128::from(b0)); + let top = big + .0 + .wrapping_sub(small.0) + .wrapping_sub(u128::from(b1 || b2)); + if top != 0 { + return saturate(negative, u128::MAX); + } + saturate( + negative, + shift_round(mid, lo, T::frac_bits(), negative, mode), + ) +} + +/// Square root, 0 for `x ≤ 0`, rounded to nearest like [`sqrt`](crate::sqrt). +#[must_use] +#[cfg_attr( + all(feature = "verify-no-panic", not(debug_assertions)), + no_panic::no_panic +)] +pub fn sqrt_or_zero(x: T) -> T { + if x > T::zero() { + x.sqrt_round() + } else { + T::zero() + } +} diff --git a/src/ops/circular.rs b/src/ops/circular.rs index 679ce79..abf13d1 100644 --- a/src/ops/circular.rs +++ b/src/ops/circular.rs @@ -1,14 +1,25 @@ -//! Trigonometric functions via circular CORDIC. +//! Trigonometric functions. -use crate::bounded::{NonNegative, UnitInterval}; +use crate::bounded::UnitInterval; use crate::error::{Error, Result}; -use crate::kernel::circular_vectoring; -use crate::ops::algebraic::sqrt_nonneg; -use crate::tables::chebyshev::{COS_Q_HI, COS_Q_LO, SIN_P_HI, SIN_P_LO, horner}; +use crate::eval::{W, atan_ratio, beyond, fixed, pi, sin_cos_octant, tiered, work}; +use crate::real::Work; use crate::traits::CordicNumber; -/// `1/(2π)` as I1F63, for the angle reduction multiply in [`sin_cos`]. -const FRAC_1_2PI_I1F63: i64 = 0x145F_306D_C9C8_82A5; +/// Sine and cosine of `angle`, reduced by 2π exactly, in `as_work`'s working type. +fn sin_cos_work(angle: T, _as_work: fn(T) -> V) -> (V, V) { + let raw = angle.raw_i128(); + let (oct, w) = V::octant(raw.unsigned_abs(), T::frac_bits(), true); + let (s, c) = sin_cos_octant(oct, w); + (if raw < 0 { -s } else { s }, c) +} + +/// `√(1 − a²)` for `a` in `[0, 1]`, from `(1 − a)(1 + a)` so it keeps its +/// bits near 1. +fn cofactor(a: V) -> V { + let one = V::one(); + ((one - a) * (one + a)).sqrt() +} /// Sine and cosine of any angle, cheaper than separate calls. #[must_use] @@ -17,85 +28,13 @@ const FRAC_1_2PI_I1F63: i64 = 0x145F_306D_C9C8_82A5; no_panic::no_panic )] pub fn sin_cos(angle: T) -> (T, T) { - let pi = T::pi(); - let frac_pi_2 = T::frac_pi_2(); - let two_pi = pi + pi; - - // Reduce to [-π, π] by n = round(angle / 2π) at any magnitude. The - // reciprocal multiply leaves n off by at most one, which the clamp below - // absorbs bit for bit (docs/sin_cos.md); other types divide. - let reduced = if angle > pi || angle < -pi { - let quotient = if T::total_bits() <= 2 * T::frac_bits() { - angle.saturating_mul(T::from_i1f63(FRAC_1_2PI_I1F63)) - } else { - angle.div(two_pi) - }; - let n = quotient.round(); - // n·2π is exact and the difference small, so wrapping recovers it even - // when n·2π exceeds MAX. - angle.wrapping_sub(n.wrapping_mul(two_pi)) - } else { - angle - }; - - let reduced = if reduced > pi { - reduced.saturating_sub(two_pi) - } else if reduced < -pi { - reduced.saturating_add(two_pi) - } else { - reduced - }; - - let (reduced, negate) = if reduced > frac_pi_2 { - (reduced - pi, true) - } else if reduced < -frac_pi_2 { - (reduced + pi, true) - } else { - (reduced, false) - }; - - // Past π/4 the polynomials take π/2 − |x| and swap roles, which avoids - // cancellation near π/2. - let one = T::one(); - let frac_pi_4 = T::frac_pi_4(); - let abs_reduced = reduced.abs(); - let (poly_arg, swapped) = if abs_reduced >= frac_pi_4 { - (frac_pi_2.saturating_sub(abs_reduced), true) - } else { - (abs_reduced, false) - }; - let u = poly_arg.saturating_mul(poly_arg); - - // Minimax polynomials, multiplying by constants rather than dividing at - // each step: degree 15 and 14 from 24 fractional bits, else 9 and 8. - let (sp_val, cp_val) = if T::frac_bits() >= 24 { - let sp = horner(&SIN_P_HI, u); - let sin_approx = poly_arg.saturating_add(poly_arg.saturating_mul(u).saturating_mul(sp)); - let cp = horner(&COS_Q_HI, u); - (sin_approx, one.saturating_add(u.saturating_mul(cp))) - } else { - let sp = horner(&SIN_P_LO, u); - let sin_approx = poly_arg.saturating_add(poly_arg.saturating_mul(u).saturating_mul(sp)); - let cp = horner(&COS_Q_LO, u); - (sin_approx, one.saturating_add(u.saturating_mul(cp))) - }; - - let (sin_unsigned, cos_val) = if swapped { - (cp_val, sp_val) - } else { - (sp_val, cp_val) - }; - let sin_val = if reduced < T::zero() { - -sin_unsigned - } else { - sin_unsigned - }; - - if negate { - (-sin_val, -cos_val) - } else { - (sin_val, cos_val) + if crate::fast::applies::() { + let raw = angle.raw_i128(); + let (s, c) = crate::fast::sin_cos(raw.unsigned_abs(), T::frac_bits(), true); + let s = if raw < 0 { -s } else { s }; + return (crate::fast::round(s), crate::fast::round(c)); } + tiered!(T, |v| sin_cos_work(angle, v)) } /// Sine of any angle. @@ -120,10 +59,10 @@ pub fn cos(angle: T) -> T { sin_cos(angle).1 } -/// Tangent, as `sin(angle) / cos(angle)`. +/// Tangent, as `sin(angle) / cos(angle)` divided before rounding. /// -/// It saturates to `T::MAX` or `T::MIN` within about 4e-5 of a pole for -/// I16F16 and 5e-10 for I32F32. Check the cosine to catch near-pole angles: +/// It saturates to `T::MAX` or `T::MIN` only where the tangent itself is out +/// of range. Check the cosine to catch near-pole angles: /// /// ``` /// use fixed::types::I16F16; @@ -144,8 +83,16 @@ pub fn cos(angle: T) -> T { no_panic::no_panic )] pub fn tan(angle: T) -> T { - let (s, c) = sin_cos(angle); - s.div(c) + tiered!(T, |v| tan_work(angle, v)) +} + +fn tan_work(angle: T, as_work: fn(T) -> V) -> V { + let (s, c) = sin_cos_work(angle, as_work); + if c.is_zero() { + beyond::().with_sign(s.is_neg()) + } else { + s / c + } } /// Arcsine. Domain: `[-1, 1]`. Returns an angle in `[-π/2, π/2]`. @@ -157,40 +104,33 @@ pub fn tan(angle: T) -> T { no_panic::no_panic )] pub fn asin(x: T) -> Result { - asin_unguarded(x) -} - -/// [`asin`] without the no-panic guard, which [`acos`] calls so the link -/// check sees no guarded callee inside it. -#[inline] -fn asin_unguarded(x: T) -> Result { - let Some(unit_x) = UnitInterval::new(x) else { + if UnitInterval::new(x).is_none() { return Err(Error::domain("asin", "value in range [-1, 1]")); - }; - - if x == T::one() { - return Ok(T::frac_pi_2()); } - if x == -T::one() { - return Ok(-T::frac_pi_2()); - } - if x == T::zero() { - return Ok(T::zero()); - } - - // asin(x) = atan(x / √(1 − x²)). - let sqrt_term = sqrt_nonneg(NonNegative::one_minus_square(unit_x)); + Ok(tiered!(T, |v| { + let a = v(x).abs(); + atan_ratio(a, cofactor(a)).with_sign(x.is_negative()) + })) +} - // A root below 2^-15 puts x all but at ±1, so the result is ±π/2. - if sqrt_term < T::from_i1f63(0x0001_0000_0000_0000) { - return if x.is_positive() { - Ok(T::frac_pi_2()) - } else { - Ok(-T::frac_pi_2()) - }; +/// `asin(1 − gap)` from the gap to 1, which keeps all of its bits where +/// `1 − gap` would round. Domain: `gap` in `[0, 2]`. +/// # Errors +/// Returns `DomainError` if `gap` is outside `[0, 2]`. +#[must_use = "returns the arcsine result which should be handled"] +#[cfg_attr( + all(feature = "verify-no-panic", not(debug_assertions)), + no_panic::no_panic +)] +pub fn asin_from_gap(gap: T) -> Result { + let g = work(gap); + let two = W::::from_int(2); + if g.is_neg() || g > two { + return Err(Error::domain("asin_from_gap", "gap in range [0, 2]")); } - - Ok(atan(x.div(sqrt_term))) + let a = W::::one() - g; + let s = (g * (two - g)).sqrt(); + Ok(fixed(atan_ratio(a.abs(), s).with_sign(a.is_neg()))) } /// Arccosine. Domain: `[-1, 1]`. Returns an angle in `[0, π]`. @@ -202,70 +142,46 @@ fn asin_unguarded(x: T) -> Result { no_panic::no_panic )] pub fn acos(x: T) -> Result { - asin_unguarded(x).map(|a| T::frac_pi_2().saturating_sub(a)) + if UnitInterval::new(x).is_none() { + return Err(Error::domain("acos", "value in range [-1, 1]")); + } + Ok(tiered!(T, |v| acos_work(v(x)))) +} + +fn acos_work(x: V) -> V { + let a = x.abs(); + let near = atan_ratio(cofactor(a), a); + if x.is_neg() { pi::() - near } else { near } } -/// Arctangent of any value. Returns an angle in `(-π/2, π/2)`. +/// Arctangent of any value. Returns an angle in `[-π/2, π/2]`. #[must_use] #[cfg_attr( all(feature = "verify-no-panic", not(debug_assertions)), no_panic::no_panic )] pub fn atan(x: T) -> T { - let zero = T::zero(); - let one = T::one(); - - if x == zero { - return zero; - } - - // Past 1, atan(x) = sign(x)·π/2 − atan(1/x) keeps the CORDIC convergent. - let abs_x = x.abs(); - if abs_x > one { - let recip = one.div(x); - let atan_recip = circular_vectoring(one, recip, zero).2; + tiered!(T, |v| atan_work(v(x))) +} - if x.is_positive() { - T::frac_pi_2() - atan_recip - } else { - -T::frac_pi_2() - atan_recip - } - } else { - circular_vectoring(one, x, zero).2 - } +fn atan_work(x: V) -> V { + let (a, one) = (x.abs(), V::one()); + atan_ratio(a, one).with_sign(x.is_neg()) } -/// Four-quadrant arctangent. Returns an angle in `[-π, π]`, and 0 for (0, 0). +/// Four-quadrant arctangent. Returns an angle in `[-π, π]`: `π` along the +/// negative x-axis, and 0 for (0, 0). #[must_use] #[cfg_attr( all(feature = "verify-no-panic", not(debug_assertions)), no_panic::no_panic )] pub fn atan2(y: T, x: T) -> T { - let zero = T::zero(); - let pi = T::pi(); - let frac_pi_2 = T::frac_pi_2(); - - if x == zero { - return if y.is_negative() { - -frac_pi_2 - } else if y == zero { - zero - } else { - frac_pi_2 - }; - } - - if y == zero { - return if x.is_negative() { pi } else { zero }; - } - - let (_, _, base_angle) = circular_vectoring(x.abs(), y.abs(), zero); - - match (x.is_negative(), y.is_negative()) { - (false, false) => base_angle, - (false, true) => -base_angle, - (true, false) => pi - base_angle, - (true, true) => base_angle - pi, + let (ry, rx) = (y.raw_i128(), x.raw_i128()); + if crate::fast::applies::() && ry != 0 && rx != 0 { + let angle = crate::fast::atan2(ry, rx); + let raw = crate::fast::shift_round(angle, 125 - T::frac_bits()); + return T::from_raw_i128(if angle < 0 { -raw } else { raw }); } + tiered!(T, |v| crate::cx::atan2(v(y), v(x))) } diff --git a/src/ops/complex.rs b/src/ops/complex.rs new file mode 100644 index 0000000..e82d3df --- /dev/null +++ b/src/ops/complex.rs @@ -0,0 +1,357 @@ +//! Complex numbers over a fixed-point type, each part rounded once. + +use crate::cx::{Cx, atan2, phi}; +use crate::error::{Error, Result}; +use crate::eval::{W, atan_ratio, exp_wide, fixed, sin_cos, sin_cos_octant, tiered, work}; +use crate::real::Work; +use crate::traits::CordicNumber; +use core::ops::{Add, Div, Mul, Neg, Sub}; + +/// A complex number `re + i·im`. +#[derive(Clone, Copy, Debug, Default, PartialEq, Eq)] +pub struct Complex { + /// The real part. + pub re: T, + /// The imaginary part. + pub im: T, +} + +impl Complex { + /// `re + i·im`. + #[must_use] + pub const fn new(re: T, im: T) -> Self { + Self { re, im } + } + + fn work(self) -> Cx> { + Cx::new(work(self.re), work(self.im)) + } + + fn work_in(self, as_work: fn(T) -> V) -> Cx { + Cx::new(as_work(self.re), as_work(self.im)) + } + + fn round(z: Cx>) -> Self { + Self::new(fixed(z.re), fixed(z.im)) + } + + /// The conjugate `re − i·im`. + #[must_use] + #[cfg_attr( + all(feature = "verify-no-panic", not(debug_assertions)), + no_panic::no_panic + )] + pub fn conj(self) -> Self { + Self::new(self.re, T::zero().saturating_sub(self.im)) + } + + /// The modulus `|z|`, rounded once. + #[must_use] + #[cfg_attr( + all(feature = "verify-no-panic", not(debug_assertions)), + no_panic::no_panic + )] + pub fn norm(self) -> T { + fixed(self.work().abs()) + } + + /// The argument in `(−π, π]`, 0 at the origin. + #[must_use] + #[cfg_attr( + all(feature = "verify-no-panic", not(debug_assertions)), + no_panic::no_panic + )] + pub fn arg(self) -> T { + fixed(atan2(work(self.im), work(self.re))) + } + + /// `e^z`. + #[must_use] + #[cfg_attr( + all(feature = "verify-no-panic", not(debug_assertions)), + no_panic::no_panic + )] + pub fn exp(self) -> Self { + let (re, im) = tiered!(T, |v| self.work_in(v).exp().parts()); + Self::new(re, im) + } + + /// `(e^z − 1)/z`, which is 1 at 0. + #[must_use] + #[cfg_attr( + all(feature = "verify-no-panic", not(debug_assertions)), + no_panic::no_panic + )] + pub fn exprel(self) -> Self { + let (re, im) = tiered!(T, |v| phi(1, self.work_in(v)).parts()); + Self::new(re, im) + } + + /// `(e^z − 1 − z)/z²`, which is ½ at 0. + #[must_use] + #[cfg_attr( + all(feature = "verify-no-panic", not(debug_assertions)), + no_panic::no_panic + )] + pub fn exprel2(self) -> Self { + let (re, im) = tiered!(T, |v| phi(2, self.work_in(v)).parts()); + Self::new(re, im) + } + + /// The Faddeeva function `w(z) = e^(−z²)·erfc(−iz)`. + #[must_use] + #[cfg_attr( + all(feature = "verify-no-panic", not(debug_assertions)), + no_panic::no_panic + )] + pub fn faddeeva(self) -> Self { + let (re, im) = tiered!(T, 48, |v| crate::cx::faddeeva(self.work_in(v)).parts()); + Self::new(re, im) + } + + /// `E₁(z) = ∫_z^∞ e^(−t)/t dt` on the principal branch, from above on the negative axis. + /// # Errors + /// Returns `DomainError` at `z = 0`. + #[must_use = "returns the integral which should be handled"] + #[cfg_attr( + all(feature = "verify-no-panic", not(debug_assertions)), + no_panic::no_panic + )] + pub fn e1(self) -> Result { + let z = self.work(); + if z.is_zero() { + return Err(Error::domain("e1", "non-zero argument")); + } + let (re, im) = tiered!(T, 24, |v| crate::cx::e1(self.work_in(v)).parts()); + Ok(Self::new(re, im)) + } + + /// `Γ(a, z) = ∫_z^∞ t^(a−1)·e^(−t) dt` for real `a` in `(0, 128]`, principal branch. + /// # Errors + /// Returns `DomainError` for `a` outside `(0, 128]`. + #[must_use = "returns the integral which should be handled"] + #[cfg_attr( + all(feature = "verify-no-panic", not(debug_assertions)), + no_panic::no_panic + )] + pub fn gamma_upper(self, a: T) -> Result { + let wa = work(a); + if wa <= W::::zero() || wa > W::::from_int(128) { + return Err(Error::domain("gamma_upper", "a in (0, 128]")); + } + let (re, im) = tiered!(T, 24, |v| crate::cx::gamma_upper(v(a), self.work_in(v)) + .parts()); + Ok(Self::new(re, im)) + } + + /// The principal logarithm `ln|z| + i·arg z`, `arg z` in `(−π, π]`. + /// # Errors + /// Returns `DomainError` at `z = 0`. + #[must_use = "returns the logarithm which should be handled"] + #[cfg_attr( + all(feature = "verify-no-panic", not(debug_assertions)), + no_panic::no_panic + )] + pub fn ln(self) -> Result { + let z = self.work(); + if z.is_zero() { + return Err(Error::domain("ln", "non-zero argument")); + } + let (re, im) = tiered!(T, |v| self.work_in(v).ln().parts()); + Ok(Self::new(re, im)) + } + + /// `z^a = e^(a·ln z)` for real `a`, principal branch; the phase's multiple of `π/4` reduced + /// exactly, so parts that vanish on an axis or diagonal stay 0, and near one keep their bits. + /// # Errors + /// Returns `DomainError` at `z = 0` for `a < 0`. + #[must_use = "returns the power which should be handled"] + #[cfg_attr( + all(feature = "verify-no-panic", not(debug_assertions)), + no_panic::no_panic + )] + pub fn pow(self, power: T) -> Result { + let z = self.work(); + if z.is_zero() { + return match power.partial_cmp(&T::zero()) { + Some(core::cmp::Ordering::Greater) => Ok(Self::new(T::zero(), T::zero())), + Some(core::cmp::Ordering::Equal) => Ok(Self::new(T::one(), T::zero())), + _ => Err(Error::domain("pow", "non-negative exponent at zero")), + }; + } + let (x, y) = (self.re.raw_i128(), self.im.raw_i128()); + let (ax, ay) = (x.unsigned_abs(), y.unsigned_abs()); + let (wx, wy) = (z.re.abs(), z.im.abs()); + // atan2(|y|, |x|) = j·π/4 + ψ about the nearest axis or diagonal, ψ to + // full relative precision; past π/2, π − that. + let (line, psi) = if ax == ay { + (1u8, W::::zero()) + } else if wy.mul_int(5) <= wx.mul_int(2) { + (0, atan_ratio(wy, wx)) + } else if wx.mul_int(5) <= wy.mul_int(2) { + (2, -atan_ratio(wx, wy)) + } else { + let gap = wy - wx; + (1, atan_ratio(gap.abs(), wx + wy).with_sign(gap.is_neg())) + }; + let (eighths, eta) = if x < 0 { (4 - line, -psi) } else { (line, psi) }; + // a·k/8 turns, from |a|·k reduced modulo 2^(frac + 3). + let point = T::frac_bits() + 3; + let mask = u128::MAX >> (128 - point); + let turns = power.raw_i128().unsigned_abs().wrapping_mul(eighths.into()) & mask; + let (oct, frac) = W::::octant(turns, point, false); + let (s1, c1) = sin_cos_octant(oct, frac); + let s1 = if power.is_negative() { -s1 } else { s1 }; + let (s2, c2) = sin_cos(work(power) * eta); + let sine = s1 * c2 + c1 * s2; + let cosine = c1 * c2 - s1 * s2; + let sine = if y < 0 { -sine } else { sine }; + let modulus = exp_wide(work(power) * z.ln().re); + Ok(Self::round(Cx::new(cosine, sine).scale(modulus))) + } + + /// [`exprel`](Self::exprel) and [`exprel2`](Self::exprel2) weighted by `wₜ(s) = ((1 − εt)/(1 − εs))^p/(1 − εs)`: + /// `K₁ = ∫₀¹ e^(zs)·w₁(s) ds` and `K₂ = ∫₀¹ ∫₀ᵗ e^(zs)·wₜ(s) ds dt`, precise as `ε → 0` for any `p`. + /// # Errors + /// Returns `DomainError` unless `ε` is in `[0, 1)` and `p ≥ 0`. + #[must_use = "returns the integrals which should be handled"] + #[cfg_attr( + all(feature = "verify-no-panic", not(debug_assertions)), + no_panic::no_panic + )] + pub fn exprel_weighted(self, eps: T, p: T) -> Result<(Self, Self)> { + let zero = T::zero(); + if eps < zero || eps >= T::one() || p < zero { + return Err(Error::domain("exprel_weighted", "eps in [0, 1) and p >= 0")); + } + let ((r1, i1), (r2, i2)) = tiered!(T, 16, |v| { + let z = self.work_in(v); + let (k1, k2) = if eps == zero { + (phi(1, z), phi(2, z)) + } else { + crate::cx::exprel_weighted(z, v(eps), v(p)) + }; + (k1.parts(), k2.parts()) + }); + Ok((Self::new(r1, i1), Self::new(r2, i2))) + } + + /// The error function `erf z`. + #[must_use] + #[cfg_attr( + all(feature = "verify-no-panic", not(debug_assertions)), + no_panic::no_panic + )] + pub fn erf(self) -> Self { + let (re, im) = tiered!(T, 48, |v| crate::cx::erf(self.work_in(v)).parts()); + Self::new(re, im) + } +} + +impl Complex { + /// `self + o`, each part saturating; the `+` operator. + #[must_use] + #[cfg_attr( + all(feature = "verify-no-panic", not(debug_assertions)), + no_panic::no_panic + )] + pub fn saturating_add(self, o: Self) -> Self { + Self::new(self.re.saturating_add(o.re), self.im.saturating_add(o.im)) + } + + /// `self − o`, each part saturating; the `−` operator. + #[must_use] + #[cfg_attr( + all(feature = "verify-no-panic", not(debug_assertions)), + no_panic::no_panic + )] + pub fn saturating_sub(self, o: Self) -> Self { + Self::new(self.re.saturating_sub(o.re), self.im.saturating_sub(o.im)) + } + + /// `−self`, each part saturating; unary `−`. + #[must_use] + #[cfg_attr( + all(feature = "verify-no-panic", not(debug_assertions)), + no_panic::no_panic + )] + pub fn saturating_neg(self) -> Self { + Self::new(T::zero(), T::zero()).saturating_sub(self) + } + + /// `self·o`, each part formed exactly and rounded once; the `*` operator. + #[must_use] + #[cfg_attr( + all(feature = "verify-no-panic", not(debug_assertions)), + no_panic::no_panic + )] + pub fn saturating_mul(self, o: Self) -> Self { + let (xr, xi, yr, yi) = (work(self.re), work(self.im), work(o.re), work(o.im)); + Self::new( + fixed(W::::dot2(xr, yr, -xi, yi)), + fixed(W::::dot2(xr, yi, xi, yr)), + ) + } + + /// `self/o` from exact sums of products, so no `o ≠ 0` overflows, else saturating; `/`. + #[must_use] + #[cfg_attr( + all(feature = "verify-no-panic", not(debug_assertions)), + no_panic::no_panic + )] + pub fn saturating_div(self, o: Self) -> Self { + let (xr, xi, yr, yi) = (work(self.re), work(self.im), work(o.re), work(o.im)); + if yr.is_zero() && yi.is_zero() { + let edge = |x: T| { + if x == T::zero() { + x + } else if x.is_negative() { + T::min_value() + } else { + T::max_value() + } + }; + return Self::new(edge(self.re), edge(self.im)); + } + let norm = W::::dot2(yr, yr, yi, yi); + Self::new( + fixed(W::::dot2(xr, yr, xi, yi) / norm), + fixed(W::::dot2(xi, yr, -xr, yi) / norm), + ) + } +} + +impl Add for Complex { + type Output = Self; + fn add(self, o: Self) -> Self { + self.saturating_add(o) + } +} + +impl Sub for Complex { + type Output = Self; + fn sub(self, o: Self) -> Self { + self.saturating_sub(o) + } +} + +impl Neg for Complex { + type Output = Self; + fn neg(self) -> Self { + self.saturating_neg() + } +} + +impl Mul for Complex { + type Output = Self; + fn mul(self, o: Self) -> Self { + self.saturating_mul(o) + } +} + +impl Div for Complex { + type Output = Self; + fn div(self, o: Self) -> Self { + self.saturating_div(o) + } +} diff --git a/src/ops/elliptic.rs b/src/ops/elliptic.rs new file mode 100644 index 0000000..9f07ad8 --- /dev/null +++ b/src/ops/elliptic.rs @@ -0,0 +1,837 @@ +//! Elliptic integrals and Jacobi elliptic functions of an exactly held [`Modulus`]. + +use crate::error::{Error, Result}; +use crate::eval::{ + W, atan_ratio, by_width, fixed, half_pi, log1p, pi, sin_cos, sinh_cosh, tiered, work, +}; +use crate::real::Work; +use crate::series::ODD; +use crate::traits::CordicNumber; + +/// An elliptic modulus, from the parameter `m` or the complement `k'`. +#[derive(Clone, Copy, Debug, PartialEq, Eq)] +pub struct Modulus { + value: T, + complement: bool, +} + +impl Modulus { + /// From `m = k²`, in `[0, 1]`. + #[must_use] + #[cfg_attr( + all(feature = "verify-no-panic", not(debug_assertions)), + no_panic::no_panic + )] + pub fn from_parameter(m: T) -> Option { + (m >= T::zero() && m <= T::one()).then_some(Self { + value: m, + complement: false, + }) + } + + /// From the complement `k' = √(1 − m)`, in `[0, 1]`, exact where `m` is near 1. + #[must_use] + #[cfg_attr( + all(feature = "verify-no-panic", not(debug_assertions)), + no_panic::no_panic + )] + pub fn from_complement(kc: T) -> Option { + (kc >= T::zero() && kc <= T::one()).then_some(Self { + value: kc, + complement: true, + }) + } + + fn at_separatrix(self) -> bool { + if self.complement { + self.value == T::zero() + } else { + self.value == T::one() + } + } + + /// The parameter `m`, rounded. + #[must_use] + #[cfg_attr( + all(feature = "verify-no-panic", not(debug_assertions)), + no_panic::no_panic + )] + pub fn parameter(self) -> T { + fixed(parts(self).m) + } + + /// The complement `k'`, rounded. + #[must_use] + #[cfg_attr( + all(feature = "verify-no-panic", not(debug_assertions)), + no_panic::no_panic + )] + pub fn complement(self) -> T { + fixed(parts(self).kc) + } +} + +/// `m`, `k'` and `k'²` in the working arithmetic, each from the given value. +struct Parts { + m: V, + kc: V, + kc2: V, +} + +fn parts(modulus: Modulus) -> Parts> { + parts_in(modulus, work) +} + +fn parts_in(modulus: Modulus, as_work: fn(T) -> V) -> Parts { + let one = V::one(); + let v = as_work(modulus.value); + if modulus.complement { + Parts { + m: (one - v) * (one + v), + kc: v, + kc2: v.square(), + } + } else { + let kc2 = one - v; + Parts { + m: v, + kc: kc2.sqrt(), + kc2, + } + } +} + +/// `R_C(1, 1 + e)`, `atan √e/√e` or `atanh √−e/√−e`, from `e` and a maybe tiny `1 + e`. +fn rc_one(e: V, one_plus_e: V) -> V { + // Σ (−e)^k/(2k + 1), as many terms as e's size needs below 2^−4. + let size = -e.exponent(); + if size >= 4 { + let terms = V::PREC / size.cast_unsigned() + 1; + return V::horner(e, &ODD, 0, 1, terms, true); + } + let t = e.abs().sqrt(); + if t.is_zero() { + return V::one(); + } + if e.is_neg() { + // 1 − t = (1 + e)/(1 + t), so atanh t = log1p(2t(1 + t)/(1 + e))/2. + let one = V::one(); + log1p((t * (one + t) / one_plus_e).ldexp(1)).ldexp(-1) / t + } else { + atan_ratio(t, V::one()) / t + } +} + +/// `R_C(x, y)` for `x ≥ 0` and `y > 0`, by ratios so `x` near `y` does not cancel. +fn rc(x: V, y: V) -> V { + if x.is_zero() { + return half_pi::() / y.sqrt(); + } + rc_one((y - x) / x, y / x) / x.sqrt() +} + +/// Whether deviations at most `spread·4^−M/A_M` leave a closing series of this `order` exact. +fn converged(spread: V, scale: V, mean: V, order: i32) -> bool { + let bits = (V::PREC.cast_signed() + 4 + order - 1) / order; + (spread * scale).ldexp(bits) < mean.abs() +} + +/// The deviations `(A₀ − v₀)/(4^M A_M)` after `M` duplication steps. +fn deviation(a0: V, v0: V, scale: V, a: V) -> V { + (a0 - v0) * scale / a +} + +/// `R_F(x, y, z)` by duplication, any `roots` standing in for the first square roots. +fn rf(x0: V, y0: V, z0: V, roots: Option<[V; 3]>) -> V { + let (mut vx, mut vy, mut vz) = (x0, y0, z0); + let a0 = (vx + vy + vz).div_int(3); + let mut mean = a0; + let mut scale = V::one(); + let spread = (a0 - x0).abs().max((a0 - y0).abs()).max((a0 - z0).abs()); + // Octaves apart, about log₂log₂ of the ratio steps to meet; each then quarters the rest. + for step in 0..by_width::([13, 18, 23]) { + if converged(spread, scale, mean, 8) { + break; + } + let [sx, sy, sz] = match roots { + Some(r) if step == 0 => r.map(Work::abs), + _ => [vx.sqrt(), vy.sqrt(), vz.sqrt()], + }; + let lam = sx * sy + sx * sz + sy * sz; + mean = (mean + lam).ldexp(-2); + vx = (vx + lam).ldexp(-2); + vy = (vy + lam).ldexp(-2); + vz = (vz + lam).ldexp(-2); + scale = scale.ldexp(-2); + } + let dx = deviation(a0, x0, scale, mean); + let dy = deviation(a0, y0, scale, mean); + let dz = -(dx + dy); + let e2 = dx * dy - dz.square(); + let e3 = dx * dy * dz; + // DLMF 19.36.1, through the seventh order. + let one = V::one(); + let series = one - e2.div_int(10) + e3.div_int(14) + e2.square().div_int(24) + - (e2 * e3).mul_int(3).div_int(44) + - (e2.square() * e2).mul_int(5).div_int(208) + + e3.square().mul_int(3).div_int(104) + + (e2.square() * e3).div_int(16); + series / mean.sqrt() +} + +/// The closing series of `R_J` and `R_D` through degree 7, DLMF 19.36.2. +fn rj_series(e2: V, e3: V, e4: V, e5: V) -> V { + let low = + V::one() - e2.mul_int(3).div_int(14) + e3.div_int(6) + e2.square().mul_int(9).div_int(88) + - e4.mul_int(3).div_int(22) + - (e2 * e3).mul_int(9).div_int(52) + + e5.mul_int(3).div_int(26); + let high = -(e2.square() * e2).div_int(16) + + e3.square().mul_int(3).div_int(40) + + (e2 * e4).mul_int(3).div_int(20) + + (e2.square() * e3).mul_int(45).div_int(272) + - (e3 * e4 + e2 * e5).mul_int(9).div_int(68); + low + high +} + +/// `R_J(x, y, z, p)` for `p > 0`. A `p` far above the rest, which duplication +/// brings down only fourfold a step, DLMF 19.21.12 trades for `q` near the least, +/// `x`: `(p − x)R_J(p) + (q − x)R_J(q) = 3R_F − 3R_C(yz/x, pq/x)`. +fn rj(x0: V, y0: V, z0: V, p0: V, roots: Option<[V; 4]>) -> V { + let mut sorted = [(x0, 0), (y0, 1), (z0, 2)]; + sorted.sort_unstable_by_key(|entry| entry.0); + let [(lo, _), (mid, _), (hi, _)] = sorted; + if p0 <= hi.ldexp(4) { + return rj_duplicate(x0, y0, z0, p0, roots); + } + let gap = (mid - lo) * (hi - lo) / (p0 - lo); + let near = lo + gap; + let picked = roots.map(|picked| { + let at = |i: usize| picked.get(i).copied().unwrap_or_else(V::zero); + [at(sorted[0].1), at(sorted[1].1), at(sorted[2].1)] + }); + let near_roots = picked.map(|[a, b, c]| [a, b, c, near.sqrt()]); + let mut rest = + rf(lo, mid, hi, picked).mul_int(3) - gap * rj_duplicate(lo, mid, hi, near, near_roots); + if !lo.is_zero() { + let xi = mid * hi / lo; + let excess = (p0 - mid) * (p0 - hi) / ((p0 - lo) * xi); + rest = rest - (rc_one(excess, p0 * near / (lo * xi)) / xi.sqrt()).mul_int(3); + } + rest / (p0 - lo) +} + +/// [`rj`] by duplication, its `R_C` terms `4^(−3m)·δ/d_m²` from one `δ = Π(p − v)`. +fn rj_duplicate(x0: V, y0: V, z0: V, p0: V, roots: Option<[V; 4]>) -> V { + let (mut vx, mut vy, mut vz, mut vp) = (x0, y0, z0, p0); + let a0 = (vx + vy + vz + vp.ldexp(1)).div_int(5); + let delta = (vp - vx) * (vp - vy) * (vp - vz); + let mut mean = a0; + let mut scale = V::one(); + let mut sum = V::zero(); + let spread = [x0, y0, z0, p0] + .iter() + .fold(V::zero(), |m, &v| m.max((a0 - v).abs())); + for step in 0..by_width::([15, 21, 28]) { + if converged(spread, scale, mean, 8) { + break; + } + let [sx, sy, sz, sp] = match roots { + Some(r) if step == 0 => r.map(Work::abs), + _ => [vx.sqrt(), vy.sqrt(), vz.sqrt(), vp.sqrt()], + }; + let lam = sx * sy + sx * sz + sy * sz; + let inv = V::one() / ((sp + sx) * (sp + sy) * (sp + sz)); + let ex = delta * scale.square() * scale * inv.square(); + // 1 + ex = 2√vp(vp + λ)/den, as Π(√vp + √v) + Π(√vp − √v) = 2√vp(vp + λ). + let one_plus_e = (sp * (vp + lam) * inv).ldexp(1); + sum = sum + scale * rc_one(ex, one_plus_e) * inv; + mean = (mean + lam).ldexp(-2); + vx = (vx + lam).ldexp(-2); + vy = (vy + lam).ldexp(-2); + vz = (vz + lam).ldexp(-2); + vp = (vp + lam).ldexp(-2); + scale = scale.ldexp(-2); + } + let dx = deviation(a0, x0, scale, mean); + let dy = deviation(a0, y0, scale, mean); + let dz = deviation(a0, z0, scale, mean); + let dp = -(dx + dy + dz).ldexp(-1); + let e2 = dx * dy + dx * dz + dy * dz - dp.square().mul_int(3); + let xyz = dx * dy * dz; + let e3 = xyz + (e2 * dp).ldexp(1) + (dp.square() * dp).ldexp(2); + let e4 = (xyz.ldexp(1) + e2 * dp + (dp.square() * dp).mul_int(3)) * dp; + let e5 = xyz * dp.square(); + scale * rj_series(e2, e3, e4, e5) / (mean * mean.sqrt()) + sum.mul_int(6) +} + +/// `R_D(x, y, z)` by duplication. +fn rd(x0: V, y0: V, z0: V) -> V { + let (mut vx, mut vy, mut vz) = (x0, y0, z0); + let a0 = (vx + vy + vz.mul_int(3)).div_int(5); + let mut mean = a0; + let mut scale = V::one(); + let mut sum = V::zero(); + let spread = [x0, y0, z0] + .iter() + .fold(V::zero(), |m, &v| m.max((a0 - v).abs())); + for _ in 0..by_width::([15, 21, 28]) { + if converged(spread, scale, mean, 8) { + break; + } + let (sx, sy, sz) = (vx.sqrt(), vy.sqrt(), vz.sqrt()); + let lam = sx * sy + sx * sz + sy * sz; + sum = sum + scale / (sz * (vz + lam)); + mean = (mean + lam).ldexp(-2); + vx = (vx + lam).ldexp(-2); + vy = (vy + lam).ldexp(-2); + vz = (vz + lam).ldexp(-2); + scale = scale.ldexp(-2); + } + let dx = deviation(a0, x0, scale, mean); + let dy = deviation(a0, y0, scale, mean); + let dz = -(dx + dy).div_int(3); + let xy = dx * dy; + let z2 = dz.square(); + let e2 = xy - z2.mul_int(6); + let e3 = (xy.mul_int(3) - z2.mul_int(8)) * dz; + let e4 = (xy - z2) * z2.mul_int(3); + let e5 = xy * z2 * dz; + scale * rj_series(e2, e3, e4, e5) / (mean * mean.sqrt()) + sum.mul_int(3) +} + +/// The arithmetic-geometric mean of 1 and `b`, for `b` in `(0, 1]`. +fn agm(b: V) -> V { + let (mut a, mut b) = (V::one(), b); + for _ in 0..32 { + if V::negligible(&(a - b), &a) { + break; + } + (a, b) = ((a + b).ldexp(-1), (a * b).sqrt()); + } + a +} + +/// `K(m) = π/(2·AGM(1, k'))`, for `k' > 0`. +fn complete_k(kc: V) -> V { + half_pi::() / agm(kc) +} + +const DEPTH: usize = 40; + +/// The AGM of 1 and `k′`, each level's means kept, `mean` its limit. +struct Means { + arith: [V; DEPTH], + geo: [V; DEPTH], + used: usize, + mean: V, +} + +fn means(kc: V) -> Means { + let one = V::one(); + let mut out = Means { + arith: [V::zero(); DEPTH], + geo: [V::zero(); DEPTH], + used: 0, + mean: one, + }; + let (mut a, mut b) = (one, kc); + for (i, (x, y)) in out.arith.iter_mut().zip(out.geo.iter_mut()).enumerate() { + (*x, *y) = (a, b); + out.used = i + 1; + out.mean = (a + b).ldexp(-1); + if (a - b).abs().ldexp(V::PREC.cast_signed() / 2 + 2) <= a { + break; + } + b = (a * b).sqrt(); + a = out.mean; + } + out +} + +/// `(sn, cn, dn)` of `v` in `[0, K/2]` by Bulirsch: `sin_cos` of `c·v`, then rational steps up. +fn sn_cn_dn_core(arg: V, means: &Means) -> (V, V, V) { + let one = V::one(); + let (sine, cosine) = sin_cos(arg * means.mean); + if sine.is_zero() { + return (sine, cosine, one); + } + let mut ratio = cosine / sine; + let mut cot = means.mean * ratio; + let mut dn = one; + for i in (0..means.used).rev() { + let at = |t: &[V; DEPTH]| t.get(i).copied().unwrap_or(one); + ratio = ratio * cot; + cot = cot * dn; + dn = (at(&means.geo) + ratio) / (at(&means.arith) + ratio); + ratio = cot / at(&means.arith); + } + let sn = one / (cot.square() + one).sqrt(); + (sn, cot * sn, dn) +} + +/// `u = (2^124·c₀ + 2^62·c₁ + c₂)·period + rest`, the count in parts so a 128-bit `u` fits. +fn reduce(u: V, period: V) -> ([i64; 3], V) { + let cycles = u / period; + let top = cycles.ldexp(-124).round_int(); + let cycles = cycles - V::from_int(top).ldexp(124); + let high = cycles.ldexp(-62).round_int(); + let count = [ + top, + high, + (cycles - V::from_int(high).ldexp(62)).round_int(), + ]; + (count, u - times(count, period)) +} + +/// `count·unit` for a count in [`reduce`]'s parts. +fn times([top, high, low]: [i64; 3], unit: V) -> V { + unit.mul_int(top).ldexp(124) + unit.mul_int(high).ldexp(62) + unit.mul_int(low) +} + +/// `(sn, cn, dn, am)` of `u` for the modulus parts. +fn jacobi_work(u: V, parts_: &Parts) -> (V, V, V, V) { + let one = V::one(); + if parts_.kc.is_zero() { + // The separatrix: sn = tanh u, cn = dn = sech u, am = atan(sinh u). + if u.abs() > V::from_int(crate::eval::HUGE) { + let edge = one.with_sign(u.is_neg()); + return ( + edge, + V::zero(), + V::zero(), + half_pi::().with_sign(u.is_neg()), + ); + } + let (sh, ch) = sinh_cosh(u.abs()); + let sech = one / ch; + let am = atan_ratio(sh, one).with_sign(u.is_neg()); + return ((sh / ch).with_sign(u.is_neg()), sech, sech, am); + } + let levels = means(parts_.kc); + let quarter = half_pi::() / levels.mean; + let (count, rest) = reduce(u, quarter.ldexp(1)); + let dist = rest.abs(); + let (sn, cn, dn) = if dist <= quarter.ldexp(-1) { + sn_cn_dn_core(dist, &levels) + } else { + // sn(K − w) = cn/dn, cn(K − w) = kk'·sn/dn and dn(K − w) = kk'/dn. + let (sine, cosine, delta) = sn_cn_dn_core(quarter - dist, &levels); + (cosine / delta, parts_.kc * sine / delta, parts_.kc / delta) + }; + let am = atan_ratio(sn, cn.abs()); + let am = if cn.is_neg() { pi::() - am } else { am }; + let flip = count[2] % 2 != 0; + let sign = |x: V, neg: bool| if neg { -x } else { x }; + let winding = times(count, pi::()); + ( + sign(sign(sn, rest.is_neg()), flip), + sign(cn, flip), + dn, + sign(am, rest.is_neg()) + winding, + ) +} + +/// The Jacobi elliptic functions `(sn, cn, dn)` of `u`, any `u`. +#[must_use] +#[cfg_attr( + all(feature = "verify-no-panic", not(debug_assertions)), + no_panic::no_panic +)] +pub fn jacobi_sn_cn_dn(u: T, modulus: Modulus) -> (T, T, T) { + // am, which grows with u, joins the check: reducing u loses bits in proportion. + let ((sn, cn), (dn, _)) = tiered!(T, |v| { + let (sn, cn, dn, am) = jacobi_work(v(u), &parts_in(modulus, v)); + ((sn, cn), (dn, am)) + }); + (sn, cn, dn) +} + +/// The Jacobi amplitude `am(u)`, continuous: `am(u + 2K) = am(u) + π`. +#[must_use] +#[cfg_attr( + all(feature = "verify-no-panic", not(debug_assertions)), + no_panic::no_panic +)] +pub fn jacobi_am(u: T, modulus: Modulus) -> T { + tiered!(T, |v| jacobi_work(v(u), &parts_in(modulus, v)).3) +} + +/// `R_F(cn², dn², 1)`, finite unless `cn` and `dn` are both 0. +fn rf_sq(cn: V, dn: V) -> Option { + if cn.is_zero() && dn.is_zero() { + return None; + } + Some(rf( + cn.square(), + dn.square(), + V::one(), + Some([cn, dn, V::one()]), + )) +} + +/// The `u` in `(−2K, 2K]` with these [`jacobi_sn_cn_dn`] values, all given so none is derived. +/// # Errors +/// Returns `DomainError` when `cn` and `dn` are both 0, where `u` is infinite. +#[must_use = "returns the argument which should be handled"] +#[cfg_attr( + all(feature = "verify-no-panic", not(debug_assertions)), + no_panic::no_panic +)] +pub fn jacobi_inverse(sn: T, cn: T, dn: T, modulus: Modulus) -> Result { + if cn == T::zero() && dn == T::zero() { + return Err(Error::domain("jacobi_inverse", "cn or dn non-zero")); + } + Ok(tiered!(T, |v| inverse_work( + v(sn), + v(cn), + v(dn), + &parts_in(modulus, v) + ))) +} + +fn inverse_work(sw: V, cw: V, dw: V, parts_: &Parts) -> V { + let part = sw.abs() * rf_sq(cw, dw).unwrap_or_else(V::zero); + let arg = if cw.is_neg() { + if parts_.kc.is_zero() { + crate::eval::beyond() + } else { + complete_k(parts_.kc).ldexp(1) - part + } + } else { + part + }; + arg.with_sign(sw.is_neg()) +} + +/// The complete elliptic integral of the first kind `K(m)`. +/// # Errors +/// Returns `DomainError` at `m = 1`, where `K` is infinite. +#[must_use = "returns the integral which should be handled"] +#[cfg_attr( + all(feature = "verify-no-panic", not(debug_assertions)), + no_panic::no_panic +)] +pub fn ellip_k(modulus: Modulus) -> Result { + if modulus.at_separatrix() { + return Err(Error::domain("ellip_k", "m below 1")); + } + Ok(tiered!(T, |v| complete_k(parts_in(modulus, v).kc))) +} + +/// The complete elliptic integral of the second kind `E(m)`, 1 at `m = 1`. +#[must_use] +#[cfg_attr( + all(feature = "verify-no-panic", not(debug_assertions)), + no_panic::no_panic +)] +pub fn ellip_e(modulus: Modulus) -> T { + if modulus.at_separatrix() { + return T::one(); + } + tiered!(T, |v| complete_e(&parts_in(modulus, v))) +} + +fn complete_e(p: &Parts) -> V { + let (one, zero) = (V::one(), V::zero()); + let f = rf(zero, p.kc2, one, Some([zero, p.kc, one])); + f - (p.m * rd(zero, p.kc2, one)).div_int(3) +} + +/// `Π(n; φ|m) = ∫₀^φ dθ/((1 − n·sin²θ)√(1 − m·sin²θ))`, `φ` in `(−π, π]` by `(sn, cn, dn)`. +/// # Errors +/// Returns `DomainError` when `1 − n·sn² ≤ 0`, or `cn` and `dn` are both 0. +#[must_use = "returns the integral which should be handled"] +#[cfg_attr( + all(feature = "verify-no-panic", not(debug_assertions)), + no_panic::no_panic +)] +pub fn ellip_pi(n: T, sn: T, cn: T, dn: T, modulus: Modulus) -> Result { + let (nw, sw) = (work(n), work(sn)); + let one = W::::one(); + let pole = one - nw * sw.square(); + // Past π/2 the integral crosses its pole unless n < 1. + let reachable = !cn.is_negative() || nw < one; + if (cn == T::zero() && dn == T::zero()) || pole.is_neg() || pole.is_zero() || !reachable { + return Err(Error::domain( + "ellip_pi", + "1 - n sn^2 > 0, n < 1 past pi/2, cn or dn non-zero", + )); + } + Ok(tiered!(T, |v| { + let (s_v, n_v) = (v(sn), v(n)); + let gap = v(T::one()) - n_v * s_v.square(); + pi_work(n_v, s_v, v(cn), v(dn), gap, &parts_in(modulus, v)) + })) +} + +/// `Π(n; φ|m)` from `φ`'s `(sn, cn, dn)` and `1 − n·sn²`. +fn pi_work(nw: V, sw: V, cw: V, dw: V, pole: V, parts_: &Parts) -> V { + let one = V::one(); + let s2 = sw.square(); + let rf_value = rf_sq(cw, dw).unwrap_or_else(V::zero); + let rj_value = rj( + cw.square(), + dw.square(), + one, + pole, + Some([cw, dw, one, pole.sqrt()]), + ); + let part = sw.abs() * (rf_value + (nw * s2 * rj_value).div_int(3)); + let total = if cw.is_neg() { + if parts_.kc.is_zero() { + crate::eval::beyond() + } else { + complete_pi(nw, parts_).ldexp(1) - part + } + } else { + part + }; + total.with_sign(sw.is_neg()) +} + +fn complete_pi(n: V, p: &Parts) -> V { + let (one, zero) = (V::one(), V::zero()); + let f = rf(zero, p.kc2, one, Some([zero, p.kc, one])); + let pole = one - n; + f + (n * rj(zero, p.kc2, one, pole, Some([zero, p.kc, one, pole.sqrt()]))).div_int(3) +} + +/// `∫ dv/(1 − n·sn²v)` from `u` to `u + step`, the change in `Π(n; am v|m)`, from the ends +/// reduced to `[−K, K]` with `2Π(n|m)` a period, so a small step keeps its bits. +/// # Errors +/// Returns `DomainError` unless `n < 1`. +#[must_use = "returns the integral which should be handled"] +#[cfg_attr( + all(feature = "verify-no-panic", not(debug_assertions)), + no_panic::no_panic +)] +pub fn ellip_pi_increment(n: T, u: T, step: T, modulus: Modulus) -> Result { + if n >= T::one() { + return Err(Error::domain("ellip_pi_increment", "n below 1")); + } + let (rise, _) = tiered!(T, |v| increment_work( + v(n), + v(u), + v(step), + &parts_in(modulus, v) + )); + Ok(rise) +} + +/// The increment, and the size its error scales with: the ends' integrals, `u`, and `K/(1 − n·sn²)`. +fn increment_work(n: V, u: V, s: V, p: &Parts) -> (V, V) { + let one = V::one(); + let local = |v: V| { + let (sn, cn, dn, _) = jacobi_work(v, p); + let gap = cn.square() + (one - n) * sn.square(); + (pi_work(n, sn, cn, dn, gap, p), one / gap) + }; + let reach = u.abs().max((u + s).abs()); + let ((from, near), (to, far), whole, scale) = if p.kc.is_zero() { + // Past the cap sech² v is below the working precision and the integrand 1/(1 − n). + let cap = V::from_int(i64::from(by_width::([29, 59, 88]))); + if let Some(rise) = gauss(n, u, s, p) { + return (rise, rise.abs().max(reach)); + } + let capped = |v: V| { + let edge = cap.with_sign(v.is_neg()); + if v.abs() > cap { + let (at, steep) = local(edge); + (at + (v - edge) / (one - n), steep) + } else { + local(v) + } + }; + (capped(u), capped(u + s), V::zero(), V::zero()) + } else { + let quarter = complete_k(p.kc); + let period = quarter.ldexp(1); + let (_, start) = reduce(u, period); + if let Some(rise) = gauss(n, start, s, p) { + return (rise, rise.abs().max(reach)); + } + let (count, end) = reduce(start + s, period); + let whole = if count == [0; 3] { + V::zero() + } else { + times(count, complete_pi(n, p).ldexp(1)) + }; + (local(start), local(end), whole, quarter) + }; + let size = from.abs().max(to.abs()).max(whole.abs()).max(reach); + (to - from + whole, size.max(scale * near.max(far))) +} + +/// The increment by five-point Gauss–Legendre where the step is too short for a difference not to +/// cancel, so far inside the nearest zero of `1 − n·sn²`, by two derivatives, that it is exact. +fn gauss(n: V, from: V, step: V, p: &Parts) -> Option { + let one = V::one(); + let half = step.ldexp(-1); + let mid = from + half; + let (sn, cn, dn, _) = jacobi_work(mid, p); + let gap = cn.square() + (one - n) * sn.square(); + let slope = (n * sn * cn * dn).ldexp(1); + let bend = (n * ((cn * dn).square() - (sn * dn).square() - p.m * (sn * cn).square())).ldexp(1); + let room = (gap / slope).abs().min((gap / bend).abs().sqrt()).min(one); + if step + .abs() + .ldexp(i32::try_from(by_width::([7, 14, 20])).unwrap_or(0)) + > room + { + return None; + } + let at = |v: V| { + let (sine, cosine, _, _) = jacobi_work(v, p); + one / (cosine.square() + (one - n) * sine.square()) + }; + let root = V::from_int(10).div_int(7).sqrt().ldexp(1); + let (inner, outer) = ( + (V::from_int(5) - root).sqrt().div_int(3), + (V::from_int(5) + root).sqrt().div_int(3), + ); + let lean = V::from_int(70).sqrt().mul_int(13); + let pair = |x: V| at(mid - half * x) + at(mid + half * x); + let sum = V::from_int(128).div_int(225) / gap + + (V::from_int(322) + lean).div_int(900) * pair(inner) + + (V::from_int(322) - lean).div_int(900) * pair(outer); + Some(half * sum) +} + +/// The complete elliptic integral of the third kind `Π(n|m)`. +/// # Errors +/// Returns `DomainError` for `n ≥ 1`, or at `m = 1`. +#[must_use = "returns the integral which should be handled"] +#[cfg_attr( + all(feature = "verify-no-panic", not(debug_assertions)), + no_panic::no_panic +)] +pub fn ellip_pi_complete(n: T, modulus: Modulus) -> Result { + if n >= T::one() || modulus.at_separatrix() { + return Err(Error::domain( + "ellip_pi_complete", + "n below 1 and m below 1", + )); + } + Ok(tiered!(T, |v| complete_pi(v(n), &parts_in(modulus, v)))) +} + +/// Carlson's `R_F(x, y, z)`. Domain: `x, y, z ≥ 0`, at most one of them 0. +/// # Errors +/// Returns `DomainError` outside the domain. +#[must_use = "returns the integral which should be handled"] +#[cfg_attr( + all(feature = "verify-no-panic", not(debug_assertions)), + no_panic::no_panic +)] +pub fn carlson_rf(x: T, y: T, z: T) -> Result { + check_rf(&[x, y, z], "carlson_rf")?; + Ok(tiered!(T, |v| rf(v(x), v(y), v(z), None))) +} + +/// [`carlson_rf`] of the squares of `rx`, `ry` and `rz`, for arguments too small to hold. +/// # Errors +/// Returns `DomainError` if two of the roots are 0. +#[must_use = "returns the integral which should be handled"] +#[cfg_attr( + all(feature = "verify-no-panic", not(debug_assertions)), + no_panic::no_panic +)] +pub fn carlson_rf_roots(rx: T, ry: T, rz: T) -> Result { + let zeros = [rx, ry, rz].iter().filter(|&&r| r == T::zero()).count(); + if zeros > 1 { + return Err(Error::domain( + "carlson_rf_roots", + "x, y, z >= 0 with at most one zero", + )); + } + Ok(tiered!(T, |v| { + let r = [v(rx), v(ry), v(rz)]; + let [a, b, c] = r.map(Work::square); + rf(a, b, c, Some(r)) + })) +} + +fn check_rf(v: &[T; 3], name: &'static str) -> Result<()> { + let zeros = v.iter().filter(|&&x| x == T::zero()).count(); + if v.iter().any(|x| x.is_negative()) || zeros > 1 { + return Err(Error::domain(name, "x, y, z >= 0 with at most one zero")); + } + Ok(()) +} + +/// Carlson's `R_J(x, y, z, p)`. Domain: `x, y, z ≥ 0`, at most one 0, and `p > 0`. +/// # Errors +/// Returns `DomainError` outside the domain. +#[must_use = "returns the integral which should be handled"] +#[cfg_attr( + all(feature = "verify-no-panic", not(debug_assertions)), + no_panic::no_panic +)] +pub fn carlson_rj(x: T, y: T, z: T, p: T) -> Result { + check_rf(&[x, y, z], "carlson_rj")?; + if p <= T::zero() { + return Err(Error::domain("carlson_rj", "p > 0")); + } + Ok(tiered!(T, |v| rj(v(x), v(y), v(z), v(p), None))) +} + +/// [`carlson_rj`] of the squares of `rx`, `ry`, `rz` and `rp`. +/// # Errors +/// Returns `DomainError` if two of `rx`, `ry`, `rz` are 0, or `rp` is. +#[must_use = "returns the integral which should be handled"] +#[cfg_attr( + all(feature = "verify-no-panic", not(debug_assertions)), + no_panic::no_panic +)] +pub fn carlson_rj_roots(rx: T, ry: T, rz: T, rp: T) -> Result { + let zeros = [rx, ry, rz].iter().filter(|&&r| r == T::zero()).count(); + if zeros > 1 { + return Err(Error::domain( + "carlson_rj_roots", + "x, y, z >= 0 with at most one zero", + )); + } + if rp == T::zero() { + return Err(Error::domain("carlson_rj_roots", "rp non-zero")); + } + Ok(tiered!(T, |v| { + let roots = [v(rx), v(ry), v(rz), v(rp)]; + let [a, b, c, d] = roots.map(Work::square); + rj(a, b, c, d, Some(roots)) + })) +} + +/// Carlson's `R_C(x, y)`. Domain: `x ≥ 0`, `y > 0`. +/// # Errors +/// Returns `DomainError` outside the domain. +#[must_use = "returns the integral which should be handled"] +#[cfg_attr( + all(feature = "verify-no-panic", not(debug_assertions)), + no_panic::no_panic +)] +pub fn carlson_rc(x: T, y: T) -> Result { + if x < T::zero() || y <= T::zero() { + return Err(Error::domain("carlson_rc", "x >= 0, y > 0")); + } + Ok(tiered!(T, |v| rc(v(x), v(y)))) +} + +/// Carlson's `R_D(x, y, z)`. Domain: `x, y ≥ 0`, not both 0, and `z > 0`. +/// # Errors +/// Returns `DomainError` outside the domain. +#[must_use = "returns the integral which should be handled"] +#[cfg_attr( + all(feature = "verify-no-panic", not(debug_assertions)), + no_panic::no_panic +)] +pub fn carlson_rd(x: T, y: T, z: T) -> Result { + let zero = T::zero(); + if x < zero || y < zero || z <= zero || (x == zero && y == zero) { + return Err(Error::domain( + "carlson_rd", + "x, y >= 0 not both zero, z > 0", + )); + } + Ok(tiered!(T, |v| rd(v(x), v(y), v(z)))) +} diff --git a/src/ops/exponential.rs b/src/ops/exponential.rs index 326722f..6dcbf46 100644 --- a/src/ops/exponential.rs +++ b/src/ops/exponential.rs @@ -1,127 +1,38 @@ #![doc = include_str!("../../docs/exponential.md")] -use crate::bounded::{NormalizedLnArg, OpenUnitInterval}; +use crate::consts::{FRAC_1_LN_10, LN_2}; use crate::error::{Error, Result}; -use crate::ops::hyperbolic::atanh_open; +use crate::eval::{ + HUGE, W, exp as exp_work, exp_wide, expm1 as expm1_work, fixed, ln as ln_work, + log1p as log1p_work, log1p_minus, log1p_minus_ratio, log1p_ratio as log1p_ratio_work, + log2 as log2_work, phi, phis, tiered, work, +}; +use crate::real::Work; use crate::traits::CordicNumber; -/// `1/ln 2 − 1` as I1F63 (`1/ln 2` itself exceeds 1). -const FRAC_1_LN2_MINUS_1_I1F63: i64 = 0x38AA_3B29_5C17_F0BC; - -/// Right-shifts a non-negative value by `n`, rounding to nearest. A plain `>>` -/// would bias small `exp` results, only a few ulps, low by up to an ulp. -fn shr_rounded(x: T, n: u32) -> T { - if n == 0 { - return x; - } - let ulp = T::one() >> T::frac_bits(); - let shifted = x >> n; - let rem = x.saturating_sub(shifted << n); - let half = ulp << (n - 1); - if rem >= half { - shifted.saturating_add(ulp) - } else { - shifted - } -} - /// Exponential function e^x. Saturates to `T::MAX` above `ln(T::MAX)` and to -/// zero far below 0; the [module docs](self) list the thresholds. +/// zero where e^x rounds below half an ulp; the [module docs](self) list the +/// thresholds. #[must_use] #[cfg_attr( all(feature = "verify-no-panic", not(debug_assertions)), no_panic::no_panic )] pub fn exp(x: T) -> T { - let zero = T::zero(); - let one = T::one(); - let ln2 = T::ln_2(); - - if x == zero { - return one; - } - - // exp(x) = 2^k·exp(r) with r in [0, ln 2). The estimate of k = ⌊x / ln 2⌋ - // is within one, so one correction below fixes it (docs/exp.md). - let (int_bits, frac_bits) = ( - (T::total_bits() - T::frac_bits()).cast_signed(), - T::frac_bits().cast_signed(), - ); - let quotient = if T::total_bits() <= 2 * T::frac_bits() { - x.saturating_add(x.saturating_mul(T::from_i1f63(FRAC_1_LN2_MINUS_1_I1F63))) - } else { - x.div(ln2) - }; - let mut scale = quotient.to_i32(); - - // The result exceeds MAX once k ≥ int_bits − 1 and rounds to zero once - // k ≤ −(frac_bits + 2); these exits allow one of slack before correction. - if scale >= int_bits { - return T::max_value(); - } - if scale < -(frac_bits + 2) { - return zero; - } - - let mut r = x.saturating_sub(ln2.mul_int(scale)); - if r < zero { - scale -= 1; - r = r.saturating_add(ln2); - } else if r >= ln2 { - scale += 1; - r = r.saturating_sub(ln2); - } - if scale >= int_bits - 1 { - return T::max_value(); - } - if scale <= -(frac_bits + 2) { - return zero; - } - - // Factored Taylor series, of degree 7, 12 or 19 by precision (docs/exp.md). - let mut p = one; - if T::frac_bits() >= 40 { - p = one.saturating_add(r.div_int(19).saturating_mul(p)); - p = one.saturating_add(r.div_int(18).saturating_mul(p)); - p = one.saturating_add(r.div_int(17).saturating_mul(p)); - p = one.saturating_add(r.div_int(16).saturating_mul(p)); - p = one.saturating_add(r.div_int(15).saturating_mul(p)); - p = one.saturating_add(r.div_int(14).saturating_mul(p)); - p = one.saturating_add(r.div_int(13).saturating_mul(p)); - } - if T::frac_bits() >= 24 { - p = one.saturating_add(r.div_int(12).saturating_mul(p)); - p = one.saturating_add(r.div_int(11).saturating_mul(p)); - p = one.saturating_add(r.div_int(10).saturating_mul(p)); - p = one.saturating_add(r.div_int(9).saturating_mul(p)); - p = one.saturating_add(r.div_int(8).saturating_mul(p)); - } - p = one.saturating_add(r.div_int(7).saturating_mul(p)); - p = one.saturating_add(r.div_int(6).saturating_mul(p)); - p = one.saturating_add(r.div_int(5).saturating_mul(p)); - p = one.saturating_add(r.div_int(4).saturating_mul(p)); - p = one.saturating_add(r.div_int(3).saturating_mul(p)); - p = one.saturating_add(r.div_int(2).saturating_mul(p)); - let exp_r = one.saturating_add(r.saturating_mul(p)); - - if scale > 0 { - let shift = scale.unsigned_abs(); - // A left shift past the headroom would wrap. - let headroom = T::max_value() >> shift; - if exp_r > headroom { - T::max_value() - } else { - exp_r << shift + if crate::fast::applies::() + && let Some((v, q)) = crate::fast::exp(x.raw_i128(), T::frac_bits()) + { + // v·2^(q − 126) to 2^−121, keeping 24 guard bits below 2^(97 − frac). + let s = 126 - T::frac_bits().cast_signed() - q; + if s >= 30 { + return T::from_raw_i128(crate::fast::shift_round(v, s.cast_unsigned())); } - } else { - shr_rounded(exp_r, scale.unsigned_abs()) } + tiered!(T, |v| exp_wide(v(x))) } -/// Power `base^exponent`, as `exp(exponent · ln|base|)`, saturating like [`exp`]. -/// -/// A negative base takes its sign from the exponent's parity. The relative -/// error grows with `|exponent|`, which scales the logarithm's rounding error. +/// Power `base^exponent`, as `exp(exponent · ln|base|)`, saturating like +/// [`exp`]; a negative base takes its sign from the exponent's parity. /// # Errors /// Returns `DomainError` for a zero base with a negative exponent, or a /// negative base with a non-integer exponent, which has no real result. @@ -145,9 +56,9 @@ pub fn pow(base: T, exponent: T) -> Result { )) }; } - let magnitude = exp(exponent.saturating_mul(ln_positive(base.abs()))); + let magnitude = exp_wide::>(work(exponent) * ln_work(work(base).abs())); if !base.is_negative() { - return Ok(magnitude); + return Ok(fixed(magnitude)); } // The shift isolates the integer part exactly at any magnitude. let integer_part = exponent >> T::frac_bits(); @@ -155,7 +66,7 @@ pub fn pow(base: T, exponent: T) -> Result { return Err(Error::domain("pow", "integer exponent for a negative base")); } let odd = (integer_part >> 1) << 1 != integer_part; - Ok(if odd { -magnitude } else { magnitude }) + Ok(fixed(if odd { -magnitude } else { magnitude })) } /// Natural logarithm. Domain: `x > 0`. @@ -170,46 +81,12 @@ pub fn ln(x: T) -> Result { if x <= T::zero() { return Err(Error::domain("ln", "positive value")); } - Ok(ln_positive(x)) -} - -/// Natural logarithm of a positive value. The caller must ensure `x > 0`. -pub(crate) fn ln_positive(x: T) -> T { - let zero = T::zero(); - let one = T::one(); - let two = T::two(); - - if x == one { - return zero; + if crate::fast::applies::() { + let v = crate::fast::ln(x.raw_i128(), T::frac_bits()); + let raw = crate::fast::shift_round(v, 119 - T::frac_bits()); + return Ok(T::from_raw_i128(if v < 0 { -raw } else { raw })); } - - let ln2 = T::ln_2(); - - // ln(x) = ln(x·2^-k) + k·ln 2, with x·2^-k in [0.5, 2] chosen from the - // leading bit e = ⌊log₂ x⌋; a value landing exactly on 2 stops there. - let e = x.checked_int_log2().unwrap_or(0); - let (normalized, k) = if e >= 1 { - let candidate = x >> (e - 1).unsigned_abs(); - if candidate == two { - (candidate, e - 1) - } else { - (x >> e.unsigned_abs(), e) - } - } else if e <= -2 { - (x << (-1 - e).unsigned_abs(), e + 1) - } else { - (x, 0) - }; - let k_ln2 = ln2.mul_int(k); - - // ln(x) = 2·atanh((x − 1)/(x + 1)), whose argument is within 1/3 of 0. - let norm = NormalizedLnArg::from_normalized(normalized); - let arg = OpenUnitInterval::from_normalized_ln_arg(norm); - - let atanh_val = atanh_open(arg); - let ln_normalized = atanh_val.saturating_add(atanh_val); - - ln_normalized.saturating_add(k_ln2) + Ok(tiered!(T, |v| ln_work(v(x)))) } /// Base-2 logarithm. Domain: `x > 0`. @@ -221,9 +98,10 @@ pub(crate) fn ln_positive(x: T) -> T { no_panic::no_panic )] pub fn log2(x: T) -> Result { - let ln_x = ln(x)?; - let ln_2 = T::ln_2(); - Ok(ln_x.div(ln_2)) + if x <= T::zero() { + return Err(Error::domain("log2", "positive value")); + } + Ok(tiered!(T, |v| log2_work(v(x)))) } /// Base-10 logarithm. Domain: `x > 0`. @@ -235,13 +113,14 @@ pub fn log2(x: T) -> Result { no_panic::no_panic )] pub fn log10(x: T) -> Result { - let ln_x = ln(x)?; - let ln_10 = T::ln_10(); - Ok(ln_x.div(ln_10)) + if x <= T::zero() { + return Err(Error::domain("log10", "positive value")); + } + Ok(tiered!(T, |v| log10_work(v(x)))) } -/// Power of two 2^x, as `exp(x · ln 2)`. Saturates to `T::MAX` above -/// `log2(T::MAX)` and to zero far below 0; the [module docs](self) list the +/// Power of two 2^x. Saturates to `T::MAX` above `log2(T::MAX)` and to zero +/// where 2^x rounds below half an ulp; the [module docs](self) list the /// thresholds. #[must_use] #[cfg_attr( @@ -249,6 +128,145 @@ pub fn log10(x: T) -> Result { no_panic::no_panic )] pub fn pow2(x: T) -> T { - let ln_2 = T::ln_2(); - exp(x.saturating_mul(ln_2)) + // 2^x = 2^k · e^(r·ln 2) with k = ⌊x⌋ and r its exact fractional part. + let k = x.raw_i128() >> T::frac_bits(); + if k > 256 { + return T::max_value(); + } + if k < -256 { + return T::zero(); + } + let r = W::::from_raw(x.raw_i128() - (k << T::frac_bits()), T::frac_bits()); + let k = i32::try_from(k).unwrap_or(0); + fixed(exp_work(r * W::::constant(&LN_2)).ldexp(k)) +} + +/// `e^x − 1`, to full relative precision near 0, saturating like [`exp`]. +#[must_use] +#[cfg_attr( + all(feature = "verify-no-panic", not(debug_assertions)), + no_panic::no_panic +)] +pub fn expm1(x: T) -> T { + let v = work(x); + if v > W::::from_int(HUGE) { + return T::max_value(); + } + if v < W::::from_int(-HUGE) { + return -T::one(); + } + tiered!(T, |v| expm1_work(v(x))) +} + +/// `(e^x − 1)/x`, which is 1 at 0, saturating for large `x`. +#[must_use] +#[cfg_attr( + all(feature = "verify-no-panic", not(debug_assertions)), + no_panic::no_panic +)] +pub fn exprel(x: T) -> T { + tiered!(T, |v| phi(1, v(x))) +} + +/// `(e^x − 1 − x)/x²`, which is ½ at 0, saturating for large `x`. +#[must_use] +#[cfg_attr( + all(feature = "verify-no-panic", not(debug_assertions)), + no_panic::no_panic +)] +pub fn exprel2(x: T) -> T { + tiered!(T, |v| phi(2, v(x))) +} + +/// `[E₁, E₂, E₃, E₄, E₅, e^(−x)]`, with `Eₖ(x) = ∫₀¹ s^(k−1)·e^(−xs) ds`. +#[must_use] +#[cfg_attr( + all(feature = "verify-no-panic", not(debug_assertions)), + no_panic::no_panic +)] +pub fn exp_moments(x: T) -> [T; 6] { + if work(x) < W::::from_int(-HUGE) { + return [T::max_value(); 6]; + } + tiered!(T, |v| moments_work(v(x))) +} + +fn moments_work(v: V) -> [V; 6] { + let (decay, parts) = if v > V::from_int(HUGE) { + let mut power = V::one(); + ( + V::zero(), + [0; 5].map(|_| { + power = power / v; + power + }), + ) + } else { + let decay = exp_work(-v); + (decay, phis(v, decay).map(|p| decay * p)) + }; + let mut out = [decay; 6]; + for ((slot, part), factorial) in out.iter_mut().zip(parts).zip([1, 1, 2, 6, 24]) { + *slot = part.mul_int(factorial); + } + out +} + +fn log10_work(x: V) -> V { + ln_work(x) * V::constant(&FRAC_1_LN_10) +} + +/// `ln(1 + v)`, to full relative precision near 0. Domain: `v > −1`. +/// # Errors +/// Returns `DomainError` if `v ≤ −1`. +#[must_use = "returns the logarithm which should be handled"] +#[cfg_attr( + all(feature = "verify-no-panic", not(debug_assertions)), + no_panic::no_panic +)] +pub fn log1p(v: T) -> Result { + above_minus_one(v, "log1p").map(|_| tiered!(T, |w| log1p_work(w(v)))) +} + +/// `ln(1 + z)/z`, which is 1 at 0. Domain: `z > −1`. +/// # Errors +/// Returns `DomainError` if `z ≤ −1`. +#[must_use = "returns the ratio which should be handled"] +#[cfg_attr( + all(feature = "verify-no-panic", not(debug_assertions)), + no_panic::no_panic +)] +pub fn log1p_ratio(z: T) -> Result { + above_minus_one(z, "log1p_ratio").map(|_| tiered!(T, |w| log1p_ratio_work(w(z)))) +} + +/// `ln(1 + v) − v`, without cancelling near 0. Domain: `v > −1`. +/// # Errors +/// Returns `DomainError` if `v ≤ −1`. +#[must_use = "returns the residual which should be handled"] +#[cfg_attr( + all(feature = "verify-no-panic", not(debug_assertions)), + no_panic::no_panic +)] +pub fn log1p_residual(v: T) -> Result { + above_minus_one(v, "log1p_residual").map(|_| tiered!(T, |w| log1p_minus(w(v)))) +} + +/// `(ln(1 + u) − u)/u²`, which is −½ at 0. Domain: `u > −1`. +/// # Errors +/// Returns `DomainError` if `u ≤ −1`. +#[must_use = "returns the ratio which should be handled"] +#[cfg_attr( + all(feature = "verify-no-panic", not(debug_assertions)), + no_panic::no_panic +)] +pub fn log1p_residual_ratio(u: T) -> Result { + above_minus_one(u, "log1p_residual_ratio").map(|_| tiered!(T, |w| log1p_minus_ratio(w(u)))) +} + +fn above_minus_one(v: T, name: &'static str) -> Result> { + if v <= -T::one() { + return Err(Error::domain(name, "value > -1")); + } + Ok(work(v)) } diff --git a/src/ops/hyperbolic.rs b/src/ops/hyperbolic.rs index b4a7176..7b62148 100644 --- a/src/ops/hyperbolic.rs +++ b/src/ops/hyperbolic.rs @@ -1,19 +1,22 @@ #![doc = include_str!("../../docs/hyperbolic.md")] -use crate::bounded::{AtLeastOne, NonNegative, OpenUnitInterval, UnitInterval}; +use crate::bounded::OpenUnitInterval; use crate::error::{Error, Result}; -use crate::kernel::hyperbolic_vectoring; -use crate::ops::algebraic::sqrt_nonneg; -use crate::ops::exponential::ln_positive; +use crate::eval::{HUGE, W, expm1, fixed, log1p, sinh_cosh as sinh_cosh_work, tiered, work}; +use crate::real::Work; use crate::traits::CordicNumber; -/// The fractional part of 1.1182, the sum of the atanh table, which bounds the -/// polynomial's argument; I1F63 cannot hold 1.1182 itself. -const HYPERBOLIC_CONVERGENCE_LIMIT_FRAC_I1F63: i64 = 0x0F22_3D70_A3D7_0A3D; +/// `|x|` in the working type, and whether it is past [`HUGE`]. +fn magnitude(x: T) -> (W, bool) { + let a = work(x).abs(); + let huge = a > W::::from_int(HUGE); + (a, huge) +} -/// 0.75, above which atanh moves its argument into the CORDIC's convergent -/// range with `atanh(x) = atanh(0.5) + atanh((x − 0.5) / (1 − 0.5x))`. -const ATANH_REDUCTION_THRESHOLD_I1F63: i64 = 0x6000_0000_0000_0000; +/// `expm1(2|x|)`, from which tanh and coth follow without cancelling. +fn expm1_double(a: W) -> W { + expm1(a.ldexp(1)) +} /// Hyperbolic sine and cosine, cheaper than separate calls. Both saturate /// together for large `|x|`, as the [module docs](self) describe. @@ -23,60 +26,18 @@ const ATANH_REDUCTION_THRESHOLD_I1F63: i64 = 0x6000_0000_0000_0000; no_panic::no_panic )] pub fn sinh_cosh(x: T) -> (T, T) { - let one = T::one(); - let limit = one.saturating_add(T::from_i1f63(HYPERBOLIC_CONVERGENCE_LIMIT_FRAC_I1F63)); - - // Halve until within the limit, then double back; total_bits halvings - // reach zero, so the bound never cuts the loop short. - let mut reduced = x; - let mut depth: u32 = 0; - while reduced.abs() > limit && depth < T::total_bits() { - reduced = reduced >> 1; - depth += 1; - } - - // Factored Taylor series in u = x², with divisors (2k)(2k+1) for sinh and - // (2k−1)(2k) for cosh; docs/sinh_cosh.md gives the degrees and errors. - let u = reduced.saturating_mul(reduced); - - let mut sp = one; - let mut cp = one; - if T::frac_bits() >= 40 { - sp = one.saturating_add(u.div_int(420).saturating_mul(sp)); - sp = one.saturating_add(u.div_int(342).saturating_mul(sp)); - sp = one.saturating_add(u.div_int(272).saturating_mul(sp)); - sp = one.saturating_add(u.div_int(210).saturating_mul(sp)); - cp = one.saturating_add(u.div_int(462).saturating_mul(cp)); - cp = one.saturating_add(u.div_int(380).saturating_mul(cp)); - cp = one.saturating_add(u.div_int(306).saturating_mul(cp)); - cp = one.saturating_add(u.div_int(240).saturating_mul(cp)); + if magnitude(x).1 { + let s = if x.is_negative() { + T::min_value() + } else { + T::max_value() + }; + return (s, T::max_value()); } - if T::frac_bits() >= 24 { - sp = one.saturating_add(u.div_int(156).saturating_mul(sp)); - sp = one.saturating_add(u.div_int(110).saturating_mul(sp)); - cp = one.saturating_add(u.div_int(182).saturating_mul(cp)); - cp = one.saturating_add(u.div_int(132).saturating_mul(cp)); - } - sp = one.saturating_add(u.div_int(72).saturating_mul(sp)); - sp = one.saturating_add(u.div_int(42).saturating_mul(sp)); - sp = one.saturating_add(u.div_int(20).saturating_mul(sp)); - sp = one.saturating_add(u.div_int(6).saturating_mul(sp)); - cp = one.saturating_add(u.div_int(90).saturating_mul(cp)); - cp = one.saturating_add(u.div_int(56).saturating_mul(cp)); - cp = one.saturating_add(u.div_int(30).saturating_mul(cp)); - cp = one.saturating_add(u.div_int(12).saturating_mul(cp)); - cp = one.saturating_add(u.div_int(2).saturating_mul(cp)); - let (mut sh, mut ch) = (reduced.saturating_mul(sp), cp); - - // sinh(2x) = 2·sinh(x)·cosh(x) and cosh(2x) = cosh²(x) + sinh²(x). - for _ in 0..depth { - let new_sh = sh.saturating_mul(ch).saturating_mul(T::two()); - let new_ch = ch.saturating_mul(ch).saturating_add(sh.saturating_mul(sh)); - sh = new_sh; - ch = new_ch; - } - - (sh, ch) + tiered!(T, |v| { + let (s, c) = sinh_cosh_work(v(x).abs()); + (s.with_sign(x.is_negative()), c) + }) } /// Hyperbolic sine. Saturates to `T::MAX` or `T::MIN` past `±asinh(T::MAX)`; @@ -103,15 +64,22 @@ pub fn cosh(x: T) -> T { sinh_cosh(x).1 } -/// Hyperbolic tangent. Result in `(-1, 1)`. +/// Hyperbolic tangent. Result in `[-1, 1]`. #[must_use] #[cfg_attr( all(feature = "verify-no-panic", not(debug_assertions)), no_panic::no_panic )] pub fn tanh(x: T) -> T { - let (s, c) = sinh_cosh(x); - s.div(c) + if magnitude(x).1 { + return if x.is_negative() { -T::one() } else { T::one() }; + } + tiered!(T, |v| tanh_work(v(x))) +} + +fn tanh_work(x: V) -> V { + let u = expm1(x.abs().ldexp(1)); + (u / (u + V::from_int(2))).with_sign(x.is_neg()) } /// Hyperbolic cotangent. Domain: `x ≠ 0`. @@ -126,8 +94,15 @@ pub fn coth(x: T) -> Result { if x == T::zero() { return Err(Error::domain("coth", "non-zero value")); } - let (s, c) = sinh_cosh(x); - Ok(c.div(s)) + let (a, huge) = magnitude(x); + let one = W::::one(); + let c = if huge { + one + } else { + let u = expm1_double::(a); + (u + one.ldexp(1)) / u + }; + Ok(fixed(c.with_sign(x.is_negative()))) } /// Inverse hyperbolic sine. Accepts any value. @@ -137,36 +112,14 @@ pub fn coth(x: T) -> Result { no_panic::no_panic )] pub fn asinh(x: T) -> T { - let zero = T::zero(); - let one = T::one(); - - if x == zero { - return zero; - } - - let abs_x = x.abs(); - if abs_x <= one { - // asinh(x) = atanh(x / √(1 + x²)), whose argument is at most 1/√2. - let sqrt_term = sqrt_nonneg(NonNegative::one_plus_square(x)); - let arg = OpenUnitInterval::from_div_by_sqrt_one_plus_square(x, sqrt_term); - return atanh_open(arg); - } + tiered!(T, |v| asinh_work(v(x))) +} - // Past 1 that argument nears 1, where each atanh reduction step triples - // the error, so use sign(x)·ln(|x|·(1 + √(1 + (1/|x|)²))). - let recip = one.div(abs_x); - let factor = one.saturating_add(sqrt_nonneg(NonNegative::one_plus_square(recip))); - let magnitude = if abs_x > T::max_value() >> 2 { - // |x|·factor would overflow, as factor < 2.42. - ln_positive(abs_x).saturating_add(ln_positive(factor)) - } else { - ln_positive(abs_x.saturating_mul(factor)) - }; - if x.is_negative() { - -magnitude - } else { - magnitude - } +/// `asinh(z) = log1p(z + z²/(1 + √(1 + z²)))`, exact in form for any z. +fn asinh_work(x: V) -> V { + let (z, one) = (x.abs(), V::one()); + let z2 = z.square(); + log1p(z + z2 / (one + (one + z2).sqrt())).with_sign(x.is_neg()) } /// Inverse hyperbolic cosine. Domain: `x ≥ 1`. @@ -178,32 +131,16 @@ pub fn asinh(x: T) -> T { no_panic::no_panic )] pub fn acosh(x: T) -> Result { - let at_least_one = AtLeastOne::new(x).ok_or_else(|| Error::domain("acosh", "value >= 1"))?; - let zero = T::zero(); - let one = T::one(); - - if x == one { - return Ok(zero); - } - - if x < one.saturating_add(T::half()) { - // acosh(x) = atanh(√(x² − 1) / x), whose argument is under 0.75 here. - let sqrt_term = sqrt_nonneg(NonNegative::square_minus_one(at_least_one)); - let arg = OpenUnitInterval::from_sqrt_square_minus_one_div(sqrt_term, at_least_one); - return Ok(atanh_open(arg)); + if x < T::one() { + return Err(Error::domain("acosh", "value >= 1")); } + Ok(tiered!(T, |v| acosh_work(v(x)))) +} - // Past 1.5 that argument nears 1, where each atanh reduction step triples - // the error, so use ln(x·(1 + √(1 − (1/x)²))). - let recip = UnitInterval::from_reciprocal(at_least_one); - let factor = one.saturating_add(sqrt_nonneg(NonNegative::one_minus_square(recip))); - let result = if x > T::max_value() >> 1 { - // x·factor would overflow, as factor < 2. - ln_positive(x).saturating_add(ln_positive(factor)) - } else { - ln_positive(x.saturating_mul(factor)) - }; - Ok(result) +/// `acosh(1 + t) = log1p(t + √(t(t + 2)))`, with t exact. +fn acosh_work(x: V) -> V { + let t = x - V::one(); + log1p(t + (t * (t + V::from_int(2))).sqrt()) } /// Inverse hyperbolic tangent. Domain: `(-1, 1)`. @@ -228,45 +165,15 @@ pub fn atanh(x: T) -> Result { no_panic::no_panic )] pub fn atanh_open(x: OpenUnitInterval) -> T { - atanh_core(x.get()) + tiered!(T, |v| atanh_work(v(x.get()))) } -/// atanh for `|x| < 1`, which the caller must ensure. -fn atanh_core(x: T) -> T { - let zero = T::zero(); - let one = T::one(); - - if x == zero { - return zero; - } - - let threshold = T::from_i1f63(ATANH_REDUCTION_THRESHOLD_I1F63); - - if x.abs() <= threshold { - let (_, _, z) = hyperbolic_vectoring(one, x, zero); - return z; - } - - let half = T::half(); - let atanh_half = T::from_i1f63(crate::tables::hyperbolic::ATANH_HALF); - let sign = if x.is_negative() { -one } else { one }; - let mut abs_x = x.abs(); - - // Each step takes atanh(0.5) off and roughly halves the distance to the - // threshold, so frac_bits steps suffice. - let mut accumulated = zero; - let mut i: u32 = 0; - while abs_x > threshold && i < T::frac_bits() { - let numerator = abs_x.saturating_sub(half); - let denominator = one.saturating_sub(half.saturating_mul(abs_x)); - abs_x = numerator.div(denominator); - accumulated = accumulated.saturating_add(atanh_half); - i += 1; - } - - let (_, _, z) = hyperbolic_vectoring(one, abs_x, zero); - - sign.saturating_mul(accumulated.saturating_add(z)) +/// `atanh(a) = log1p(2a/(1 − a))/2`, with 1 − a exact. +fn atanh_work(x: V) -> V { + let a = x.abs(); + log1p((a / (V::one() - a)).ldexp(1)) + .ldexp(-1) + .with_sign(x.is_neg()) } /// Inverse hyperbolic cotangent. Domain: `|x| > 1`. @@ -278,12 +185,13 @@ fn atanh_core(x: T) -> T { no_panic::no_panic )] pub fn acoth(x: T) -> Result { - let one = T::one(); - - if x.abs() <= one { + let v = work(x); + let a = v.abs(); + let one = W::::one(); + if a <= one { return Err(Error::domain("acoth", "|value| > 1")); } - - let recip = one.div(x); - Ok(atanh_core(recip)) + // acoth(a) = log1p(2/(a − 1))/2, with a − 1 exact. + let r = log1p((one / (a - one)).ldexp(1)).ldexp(-1); + Ok(fixed(r.with_sign(v.is_neg()))) } diff --git a/src/ops/interval.rs b/src/ops/interval.rs new file mode 100644 index 0000000..c831744 --- /dev/null +++ b/src/ops/interval.rs @@ -0,0 +1,482 @@ +//! Intervals certain to hold the exact value, by [`ERROR_ULPS`](crate::ERROR_ULPS). +//! An end at `T::MIN` or `T::MAX` stands for everything past it. + +use crate::error::Result; +use crate::eval::{W, pi, stumpff as stumpff_work, turn_fraction, work}; +use crate::ops::algebraic::hypot; +use crate::ops::arith::sqrt_or_zero; +use crate::ops::circular::{atan, atan2, sin_cos}; +use crate::ops::exponential::{ + exp, exp_moments, exprel, exprel2, log1p, log1p_ratio, log1p_residual_ratio, +}; +use crate::ops::hyperbolic::tanh; +use crate::ops::ratio::{atan_ratio, atanh_ratio, stumpff, tan_ratio, tanh_ratio}; +use crate::ops::turns::sin_cos_turns; +use crate::real::Work; +use crate::traits::CordicNumber; + +/// `[y − ulp, y + ulp]`, which holds the exact value of any function giving `y`. +#[must_use] +#[cfg_attr( + all(feature = "verify-no-panic", not(debug_assertions)), + no_panic::no_panic +)] +pub fn enclose(y: T) -> (T, T) { + let ulp = T::from_raw_i128(1); + (y.saturating_sub(ulp), y.saturating_add(ulp)) +} + +/// `(lo, hi)` ordered. +fn ordered(a: T, b: T) -> (T, T) { + if a <= b { (a, b) } else { (b, a) } +} + +fn lesser(a: T, b: T) -> T { + ordered(a, b).0 +} + +fn greater(a: T, b: T) -> T { + ordered(a, b).1 +} + +/// `[a − ulp, b + ulp]` within `[floor, ceiling]`, holding exact values within an ulp of `a ≤ b`. +fn hull(a: T, b: T, floor: T, ceiling: T) -> (T, T) { + (greater(enclose(a).0, floor), lesser(enclose(b).1, ceiling)) +} + +/// `[f(lo) − ulp, f(hi) + ulp]` for a rising `f`, clamped to its range. +fn rising(f: fn(T) -> T, lo: T, hi: T, floor: T, ceiling: T) -> (T, T) { + let (lo, hi) = ordered(lo, hi); + hull(f(lo), f(hi), floor, ceiling) +} + +/// The range over `[lo, hi]`, ordered, of an even function monotone in `|x|`, 1 at 0, from its ends' values. +fn even( + (lo, hi): (T, T), + (at_lo, at_hi): (T, T), + rises: bool, + clamp: (T, T), +) -> (T, T) { + let zero = T::zero(); + let far = enclose( + if lo.raw_i128().unsigned_abs() > hi.raw_i128().unsigned_abs() { + at_lo + } else { + at_hi + }, + ); + let near = if lo <= zero && hi >= zero { + (T::one(), T::one()) + } else { + enclose(if lo > zero { at_lo } else { at_hi }) + }; + let (low, high) = if rises { + (near.0, far.1) + } else { + (far.0, near.1) + }; + (greater(low, clamp.0), lesser(high, clamp.1)) +} + +/// The greatest value of `T` at most `v`, then an ulp below it, for a `v` within an ulp. +fn below(v: W) -> T { + let (lo, hi) = (T::min_value().raw_i128(), T::max_value().raw_i128()); + T::from_raw_i128(v.to_raw(T::frac_bits(), lo, hi, true)).saturating_sub(T::from_raw_i128(1)) +} + +fn above(v: W) -> T { + T::zero().saturating_sub(below::(-v)) +} + +/// The range of `e^x` over `[lo, hi]`, rounded outward. +#[must_use] +#[cfg_attr( + all(feature = "verify-no-panic", not(debug_assertions)), + no_panic::no_panic +)] +pub fn exp_range(lo: T, hi: T) -> (T, T) { + rising(exp, lo, hi, T::zero(), T::max_value()) +} + +/// The range of `atan x` over `[lo, hi]`, rounded outward. +#[must_use] +#[cfg_attr( + all(feature = "verify-no-panic", not(debug_assertions)), + no_panic::no_panic +)] +pub fn atan_range(lo: T, hi: T) -> (T, T) { + rising(atan, lo, hi, T::min_value(), T::max_value()) +} + +/// The range of `tanh x` over `[lo, hi]`, rounded outward. +#[must_use] +#[cfg_attr( + all(feature = "verify-no-panic", not(debug_assertions)), + no_panic::no_panic +)] +pub fn tanh_range(lo: T, hi: T) -> (T, T) { + rising(tanh, lo, hi, -T::one(), T::one()) +} + +/// The ranges `((sin_min, sin_max), (cos_min, cos_max))` over `[lo, hi]`, +/// rounded outward; an extreme within the working error of an end counts. +#[must_use] +#[cfg_attr( + all(feature = "verify-no-panic", not(debug_assertions)), + no_panic::no_panic +)] +pub fn sin_cos_range(lo: T, hi: T) -> ((T, T), (T, T)) { + let (lo, hi) = ordered(lo, hi); + let span = (work(hi) - work(lo)) / pi::>().ldexp(1); + range_in_turns(span, turn_fraction(lo), sin_cos(lo), sin_cos(hi)) +} + +/// [`sin_cos_range`] over `[lo, hi]` in turns, each end reduced exactly. +#[must_use] +#[cfg_attr( + all(feature = "verify-no-panic", not(debug_assertions)), + no_panic::no_panic +)] +pub fn sin_cos_range_turns(lo: T, hi: T) -> ((T, T), (T, T)) { + let (lo, hi) = ordered(lo, hi); + let span = work(hi) - work(lo); + let start = lo.raw_i128().rem_euclid(T::one().raw_i128()); + let start = W::::from_raw(start, T::frac_bits()); + range_in_turns(span, start, sin_cos_turns(lo), sin_cos_turns(hi)) +} + +/// The ranges over `span` turns from `start` past a whole one, given the ends' sines and cosines. +fn range_in_turns( + span: W, + start: W, + (sl, cl): (T, T), + (sh, ch): (T, T), +) -> ((T, T), (T, T)) { + let one = T::one(); + if span >= W::::one() { + return ((-one, one), (-one, one)); + } + let slack = W::::one().ldexp(16 - W::::PREC.cast_signed()); + let reaches = |quarter: i64| { + let at = W::::from_int(quarter).ldexp(-2); + let ahead = at - start; + let ahead = if ahead.is_neg() { + ahead + W::::one() + } else { + ahead + }; + ahead <= span + slack || ahead >= W::::one() - slack + }; + let band = |a: T, b: T, top: bool, bottom: bool| { + let low = if bottom { + -one + } else { + greater(enclose(lesser(a, b)).0, -one) + }; + let high = if top { + one + } else { + lesser(enclose(greater(a, b)).1, one) + }; + (low, high) + }; + ( + band(sl, sh, reaches(1), reaches(3)), + band(cl, ch, reaches(0), reaches(2)), + ) +} + +/// The range of `e^x` past its first terms, [`exprel`], over `[lo, hi]`, rounded outward. +#[must_use] +#[cfg_attr( + all(feature = "verify-no-panic", not(debug_assertions)), + no_panic::no_panic +)] +pub fn exprel_range(lo: T, hi: T) -> (T, T) { + rising(exprel, lo, hi, T::zero(), T::max_value()) +} + +/// The range of [`exprel2`] over `[lo, hi]`, rounded outward. +#[must_use] +#[cfg_attr( + all(feature = "verify-no-panic", not(debug_assertions)), + no_panic::no_panic +)] +pub fn exprel2_range(lo: T, hi: T) -> (T, T) { + rising(exprel2, lo, hi, T::zero(), T::max_value()) +} + +/// The ranges of [`exp_moments`] over `[lo, hi]`, each falling, rounded outward. +#[must_use] +#[cfg_attr( + all(feature = "verify-no-panic", not(debug_assertions)), + no_panic::no_panic +)] +pub fn exp_moments_range(lo: T, hi: T) -> [(T, T); 6] { + let (lo, hi) = ordered(lo, hi); + let (low, high) = (exp_moments(hi), exp_moments(lo)); + let mut out = [(T::zero(), T::zero()); 6]; + for ((slot, a), b) in out.iter_mut().zip(low).zip(high) { + *slot = hull(a, b, T::zero(), T::max_value()); + } + out +} + +/// The ranges of [`log1p`] and its ratios over `[lo, hi]`, rounded outward. +fn log1p_family( + f: fn(T) -> Result, + lo: T, + hi: T, + rises: bool, +) -> Result<(T, T)> { + let (lo, hi) = ordered(lo, hi); + let (a, b) = (f(lo)?, f(hi)?); + let (min, max) = (T::min_value(), T::max_value()); + Ok(if rises { + hull(a, b, min, max) + } else { + hull(b, a, min, max) + }) +} + +/// The range of [`log1p`] over `[lo, hi]`, rounded outward. +/// # Errors +/// Returns `DomainError` unless both ends are above −1. +#[must_use = "returns the range which should be handled"] +#[cfg_attr( + all(feature = "verify-no-panic", not(debug_assertions)), + no_panic::no_panic +)] +pub fn log1p_range(lo: T, hi: T) -> Result<(T, T)> { + log1p_family(log1p, lo, hi, true) +} + +/// The range of [`log1p_ratio`], which falls, over `[lo, hi]`, rounded outward. +/// # Errors +/// Returns `DomainError` unless both ends are above −1. +#[must_use = "returns the range which should be handled"] +#[cfg_attr( + all(feature = "verify-no-panic", not(debug_assertions)), + no_panic::no_panic +)] +pub fn log1p_ratio_range(lo: T, hi: T) -> Result<(T, T)> { + log1p_family(log1p_ratio, lo, hi, false) +} + +/// The range of [`log1p_residual_ratio`], which rises, over `[lo, hi]`, rounded outward. +/// # Errors +/// Returns `DomainError` unless both ends are above −1. +#[must_use = "returns the range which should be handled"] +#[cfg_attr( + all(feature = "verify-no-panic", not(debug_assertions)), + no_panic::no_panic +)] +pub fn log1p_residual_ratio_range(lo: T, hi: T) -> Result<(T, T)> { + log1p_family(log1p_residual_ratio, lo, hi, true) +} + +/// The range of [`tanh_ratio`] over `[lo, hi]`, rounded outward. +#[must_use] +#[cfg_attr( + all(feature = "verify-no-panic", not(debug_assertions)), + no_panic::no_panic +)] +pub fn tanh_ratio_range(lo: T, hi: T) -> (T, T) { + let (lo, hi) = ordered(lo, hi); + even( + (lo, hi), + (tanh_ratio(lo), tanh_ratio(hi)), + false, + (T::zero(), T::one()), + ) +} + +/// The range of [`atan_ratio`] over `[lo, hi]`, rounded outward. +#[must_use] +#[cfg_attr( + all(feature = "verify-no-panic", not(debug_assertions)), + no_panic::no_panic +)] +pub fn atan_ratio_range(lo: T, hi: T) -> (T, T) { + let (lo, hi) = ordered(lo, hi); + even( + (lo, hi), + (atan_ratio(lo), atan_ratio(hi)), + false, + (T::zero(), T::one()), + ) +} + +/// The range of [`atanh_ratio`] over `[lo, hi]`, rounded outward. +/// # Errors +/// Returns `DomainError` unless both ends are in `(-1, 1)`. +#[must_use = "returns the range which should be handled"] +#[cfg_attr( + all(feature = "verify-no-panic", not(debug_assertions)), + no_panic::no_panic +)] +pub fn atanh_ratio_range(lo: T, hi: T) -> Result<(T, T)> { + let (lo, hi) = ordered(lo, hi); + let ends = (atanh_ratio(lo)?, atanh_ratio(hi)?); + Ok(even((lo, hi), ends, true, (T::one(), T::max_value()))) +} + +/// The range of `tan x / x` over `[lo, hi]`, rounded outward, everything if a pole may lie in it. +#[must_use] +#[cfg_attr( + all(feature = "verify-no-panic", not(debug_assertions)), + no_panic::no_panic +)] +pub fn tan_ratio_range(lo: T, hi: T) -> (T, T) { + let (lo, hi) = ordered(lo, hi); + let ((at_lo, cos_lo), (at_hi, cos_hi)) = (tan_ratio(lo), tan_ratio(hi)); + let zero = T::zero(); + // Between poles cos keeps its sign, and two poles are π apart. + if cos_lo == zero + || cos_hi == zero + || cos_lo.is_negative() != cos_hi.is_negative() + || work(hi) - work(lo) >= pi() + { + return (T::min_value(), T::max_value()); + } + let (min, max) = (T::min_value(), T::max_value()); + if lo <= zero && hi >= zero { + (T::one(), lesser(enclose(greater(at_lo, at_hi)).1, max)) + } else if hi <= zero { + hull(at_hi, at_lo, min, max) + } else { + hull(at_lo, at_hi, min, max) + } +} + +/// The range of `√x`, 0 for `x ≤ 0` like [`sqrt_or_zero`], over `[lo, hi]`, rounded outward. +#[must_use] +#[cfg_attr( + all(feature = "verify-no-panic", not(debug_assertions)), + no_panic::no_panic +)] +pub fn sqrt_range(lo: T, hi: T) -> (T, T) { + rising(sqrt_or_zero, lo, hi, T::zero(), T::max_value()) +} + +/// The end of `[lo, hi]` nearest 0, or 0 inside it, and the farthest. +fn near_far((a, b): (T, T)) -> (T, T) { + let (lo, hi) = ordered(a, b); + let zero = T::zero(); + let near = if lo > zero { + lo + } else if hi < zero { + hi + } else { + zero + }; + let far = if lo.raw_i128().unsigned_abs() > hi.raw_i128().unsigned_abs() { + lo + } else { + hi + }; + (near, far) +} + +/// The range of [`hypot`] over the box `x × y`, rounded outward. +#[must_use] +#[cfg_attr( + all(feature = "verify-no-panic", not(debug_assertions)), + no_panic::no_panic +)] +pub fn hypot_range(x: (T, T), y: (T, T)) -> (T, T) { + let ((nx, fx), (ny, fy)) = (near_far(x), near_far(y)); + hull(hypot(nx, ny), hypot(fx, fy), T::zero(), T::max_value()) +} + +/// The range of [`atan2`] over the box `y × x`, rounded outward: `[−π, π]` when the box +/// holds the origin or crosses the negative x-axis, else its corners' angles. +#[must_use] +#[cfg_attr( + all(feature = "verify-no-panic", not(debug_assertions)), + no_panic::no_panic +)] +pub fn atan2_range(y: (T, T), x: (T, T)) -> (T, T) { + let ((ylo, yhi), (xlo, xhi)) = (ordered(y.0, y.1), ordered(x.0, x.1)); + let zero = T::zero(); + let half_turn = above::(pi()); + let spans_zero = ylo <= zero && yhi >= zero; + if spans_zero && (xlo < zero && ylo < zero || xhi >= zero && xlo <= zero) { + return (-half_turn, half_turn); + } + let corners = [ + atan2(ylo, xlo), + atan2(ylo, xhi), + atan2(yhi, xlo), + atan2(yhi, xhi), + ]; + let low = corners.iter().copied().fold(T::max_value(), lesser); + let high = corners.iter().copied().fold(T::min_value(), greater); + hull(low, high, -half_turn, half_turn) +} + +/// The ranges of [`stumpff`]'s `[c₀, c₁, c₂, c₃]` over `[lo, hi]`, rounded outward. +/// +/// Each falls to `z = 4`, `c₃` everywhere; past it `|cₖ''| ≤ 2/(k + 4)!` rules an extreme out from +/// the slopes at the ends, or puts it within `2/(k + 4)!·w²/8` of them, `|c₀| ≤ 1`, +/// `|c₁| ≤ 1/√(1 + z)` and `0 ≤ c₂ ≤ 2/(4 + z)`. +#[must_use] +#[cfg_attr( + all(feature = "verify-no-panic", not(debug_assertions)), + no_panic::no_panic +)] +pub fn stumpff_range(lo: T, hi: T) -> [(T, T); 4] { + let (lo, hi) = ordered(lo, hi); + let (at_lo, at_hi) = (stumpff(lo), stumpff(hi)); + let pick = |c: [T; 4], k: usize| c.get(k).copied().unwrap_or_else(T::zero); + let mut out = [0, 1, 2, 3].map(|k| { + hull( + pick(at_hi, k), + pick(at_lo, k), + T::min_value(), + T::max_value(), + ) + }); + let (four, top) = (W::::from_int(4), work(hi)); + if top <= four { + return out; + } + let past = work(lo) >= four; + let start = work(lo).max(four); + let (from, to) = (stumpff_work(start), stumpff_work(top)); + let width = top - start; + let at = |c: &[W; 4], k: usize| c.get(k).copied().unwrap_or_else(W::::zero); + // 2z·cₖ' = cₖ₋₁ − k·cₖ, and c₀' = −c₁/2. + let slope = |c: &[W; 4], z: W, k: usize| match k { + 0 => -at(c, 1).ldexp(-1), + 1 => (at(c, 0) - at(c, 1)) / z.ldexp(1), + _ => (at(c, 1) - at(c, 2).ldexp(1)) / z.ldexp(1), + }; + let slack = W::::one().ldexp(16 - W::::PREC.cast_signed()); + for (k, (slot, factorial)) in out.iter_mut().zip([24, 120, 720]).enumerate() { + let bend = W::::from_int(2).div_int(factorial); + let first = if past { + enclose(pick(at_lo, k)) + } else { + (below::(at(&from, k)), above::(at(&from, k))) + }; + let last = enclose(pick(at_hi, k)); + let (mut low, mut high) = (lesser(first.0, last.0), greater(first.1, last.1)); + if slope(&from, start, k).abs() + slope(&to, top, k).abs() <= bend * width + slack { + let spread = above::(bend * width.square().ldexp(-3)); + // c₁'s extremes past 4 are cos s at tan s = s, at least −0.21724, at most 0.1284. + let fall = |c: i64| W::::from_int(c).div_int(10_000); + let root = W::::one() / (W::::one() + start).sqrt(); + let (floor, peak) = match k { + 0 => (-W::::one(), W::::one()), + 1 => (-root.min(fall(2173)), root.min(fall(1284))), + _ => (W::::zero(), W::::from_int(2) / (four + start)), + }; + low = greater(low.saturating_sub(spread), lesser(low, below::(floor))); + high = lesser(high.saturating_add(spread), greater(high, above::(peak))); + } + // Below 4 it falls from its value at `lo`. + *slot = (low, if past { high } else { greater(high, slot.1) }); + } + out +} diff --git a/src/ops/mod.rs b/src/ops/mod.rs index f13260e..b260429 100644 --- a/src/ops/mod.rs +++ b/src/ops/mod.rs @@ -1,11 +1,42 @@ //! The mathematical functions, each also re-exported at the crate root. pub mod algebraic; +pub mod arith; pub mod circular; +pub mod complex; +pub mod elliptic; pub mod exponential; pub mod hyperbolic; +pub mod interval; +pub mod ratio; +pub mod solve; +pub mod special; +pub mod turns; -pub use algebraic::sqrt; -pub use circular::{acos, asin, atan, atan2, cos, sin, sin_cos, tan}; -pub use exponential::{exp, ln, log2, log10, pow, pow2}; +pub use algebraic::{hypot, norm, normalize, rsqrt, scale_by, scale_exponent, sqrt, sqrt_product}; +pub use arith::{ + Rounding, div_or_saturate, div_rounded, dot, dot_rounded, mul_add, mul_add_rounded, mul_div, + mul_div_rounded, mul_rounded, sqrt_or_zero, +}; +pub use circular::{acos, asin, asin_from_gap, atan, atan2, cos, sin, sin_cos, tan}; +pub use complex::Complex; +pub use elliptic::{ + Modulus, carlson_rc, carlson_rd, carlson_rf, carlson_rf_roots, carlson_rj, carlson_rj_roots, + ellip_e, ellip_k, ellip_pi, ellip_pi_complete, ellip_pi_increment, jacobi_am, jacobi_inverse, + jacobi_sn_cn_dn, +}; +pub use exponential::{ + exp, exp_moments, expm1, exprel, exprel2, ln, log1p, log1p_ratio, log1p_residual, + log1p_residual_ratio, log2, log10, pow, pow2, +}; pub use hyperbolic::{acosh, acoth, asinh, atanh, cosh, coth, sinh, sinh_cosh, tanh}; +pub use interval::{ + atan_range, atan_ratio_range, atan2_range, atanh_ratio_range, enclose, exp_moments_range, + exp_range, exprel_range, exprel2_range, hypot_range, log1p_range, log1p_ratio_range, + log1p_residual_ratio_range, sin_cos_range, sin_cos_range_turns, sqrt_range, stumpff_range, + tan_ratio_range, tanh_range, tanh_ratio_range, +}; +pub use ratio::{atan_ratio, atanh_ratio, stumpff, tan_ratio, tanh_ratio}; +pub use solve::{kepler_universal, root_in}; +pub use special::{erf, erfc, fresnel, gamma}; +pub use turns::{atan2_turns, div_by_tau, mul_by_tau, sin_cos_turns, wrap_principal, wrap_radians}; diff --git a/src/ops/ratio.rs b/src/ops/ratio.rs new file mode 100644 index 0000000..085ab57 --- /dev/null +++ b/src/ops/ratio.rs @@ -0,0 +1,105 @@ +//! Ratios finite where their parts vanish, to full relative precision near 0. + +use crate::bounded::OpenUnitInterval; +use crate::error::{Error, Result}; +use crate::eval::{ + HUGE, atan_ratio as atan_of_ratio, beyond, expm1, log1p, sin_cos, stumpff as stumpff_work, + tiered, +}; +use crate::real::Work; +use crate::traits::CordicNumber; + +/// The Stumpff functions `[c₀, c₁, c₂, c₃]`, `cₖ(z) = Σ (−z)ⁿ/(2n + k)!`, of either sign. +#[must_use] +#[cfg_attr( + all(feature = "verify-no-panic", not(debug_assertions)), + no_panic::no_panic +)] +pub fn stumpff(z: T) -> [T; 4] { + tiered!(T, |v| stumpff_work(v(z))) +} + +/// `(tan x / x, cos x)`, the first 1 at 0 and saturating at the poles. +#[must_use] +#[cfg_attr( + all(feature = "verify-no-panic", not(debug_assertions)), + no_panic::no_panic +)] +pub fn tan_ratio(x: T) -> (T, T) { + tiered!(T, |v| tan_ratio_work(v(x))) +} + +fn tan_ratio_work(v: V) -> (V, V) { + let (s, c) = sin_cos(v); + let ratio = if v.is_zero() { + V::one() + } else if c.is_zero() { + beyond::().with_sign(s.is_neg() != v.is_neg()) + } else { + s / (v * c) + }; + (ratio, c) +} + +/// `atan z / z`, which is 1 at 0. +#[must_use] +#[cfg_attr( + all(feature = "verify-no-panic", not(debug_assertions)), + no_panic::no_panic +)] +pub fn atan_ratio(z: T) -> T { + if z == T::zero() { + return T::one(); + } + tiered!(T, |v| atan_ratio_work(v(z))) +} + +fn atan_ratio_work(z: V) -> V { + let a = z.abs(); + atan_of_ratio(a, V::one()) / a +} + +/// `atanh z / z`, which is 1 at 0. Domain: `(-1, 1)`. +/// # Errors +/// Returns `DomainError` if `|z| ≥ 1`. +#[must_use = "returns the ratio which should be handled"] +#[cfg_attr( + all(feature = "verify-no-panic", not(debug_assertions)), + no_panic::no_panic +)] +pub fn atanh_ratio(z: T) -> Result { + if OpenUnitInterval::new(z).is_none() { + return Err(Error::domain("atanh_ratio", "value in range (-1, 1)")); + } + if z == T::zero() { + return Ok(T::one()); + } + Ok(tiered!(T, |v| atanh_ratio_work(v(z)))) +} + +fn atanh_ratio_work(z: V) -> V { + let a = z.abs(); + log1p((a / (V::one() - a)).ldexp(1)).ldexp(-1) / a +} + +/// `tanh x / x`, which is 1 at 0. +#[must_use] +#[cfg_attr( + all(feature = "verify-no-panic", not(debug_assertions)), + no_panic::no_panic +)] +pub fn tanh_ratio(x: T) -> T { + if x == T::zero() { + return T::one(); + } + tiered!(T, |v| tanh_ratio_work(v(x))) +} + +fn tanh_ratio_work(x: V) -> V { + let a = x.abs(); + if a > V::from_int(HUGE) { + return V::one() / a; + } + let u = expm1(a.ldexp(1)); + u / ((u + V::from_int(2)) * a) +} diff --git a/src/ops/solve.rs b/src/ops/solve.rs new file mode 100644 index 0000000..014f486 --- /dev/null +++ b/src/ops/solve.rs @@ -0,0 +1,189 @@ +//! A bracketed root finder, and the universal-variable Kepler equation. + +use crate::error::{Error, Result}; +use crate::eval::{HUGE, W, beyond, stumpff, tiered, work}; +use crate::real::Work; +use crate::traits::CordicNumber; + +/// A bracket `(a, b)` around a root of `f` in `[lo, hi]`, where `f(lo)` and `f(hi)` differ in sign. +/// +/// ITP (Oliveira and Takahashi, 2020) is never slower than bisection: from `f(lo)` and `f(hi)` +/// at most `budget` more calls make `a` and `b`, signed like them, adjacent; an exact 0 ends both. +/// # Errors +/// Returns `DomainError` if `f(lo)` and `f(hi)` are non-zero with one sign. +#[must_use = "returns the bracket which should be handled"] +#[cfg_attr( + all(feature = "verify-no-panic", not(debug_assertions)), + no_panic::no_panic +)] +pub fn root_in T>( + mut func: F, + lo: T, + hi: T, + budget: u32, +) -> Result<(T, T)> { + let (lo, hi) = if lo <= hi { (lo, hi) } else { (hi, lo) }; + let (mut ya, mut yb) = (func(lo), func(hi)); + let zero = T::zero(); + if ya == zero { + return Ok((lo, lo)); + } + if yb == zero { + return Ok((hi, hi)); + } + if ya.is_negative() == yb.is_negative() { + return Err(Error::domain( + "root_in", + "f(lo) and f(hi) of opposite signs", + )); + } + let (mut a, mut b) = (lo.raw_i128(), hi.raw_i128()); + let at = |raw: i128| W::::from_raw(raw, 0); + let width = |from: i128, to: i128| to.abs_diff(from); + // ITP with κ₁ = 0.2/(b₀ − a₀), κ₂ = 2, n₀ = 1 and ε half an ulp. + let start = at(b) - at(a); + let kappa = W::::one().div_int(5) / start; + let n_max = 129 - width(a, b).leading_zeros().min(128); + for j in 0..budget { + let span = width(a, b); + if span <= 1 { + break; + } + let half = a.saturating_add_unsigned(span / 2); + let (wa, wb, wh) = (at(a), at(b), at(half)); + let (fa, fb) = (work(ya), work(yb)); + let falsi = wa + (wb - wa) * (fa / (fa - fb)); + let toward = wh - falsi; + let delta = kappa * (wb - wa).square(); + let truncated = if delta <= toward.abs() { + falsi + delta.with_sign(toward.is_neg()) + } else { + wh + }; + let reach = + W::::one().ldexp(n_max.saturating_sub(j + 1).cast_signed()) - (wb - wa).ldexp(-1); + let x = if (truncated - wh).abs() <= reach { + truncated + } else { + wh - reach.with_sign(toward.is_neg()) + }; + let raw = x.to_raw(0, a.saturating_add(1), b.saturating_sub(1), true); + let point = T::from_raw_i128(raw); + let y = func(point); + if y == zero { + return Ok((point, point)); + } + if y.is_negative() == ya.is_negative() { + (a, ya) = (raw, y); + } else { + (b, yb) = (raw, y); + } + } + Ok((T::from_raw_i128(a), T::from_raw_i128(b))) +} + +fn kepler_residual(chi: V, r0: V, sigma0: V, alpha: V, tau: V) -> (V, V) { + let x2 = chi.square(); + let z = alpha * x2; + // Past HUGE² e^√|z| swamps every term, its coefficient on a real orbit of χ's sign. + let huge = V::from_int(HUGE); + if z < -huge.square() { + return (beyond::().with_sign(chi.is_neg()), beyond()); + } + let [c0, c1, c2, c3] = stumpff(z); + let beta = V::one() - alpha * r0; + let f = r0 * chi + sigma0 * x2 * c2 + beta * x2 * chi * c3 - tau; + let r = r0 * c0 + sigma0 * chi * c1 + x2 * c2; + (f, r) +} + +/// The universal anomaly `χ` after time `t` on a two-body orbit. +/// +/// It solves `√μ·t = r₀χ + σ₀χ²c₂(αχ²) + (1 − αr₀)χ³c₃(αχ²)`, rising in `χ`, for `r₀ = |r⃗₀|`, +/// `σ₀ = r⃗₀·v⃗₀/√μ`, `α = 2/r₀ − v₀²/μ` and `τ = √μ·t`, by at most 256 steps of safeguarded Newton. +/// # Errors +/// Returns `DomainError` unless `r₀ > 0` and `r₀(2 − αr₀) ≥ σ₀²`, a real orbit. +#[must_use = "returns the anomaly which should be handled"] +#[cfg_attr( + all(feature = "verify-no-panic", not(debug_assertions)), + no_panic::no_panic +)] +pub fn kepler_universal(r0: T, sigma0: T, alpha: T, tau: T) -> Result { + let (r, s, a) = (work(r0), work(sigma0), work(alpha)); + let two = W::::from_int(2); + let h2 = W::::dot2(r, two - a * r, -s, s); + let slack = (r.ldexp(1) + s.square()).ldexp(8 - W::::PREC.cast_signed()); + if !r.is_neg() && !r.is_zero() && h2 >= -slack { + return Ok(tiered!(T, |v| kepler_work( + v(r0), + v(sigma0), + v(alpha), + v(tau) + ))); + } + Err(Error::domain( + "kepler_universal", + "r0 > 0 and r0 (2 - alpha r0) >= sigma0^2", + )) +} + +fn kepler_work(r0: V, sigma0: V, alpha: V, tau: V) -> V { + if tau.is_zero() { + return V::zero(); + } + let residual = |chi: V| kepler_residual(chi, r0, sigma0, alpha, tau); + let neg = tau.is_neg(); + let past = |m: V| { + let (f, _) = residual(m.with_sign(neg)); + f.is_zero() || f.is_neg() == neg + }; + let start = (tau / r0).abs(); + let (mut near, mut far) = (start, start); + if past(start) { + for _ in 0..1200 { + near = far.ldexp(-1); + if !past(near) { + break; + } + far = near; + } + } else { + for _ in 0..1200 { + far = near.ldexp(1); + if past(far) { + break; + } + near = far; + } + } + let (mut lo, mut hi) = if neg { (-far, -near) } else { (near, far) }; + let mut chi = (lo + hi).ldexp(-1); + let mut stride = hi - lo; + for _ in 0..256 { + let (f, slope) = residual(chi); + if f.is_zero() { + break; + } + if f.is_neg() { + lo = chi; + } else { + hi = chi; + } + // Bisection where Newton leaves the bracket or steps over half its last (rtsafe). + let step = f / slope; + let newton = chi - step; + let slow = step.abs().ldexp(1) > stride.abs(); + let next = if slow || slope.is_neg() || slope.is_zero() || newton <= lo || newton >= hi { + (lo + hi).ldexp(-1) + } else { + newton + }; + stride = next - chi; + let settled = V::negligible(&(next - chi), &chi.abs().max(lo.abs()).max(hi.abs())); + chi = next; + if settled || next == lo || next == hi { + break; + } + } + chi +} diff --git a/src/ops/special.rs b/src/ops/special.rs new file mode 100644 index 0000000..e5ea7db --- /dev/null +++ b/src/ops/special.rs @@ -0,0 +1,72 @@ +//! The error functions and the Fresnel integrals of a real argument. + +use crate::consts::FRAC_2_SQRT_PI; +use crate::cx::{Cx, erf as erf_work, faddeeva}; +use crate::error::{Error, Result}; +use crate::eval::{W, fixed, work}; +use crate::real::Work; +use crate::traits::CordicNumber; + +/// `erfc x` for `x ≥ 0`: `e^(−x²)·w(ix)`, which is real, past ½, else `1 − erf x`. +fn erfc_positive(x: V) -> V { + if x <= V::one().ldexp(-1) { + return V::one() - erf_work(Cx::real(x)).re; + } + crate::eval::exp_wide(-(x.square())) * faddeeva(Cx::new(V::zero(), x)).re +} + +/// The error function `erf x`, in `[−1, 1]`. +#[must_use] +#[cfg_attr( + all(feature = "verify-no-panic", not(debug_assertions)), + no_panic::no_panic +)] +pub fn erf(x: T) -> T { + fixed(erf_work(Cx::real(work(x))).re) +} + +/// The complementary error function `erfc x = 1 − erf x`, in `[0, 2]`. +#[must_use] +#[cfg_attr( + all(feature = "verify-no-panic", not(debug_assertions)), + no_panic::no_panic +)] +pub fn erfc(x: T) -> T { + let v = work(x); + let c = erfc_positive(v.abs()); + fixed(if v.is_neg() { + W::::from_int(2) - c + } else { + c + }) +} + +/// The Fresnel integrals `C(x) = ∫₀ˣ cos(πt²/2) dt` and `S(x) = ∫₀ˣ sin(πt²/2) dt`. +#[must_use] +#[cfg_attr( + all(feature = "verify-no-panic", not(debug_assertions)), + no_panic::no_panic +)] +pub fn fresnel(x: T) -> (T, T) { + let v = work(x); + let scale = v / W::::constant(&FRAC_2_SQRT_PI); + let e = erf_work(Cx::new(scale, -scale)); + let half = Cx::new(W::::one(), W::::one()).ldexp(-1); + let cs = half * e; + (fixed(cs.re), fixed(cs.im)) +} + +/// The gamma function `Γ(x)` for `x > 0`, saturating as it grows. +/// # Errors +/// Returns `DomainError` if `x ≤ 0`. +#[must_use = "returns the gamma function which should be handled"] +#[cfg_attr( + all(feature = "verify-no-panic", not(debug_assertions)), + no_panic::no_panic +)] +pub fn gamma(x: T) -> Result { + if x <= T::zero() { + return Err(Error::domain("gamma", "positive value")); + } + Ok(fixed(crate::cx::gamma(work(x)))) +} diff --git a/src/ops/turns.rs b/src/ops/turns.rs new file mode 100644 index 0000000..03e8441 --- /dev/null +++ b/src/ops/turns.rs @@ -0,0 +1,126 @@ +//! Angles in turns, and radians wrapped by exact multiples of 2π. + +use crate::consts::FRAC_1_2PI; +use crate::eval::{W, fixed, pi, sin_cos_octant, tiered, turn_fraction, work}; +use crate::real::Work; +use crate::traits::CordicNumber; + +/// Sine and cosine of `t` turns, `t·2π` radians, of any magnitude. +#[must_use] +#[cfg_attr( + all(feature = "verify-no-panic", not(debug_assertions)), + no_panic::no_panic +)] +pub fn sin_cos_turns(t: T) -> (T, T) { + if crate::fast::applies::() { + let raw = t.raw_i128(); + let (s, c) = crate::fast::sin_cos(raw.unsigned_abs(), T::frac_bits(), false); + let s = if raw < 0 { -s } else { s }; + return (crate::fast::round(s), crate::fast::round(c)); + } + tiered!(T, |v| turns_work(t, v)) +} + +fn turns_work(t: T, _as_work: fn(T) -> V) -> (V, V) { + let raw = t.raw_i128(); + let (oct, w) = V::octant(raw.unsigned_abs(), T::frac_bits(), false); + let (s, c) = sin_cos_octant(oct, w); + (if raw < 0 { -s } else { s }, c) +} + +/// The four-quadrant arctangent in turns, `atan2(y, x)/2π` in `(−½, ½]`, rounded once. +#[must_use] +#[cfg_attr( + all(feature = "verify-no-panic", not(debug_assertions)), + no_panic::no_panic +)] +pub fn atan2_turns(y: T, x: T) -> T { + tiered!(T, |v| atan2_turns_work(v(y), v(x))) +} + +fn atan2_turns_work(y: V, x: V) -> V { + crate::cx::atan2(y, x) * V::constant(&FRAC_1_2PI) +} + +/// `x/τ`, radians to turns, rounded once. +#[must_use] +#[cfg_attr( + all(feature = "verify-no-panic", not(debug_assertions)), + no_panic::no_panic +)] +pub fn div_by_tau(x: T) -> T { + tiered!(T, |v| by_constant(v(x), &FRAC_1_2PI, 0)) +} + +fn by_constant(x: V, c: &crate::consts::C, k: i32) -> V { + x * V::constant(c).ldexp(k) +} + +/// `x·τ`, turns to radians, rounded once. +#[must_use] +#[cfg_attr( + all(feature = "verify-no-panic", not(debug_assertions)), + no_panic::no_panic +)] +pub fn mul_by_tau(x: T) -> T { + tiered!(T, |v| by_constant(v(x), &crate::consts::PI, 1)) +} + +/// `r` rounded into `[low, high]`, else to the end nearer it modulo 2π. +fn round_into(r: W, low: W, high: W) -> T { + let t = fixed::(r); + if work(t) >= low && work(t) <= high { + return t; + } + let tau = pi::>().ldexp(1); + let ulp = T::from_raw_i128(1); + // The values of T just inside each end. + let first = fixed::(low); + let first = if work(first) < low { + first.saturating_add(ulp) + } else { + first + }; + let last = fixed::(high); + let last = if work(last) > high { + last.saturating_sub(ulp) + } else { + last + }; + let away = |c: T| { + let d = (work(c) - r).abs(); + if d > tau.ldexp(-1) { tau - d } else { d } + }; + if away(first) <= away(last) { + first + } else { + last + } +} + +/// `x` less the multiple of 2π that leaves it in `[0, 2π)`. +#[must_use] +#[cfg_attr( + all(feature = "verify-no-panic", not(debug_assertions)), + no_panic::no_panic +)] +pub fn wrap_radians(x: T) -> T { + let tau = pi::>().ldexp(1); + let below = tau - tau.ldexp(-(W::::PREC.cast_signed())); + round_into::(turn_fraction(x) * tau, W::::zero(), below) +} + +/// `x` less the multiple of 2π that leaves it in `(−π, π]`. +#[must_use] +#[cfg_attr( + all(feature = "verify-no-panic", not(debug_assertions)), + no_panic::no_panic +)] +pub fn wrap_principal(x: T) -> T { + let one = W::::one(); + let u = turn_fraction(x); + let u = if u > one.ldexp(-1) { u - one } else { u }; + let half = pi::>(); + let above = -half + half.ldexp(-(W::::PREC.cast_signed())); + round_into::(u * half.ldexp(1), above, half) +} diff --git a/src/real.rs b/src/real.rs new file mode 100644 index 0000000..4e8690f --- /dev/null +++ b/src/real.rs @@ -0,0 +1,1469 @@ +//! The arithmetic every function evaluates in: binary floating point with an +//! `N`-limb mantissa, rounded to nearest even each step, 1.5 to 8 times a type's width. + +use crate::consts::{C, FRAC_1_2PI, FRAC_1_2PI_LONG}; +use crate::series::Series; +use core::cmp::Ordering; +use core::fmt::Debug; +use core::ops::{Add, Div, Mul, Neg, Sub}; + +const SCRATCH: usize = 10; +type Scratch = [u64; SCRATCH]; + +fn limb(a: &[u64], i: usize) -> u64 { + a.get(i).copied().unwrap_or(0) +} + +fn set(a: &mut [u64], i: usize, v: u64) { + if let Some(x) = a.get_mut(i) { + *x = v; + } +} + +fn lo64(x: u128) -> u64 { + u64::try_from(x & u128::from(u64::MAX)).unwrap_or(0) +} + +fn hi64(x: u128) -> u64 { + u64::try_from(x >> 64).unwrap_or(0) +} + +fn u32_of(x: usize) -> u32 { + u32::try_from(x).unwrap_or(u32::MAX) +} + +fn i32_of(x: i64) -> i32 { + i32::try_from(x.clamp(i64::from(i32::MIN / 4), i64::from(i32::MAX / 4))).unwrap_or(0) +} + +fn shl(limbs: &mut [u64], len: usize, shift: u32) { + let words = (shift / 64) as usize; + let bits = shift % 64; + for i in (0..len).rev() { + let src = if i >= words { + limb(limbs, i - words) + } else { + 0 + }; + let below = if i > words { + limb(limbs, i - words - 1) + } else { + 0 + }; + let word = if bits == 0 { + src + } else { + (src << bits) | (below >> (64 - bits)) + }; + set(limbs, i, word); + } +} + +/// Shifts right by `s`, returning whether a set bit fell off the end. +fn shr(limbs: &mut [u64], len: usize, shift: u32) -> bool { + let words = (shift / 64) as usize; + let bits = shift % 64; + let mut sticky = (0..words.min(len)).any(|i| limb(limbs, i) != 0); + if words < len && bits > 0 { + sticky |= limb(limbs, words) << (64 - bits) != 0; + } + for i in 0..len { + let src = if i + words < len { + limb(limbs, i + words) + } else { + 0 + }; + let above = if i + words + 1 < len { + limb(limbs, i + words + 1) + } else { + 0 + }; + let word = if bits == 0 { + src + } else { + (src >> bits) | (above << (64 - bits)) + }; + set(limbs, i, word); + } + sticky +} + +fn leading_zeros(a: &Scratch, len: usize) -> u32 { + for i in (0..len).rev() { + let v = limb(a, i); + if v != 0 { + return u32_of(len - 1 - i) * 64 + v.leading_zeros(); + } + } + u32_of(len) * 64 +} + +fn add_into(a: &mut Scratch, b: &Scratch, len: usize) -> bool { + let mut carry = false; + for i in 0..len { + let (s1, c1) = limb(a, i).overflowing_add(limb(b, i)); + let (s2, c2) = s1.overflowing_add(u64::from(carry)); + set(a, i, s2); + carry = c1 || c2; + } + carry +} + +fn sub_into(a: &mut Scratch, b: &Scratch, len: usize) { + let mut borrow = false; + for i in 0..len { + let (d1, b1) = limb(a, i).overflowing_sub(limb(b, i)); + let (d2, b2) = d1.overflowing_sub(u64::from(borrow)); + set(a, i, d2); + borrow = b1 || b2; + } +} + +/// `|v|·2^62` for a mantissa top limb whose value is `top · 2^(e − 64)`, `|v| < 2`. +fn q62(neg: bool, e: i32, top: u64) -> i64 { + let sh = 2 - i64::from(e); + let q = u32::try_from(sh) + .ok() + .and_then(|s| top.checked_shr(s)) + .map_or(0, u64::cast_signed); + if neg { -q } else { q } +} + +fn q126(neg: bool, e: i32, top: u128) -> i128 { + let sh = 2 - i64::from(e); + let q = u32::try_from(sh) + .ok() + .and_then(|s| top.checked_shr(s)) + .map_or(0, u128::cast_signed); + if neg { -q } else { q } +} + +/// `a·b·2^−126` truncated, for values below 2 in magnitude. +pub fn mul_q126(a: i128, b: i128) -> i128 { + let (ua, ub) = (a.unsigned_abs(), b.unsigned_abs()); + let (ah, al) = (ua >> 64, ua & u128::from(u64::MAX)); + let (bh, bl) = (ub >> 64, ub & u128::from(u64::MAX)); + let ll = al * bl; + let (mid, carry) = (al * bh).overflowing_add(ah * bl); + let (lo, c2) = ll.overflowing_add(mid << 64); + let hi = ah * bh + (mid >> 64) + (u128::from(carry) << 64) + u128::from(c2); + let r = ((hi << 2) | (lo >> 126)).cast_signed(); + if (a < 0) == (b < 0) { r } else { -r } +} + +/// A signed value with 190 fractional bits in three limbs, below 2 in magnitude. +type Q190 = (bool, [u64; 3]); + +fn q190(neg: bool, e: i32, m: &[u64]) -> Q190 { + let mut ext = [0; SCRATCH]; + for (i, &v) in m.iter().rev().take(3).rev().enumerate() { + set(&mut ext, i, v); + } + let s = 2 - i64::from(e); + match u32::try_from(s) { + Ok(s) if s < 192 => { + shr(&mut ext, 3, s); + } + _ => return (false, [0; 3]), + } + (neg, [limb(&ext, 0), limb(&ext, 1), limb(&ext, 2)]) +} + +fn mul_q190(a: Q190, b: Q190) -> Q190 { + let mut p = [0u64; 6]; + for (i, &x) in a.1.iter().enumerate() { + let mut carry: u128 = 0; + for (j, &y) in b.1.iter().enumerate() { + let cur = u128::from(limb(&p, i + j)) + u128::from(x) * u128::from(y) + carry; + set(&mut p, i + j, lo64(cur)); + carry = cur >> 64; + } + set(&mut p, i + 3, lo64(carry)); + } + let w = |i: usize| (limb(&p, i) >> 62) | (limb(&p, i + 1) << 2); + (a.0 != b.0, [w(2), w(3), w(4)]) +} + +fn split190(m: [u64; 3]) -> (u128, u64) { + ((u128::from(m[2]) << 64) | u128::from(m[1]), m[0]) +} + +fn join190((hi, lo): (u128, u64)) -> [u64; 3] { + [lo, lo64(hi), hi64(hi)] +} + +fn add_q190(a: Q190, b: Q190) -> Q190 { + let ((ah, al), (bh, bl)) = (split190(a.1), split190(b.1)); + if a.0 == b.0 { + let (lo, carry) = al.overflowing_add(bl); + return ( + a.0, + join190((ah.wrapping_add(bh).wrapping_add(u128::from(carry)), lo)), + ); + } + let ((xh, xl), (yh, yl), neg) = if (ah, al) < (bh, bl) { + ((bh, bl), (ah, al), b.0) + } else { + ((ah, al), (bh, bl), a.0) + }; + let (lo, borrow) = xl.overflowing_sub(yl); + ( + neg, + join190((xh.wrapping_sub(yh).wrapping_sub(u128::from(borrow)), lo)), + ) +} + +/// `2^70/(2i + 1)`, `1/f` in Q62 at the middle of `[i/128, (i + 1)/128)`, for `i` from 64. +const RECIP_SEEDS: [u64; 64] = { + let mut t = [0u64; 64]; + let mut rest: &mut [u64] = &mut t; + let mut i = 64u128; + while let [slot, tail @ ..] = rest { + *slot = low64((1u128 << 70) / (2 * i + 1)); + rest = tail; + i += 1; + } + t +}; + +/// `1/√f` in Q62 at the middle of `[i/128, (i + 1)/128)`, for `i` from 16. +const RSQRT_SEEDS: [u64; 112] = { + let mut t = [0u64; 112]; + let mut rest: &mut [u64] = &mut t; + let mut i = 16u128; + while let [slot, tail @ ..] = rest { + *slot = low64(((1u128 << 124) / (2 * i + 1)).isqrt() << 4); + rest = tail; + i += 1; + } + t +}; + +const fn low64(x: u128) -> u64 { + let b = x.to_le_bytes(); + u64::from_le_bytes([b[0], b[1], b[2], b[3], b[4], b[5], b[6], b[7]]) +} + +/// `2^128/t` to about 2^−57, `t` in `[2^63, 2^64)`: a seed, then four steps of `g ← g(2 − fg)`. +pub fn recip_seed(t: u64) -> u128 { + let index = usize::try_from(t >> 57).unwrap_or(64).saturating_sub(64); + let mut g = RECIP_SEEDS.get(index).copied().unwrap_or(1 << 62); + for _ in 0..4 { + let fg = lo64((u128::from(t) * u128::from(g)) >> 64); + g = lo64((u128::from(g) * u128::from((2u64 << 62).wrapping_sub(fg))) >> 62); + } + u128::from(g) << 2 +} + +/// `2^128/√x` to about 2^−57, `x` in `[2^125, 2^128)`: a seed, then four steps of `g ← g(3 − fg²)/2`. +fn rsqrt_seed(x: u128) -> u128 { + let s = hi64(x); + let index = usize::try_from(s >> 57).unwrap_or(16).saturating_sub(16); + let mut g = RSQRT_SEEDS.get(index).copied().unwrap_or(1 << 62); + for _ in 0..4 { + let g2 = hi64(u128::from(g) * u128::from(g)); + let fg2 = u128::from(s) * u128::from(g2); + let rest = hi64((3u128 << 124).wrapping_sub(fg2)); + g = lo64((u128::from(g) * u128::from(rest)) >> 61); + } + u128::from(g) << 2 +} + +/// `⌊√x⌋`, `x` in `[2^125, 2^128)`: `x·g/2^128`, a Newton correction by `g`, then exact steps. +fn isqrt_top(x: u128) -> u128 { + let g = lo64(rsqrt_seed(x) >> 2); + let estimate = (u128::from(hi64(x)) * u128::from(g)) >> 62; + let error = x + .cast_signed() + .wrapping_sub(estimate.wrapping_mul(estimate).cast_signed()); + let step = ((error >> 32) * i128::from(g)) >> 95; + let mut r = estimate + .saturating_add_signed(step) + .min(u128::from(u64::MAX)); + for _ in 0..4 { + if r * r > x { + r -= 1; + } else if (r + 1).checked_mul(r + 1).is_some_and(|v| v <= x) { + r += 1; + } + } + r +} + +type Wide = (u128, u128); + +/// `x >> s` and whether a set bit fell off, for any `s`. +const fn wide_shr((hi, lo): Wide, s: u32) -> (Wide, bool) { + match s { + 0 => ((hi, lo), false), + 1..=127 => ( + (hi >> s, (lo >> s) | (hi << (128 - s))), + lo << (128 - s) != 0, + ), + 128 => ((0, hi), lo != 0), + 129..=255 => ((0, hi >> (s - 128)), lo != 0 || hi << (256 - s) != 0), + _ => ((0, 0), hi != 0 || lo != 0), + } +} + +const fn wide_shl((hi, lo): Wide, s: u32) -> Wide { + match s { + 0 => (hi, lo), + 1..=127 => ((hi << s) | (lo >> (128 - s)), lo << s), + 128..=255 => (lo << (s - 128), 0), + _ => (0, 0), + } +} + +const fn wide_leading_zeros((hi, lo): Wide) -> u32 { + if hi == 0 { + 128 + lo.leading_zeros() + } else { + hi.leading_zeros() + } +} + +/// What the functions need from the working arithmetic, for the three widths. +pub trait Work: + Copy + + Debug + + Ord + + Add + + Sub + + Mul + + Div + + Neg +{ + /// Mantissa bits. + const PREC: u32; + fn zero() -> Self; + fn one() -> Self; + fn from_int(i: i64) -> Self; + /// `raw · 2^−frac`, exact for any raw value of the type it serves. + fn from_raw(raw: i128, frac: u32) -> Self; + fn constant(c: &C) -> Self; + /// `self · 2^frac` rounded to nearest even, or toward −∞, clamped to `[lo, hi]`. + fn to_raw(self, frac: u32, lo: i128, hi: i128, floor: bool) -> i128; + fn is_zero(&self) -> bool; + fn is_neg(&self) -> bool; + /// `⌊log₂|x|⌋ + 1`. + fn exponent(&self) -> i32; + #[must_use] + fn abs(self) -> Self; + #[must_use] + fn with_sign(self, neg: bool) -> Self; + /// `self · 2^k`. + #[must_use] + fn ldexp(self, k: i32) -> Self; + #[must_use] + fn square(self) -> Self; + #[must_use] + fn mul_int(self, k: i64) -> Self; + /// Division by a positive integer, correctly rounded. + #[must_use] + fn div_int(self, k: u64) -> Self; + #[must_use] + fn sqrt(self) -> Self; + /// For `mag · 2^−frac` turns, or radians when `radians`: the octant + /// `⌊8·frac(t)⌋` and the fraction of it, measured from the octant's near + /// end to π/4's multiple, so odd octants count back from their top. + fn octant(mag: u128, frac: u32, radians: bool) -> (u32, Self); + /// [`octant`](Self::octant) of `|self|` radians, below `2^PREC`. + fn octant_of(self) -> (u32, Self); + /// [`octant`](Self::octant) of `|a·b|` radians from the exact product, below `2^300`. + fn octant_of_product(a: Self, b: Self) -> (u32, Self); + /// `a·b + c·d` from the exact products, rounded once. + fn dot2(a: Self, b: Self, c: Self, d: Self) -> Self; + /// `Σ ±table[first + step·k]·u^k` for k from 0 to `n` by Horner's rule, + /// the signs alternating when `alternate`, for `|u| < 1` and terms whose + /// sum stays near 1, in fixed point with 62, 126 or 190 fractional bits. + fn horner(u: Self, table: &Series, first: u32, step: u32, n: u32, alternate: bool) -> Self; + /// `round(16·num/den)` from the top bits, for `0 ≤ num ≤ den`, `den > 0`. + fn sixteenths(num: Self, den: Self) -> i64; + /// Two [`horner`](Self::horner) sums of one `u`, from `firsts`, side by side. + fn horner2( + u: Self, + table: &Series, + firsts: [u32; 2], + step: u32, + n: u32, + alternate: bool, + ) -> (Self, Self) { + let [a, b] = firsts; + ( + Self::horner(u, table, a, step, n, alternate), + Self::horner(u, table, b, step, n, alternate), + ) + } + + fn round_int(self) -> i64 { + let r = self.to_raw(0, i128::from(i64::MIN), i128::from(i64::MAX), false); + i64::try_from(r).unwrap_or(0) + } + + fn floor_int(self) -> i64 { + let r = self.to_raw(0, i128::from(i64::MIN), i128::from(i64::MAX), true); + i64::try_from(r).unwrap_or(0) + } + + /// Whether `term` no longer moves `sum`. + fn negligible(term: &Self, sum: &Self) -> bool { + term.is_zero() + || term.exponent() < sum.exponent().saturating_sub(Self::PREC.cast_signed() + 2) + } +} + +/// A binary floating-point value `m · 2^(e − 64N)`, `m`'s top bit set unless zero. +#[derive(Clone, Copy, Debug)] +pub struct Real { + neg: bool, + e: i32, + m: [u64; N], +} + +impl Real { + fn top(&self) -> u64 { + self.m.last().copied().unwrap_or(0) + } + + /// Rounds the top `N` of `len` normalized limbs by the rest and `sticky`. + fn pack(neg: bool, e: i32, ext: &Scratch, len: usize, sticky: bool) -> Self { + let low = len.saturating_sub(N); + let mut m = [0u64; N]; + for (i, x) in m.iter_mut().enumerate() { + *x = limb(ext, low + i); + } + let (half, rest) = if low == 0 { + (false, sticky) + } else { + let g = limb(ext, low - 1); + let below = (0..low - 1).any(|i| limb(ext, i) != 0); + (g >> 63 == 1, g << 1 != 0 || below || sticky) + }; + let odd = m.first().is_some_and(|&x| x & 1 == 1); + let mut e = e; + if half && (rest || odd) { + let mut carry = true; + for x in &mut m { + let (v, c) = x.overflowing_add(u64::from(carry)); + *x = v; + carry = c; + } + if carry { + if let Some(t) = m.last_mut() { + *t = 1 << 63; + } + e = e.saturating_add(1); + } + } + Self { neg, e, m } + } + + /// The value `X · 2^(e − 64·len)` of the integer `X` in `len` limbs. + fn normalize(neg: bool, e: i32, ext: &mut Scratch, len: usize, sticky: bool) -> Self { + let z = leading_zeros(ext, len); + if z >= u32_of(len) * 64 { + return ::zero(); + } + shl(ext, len, z); + Self::pack(neg, e.saturating_sub(z.cast_signed()), ext, len, sticky) + } + + /// `mag · 2^(e − 128)`. + fn from_u128(neg: bool, mag: u128, e: i32) -> Self { + if N == 1 { + if mag == 0 { + return ::zero(); + } + let z = mag.leading_zeros(); + return Self::round128(neg, e.saturating_sub(z.cast_signed()), mag << z, false); + } + if mag == 0 { + return ::zero(); + } + let z = mag.leading_zeros(); + let top = mag << z; + let mut m = [0u64; N]; + set(&mut m, N - 1, hi64(top)); + set(&mut m, N.saturating_sub(2), lo64(top)); + Self { + neg, + e: e.saturating_sub(z.cast_signed()), + m, + } + } + + /// One limb: the value `x · 2^(e − 128)` of a normalized `x`, rounded. + fn round128(neg: bool, e: i32, x: u128, sticky: bool) -> Self { + let mut m = [0u64; N]; + let mut top = hi64(x); + let low = lo64(x); + let mut e = e; + if low >> 63 == 1 && (low << 1 != 0 || sticky || top & 1 == 1) { + top = top.wrapping_add(1); + if top == 0 { + top = 1 << 63; + e = e.saturating_add(1); + } + } + if let Some(t) = m.last_mut() { + *t = top; + } + Self { neg, e, m } + } + + fn add1(a: Self, b: Self) -> Self { + let ma = u128::from(a.top()) << 64; + let full = u128::from(b.top()) << 64; + let d = a.e.abs_diff(b.e); + let (mb, sticky) = if d < 128 { + let mb = full >> d; + (mb, mb << d != full) + } else { + (0, true) + }; + if a.neg == b.neg { + let (sum, carry) = ma.overflowing_add(mb); + if carry { + let x = (sum >> 1) | 1 << 127; + Self::round128(a.neg, a.e.saturating_add(1), x, sticky || sum & 1 == 1) + } else { + Self::round128(a.neg, a.e, sum, sticky) + } + } else { + let diff = ma - mb - u128::from(sticky); + if diff == 0 { + return ::zero(); + } + let z = diff.leading_zeros(); + Self::round128( + a.neg, + a.e.saturating_sub(z.cast_signed()), + diff << z, + sticky, + ) + } + } + + /// The exact product of the mantissas, in `2N` limbs. + fn product(&self, o: &Self) -> Scratch { + let mut p = [0; SCRATCH]; + for i in 0..N { + let mut carry: u128 = 0; + for j in 0..N { + let cur = u128::from(limb(&p, i + j)) + + u128::from(limb(&self.m, i)) * u128::from(limb(&o.m, j)) + + carry; + set(&mut p, i + j, lo64(cur)); + carry = cur >> 64; + } + set(&mut p, i + N, lo64(carry)); + } + p + } + + /// The octant and remainder of `mag · 2^−point` turns, or radians, `mag` in ≤ 3 limbs. + fn octant_limbs(mag: &[u64], point: u32, radians: bool) -> (u32, Self) { + let width = 64 * u32_of(N + 1); + let mut z = [0; SCRATCH]; + let point = if radians { + let skip = 6usize.saturating_sub(N + 3); + for (i, &cv) in FRAC_1_2PI.2.iter().skip(skip).enumerate() { + let mut carry: u128 = 0; + for (j, &mv) in mag.iter().enumerate() { + let cur = u128::from(limb(&z, i + j)) + u128::from(cv) * u128::from(mv) + carry; + set(&mut z, i + j, lo64(cur)); + carry = cur >> 64; + } + set(&mut z, i + mag.len(), lo64(carry)); + } + 386 + point - 64 * u32_of(skip) + } else { + for (i, &mv) in mag.iter().enumerate() { + set(&mut z, i, mv); + } + point + }; + let need = width + 3; + if point < need { + shl(&mut z, SCRATCH, need - point); + } else { + shr(&mut z, SCRATCH, point - need); + } + Self::octant_from(&z) + } + + /// [`Work::octant`] in two limbs, the fraction as a Q127 integer, 1 as `2^127`. + pub fn octant_q127(mag: u128, point: u32, radians: bool) -> (u32, u128) { + let mut z = [0; SCRATCH]; + let point = if radians { + let skip = 6usize.saturating_sub(N + 3); + for (i, &cv) in FRAC_1_2PI.2.iter().skip(skip).enumerate() { + let mut carry: u128 = 0; + for (j, mv) in [lo64(mag), hi64(mag)].into_iter().enumerate() { + let cur = u128::from(limb(&z, i + j)) + u128::from(cv) * u128::from(mv) + carry; + set(&mut z, i + j, lo64(cur)); + carry = cur >> 64; + } + set(&mut z, i + 2, lo64(carry)); + } + 386 + point - 64 * u32_of(skip) + } else { + set(&mut z, 0, lo64(mag)); + set(&mut z, 1, hi64(mag)); + point + }; + let need = 64 * u32_of(N + 1) + 3; + if point < need { + shl(&mut z, SCRATCH, need - point); + } else { + shr(&mut z, SCRATCH, point - need); + } + let oct = u32::try_from(limb(&z, N + 1) & 7).unwrap_or(0); + let top = (u128::from(limb(&z, N)) << 64) | u128::from(limb(&z, N - 1)); + let frac = if oct & 1 == 1 { + (1 << 127) - (top >> 1) + } else { + top >> 1 + }; + (oct, frac) + } + + /// The octant and remainder of turns shifted to put the octant at bit `64(N + 1)`. + fn octant_from(z: &[u64]) -> (u32, Self) { + let oct = u32::try_from(limb(z, N + 1) & 7).unwrap_or(0); + let mut v = [0; SCRATCH]; + for i in 0..=N { + set(&mut v, i, limb(z, i)); + } + if oct & 1 == 1 { + if v.iter().all(|&x| x == 0) { + return (oct, ::one()); + } + let mut full = [0; SCRATCH]; + sub_into(&mut full, &v, N + 1); + v = full; + } + (oct, Self::normalize(false, 0, &mut v, N + 1, false)) + } + + fn wide(&self) -> Wide { + let (m0, m1, m2) = (limb(&self.m, 0), limb(&self.m, 1), limb(&self.m, 2)); + if N == 2 { + ((u128::from(m1) << 64) | u128::from(m0), 0) + } else { + ( + (u128::from(m2) << 64) | u128::from(m1), + u128::from(m0) << 64, + ) + } + } + + /// `x·2^(e − 256)`, `x`'s top bit set, rounded to nearest even, `sticky` for bits below. + fn round_wide(neg: bool, e: i32, (hi, lo): Wide, sticky: bool) -> Self { + let mut m = [0u64; N]; + let mut e = e; + if N == 2 { + let mut top = hi; + if lo >> 127 == 1 && (lo << 1 != 0 || sticky || top & 1 == 1) { + top = top.wrapping_add(1); + if top == 0 { + top = 1 << 127; + e = e.saturating_add(1); + } + } + set(&mut m, 0, lo64(top)); + set(&mut m, 1, hi64(top)); + } else { + let (mut top, mut low) = (hi, hi64(lo)); + let below = lo64(lo); + if below >> 63 == 1 && (below << 1 != 0 || sticky || low & 1 == 1) { + low = low.wrapping_add(1); + if low == 0 { + top = top.wrapping_add(1); + if top == 0 { + top = 1 << 127; + e = e.saturating_add(1); + } + } + } + set(&mut m, 0, low); + set(&mut m, 1, lo64(top)); + set(&mut m, 2, hi64(top)); + } + Self { neg, e, m } + } + + /// The correctly rounded sum for two limbs in 192 bits, `|big| ≥ |small|`, both non-zero. + fn add_two(big: Self, small: Self) -> Self { + let (bm, sm) = (big.wide().0, small.wide().0); + let gap = big.e.abs_diff(small.e); + // small·2^64 >> gap as its high 128 and low 64 bits, and whether bits fell off. + let (ah, al, sticky) = match gap { + 0 => (sm, 0, false), + 1..=63 => (sm >> gap, lo64(sm << (64 - gap)), false), + 64 => (sm >> 64, lo64(sm), false), + 65..=127 => (sm >> gap, lo64(sm >> (gap - 64)), sm << (192 - gap) != 0), + 128..=191 => (0, lo64(sm >> (gap - 64)), sm << (192 - gap) != 0), + _ => (0, 0, true), + }; + let (mut hi, lo, mut e, sticky) = if big.neg == small.neg { + let (sum, carry) = bm.overflowing_add(ah); + if carry { + let lo = (al >> 1) | (lo64(sum) << 63); + ( + (sum >> 1) | 1 << 127, + lo, + big.e.saturating_add(1), + sticky || al & 1 == 1, + ) + } else { + (sum, al, big.e, sticky) + } + } else { + let (low, b1) = 0u64.overflowing_sub(al); + let (low, b2) = low.overflowing_sub(u64::from(sticky)); + let high = bm.wrapping_sub(ah).wrapping_sub(u128::from(b1 || b2)); + if high == 0 && low == 0 { + return ::zero(); + } + let full = (high, u128::from(low) << 64); + let z = wide_leading_zeros(full); + let (h, l) = wide_shl(full, z); + (h, hi64(l), big.e.saturating_sub(z.cast_signed()), sticky) + }; + if lo >> 63 == 1 && (lo << 1 != 0 || sticky || hi & 1 == 1) { + hi = hi.wrapping_add(1); + if hi == 0 { + hi = 1 << 127; + e = e.saturating_add(1); + } + } + let mut m = [0u64; N]; + set(&mut m, 0, lo64(hi)); + set(&mut m, 1, hi64(hi)); + Self { neg: big.neg, e, m } + } + + /// The correctly rounded sum for two or three limbs, `|big| ≥ |small|`, both non-zero. + fn add_wide(big: Self, small: Self) -> Self { + let gap = big.e.abs_diff(small.e); + let ((ah, al), sticky) = wide_shr(small.wide(), gap); + let (bh, bl) = big.wide(); + if big.neg == small.neg { + let (lo, c1) = bl.overflowing_add(al); + let (hi, c2) = bh.overflowing_add(ah); + let (hi, c3) = hi.overflowing_add(u128::from(c1)); + if c2 || c3 { + let shifted = ((hi >> 1) | 1 << 127, (lo >> 1) | (hi << 127)); + return Self::round_wide( + big.neg, + big.e.saturating_add(1), + shifted, + sticky || lo & 1 == 1, + ); + } + return Self::round_wide(big.neg, big.e, (hi, lo), sticky); + } + // big − small, less one unit if bits fell off small, which `sticky` then marks. + let (lo, b1) = bl.overflowing_sub(al); + let (lo, b2) = lo.overflowing_sub(u128::from(sticky)); + let hi = bh.wrapping_sub(ah).wrapping_sub(u128::from(b1 || b2)); + if hi == 0 && lo == 0 { + return ::zero(); + } + let z = wide_leading_zeros((hi, lo)); + Self::round_wide( + big.neg, + big.e.saturating_sub(z.cast_signed()), + wide_shl((hi, lo), z), + sticky, + ) + } + + /// The correctly rounded product for two or three limbs of normalized mantissas. + fn mul_wide(self, o: Self) -> Option { + let neg = self.neg != o.neg; + let e = self.e.saturating_add(o.e); + let (acc, sticky) = if N == 2 { + let (a, b) = (self.wide().0, o.wide().0); + let (a1, a0, b1, b0) = ( + a >> 64, + a & u128::from(u64::MAX), + b >> 64, + b & u128::from(u64::MAX), + ); + let (mid, c1) = (a0 * b1).overflowing_add(a1 * b0); + let (lo, c2) = (a0 * b0).overflowing_add(mid << 64); + let hi = a1 * b1 + (mid >> 64) + (u128::from(c1) << 64) + u128::from(c2); + ((hi, lo), false) + } else { + let mut p = [0u64; 6]; + for i in 0..3 { + let x = u128::from(limb(&self.m, i)); + let mut carry: u128 = 0; + for j in 0..3 { + let cur = u128::from(limb(&p, i + j)) + x * u128::from(limb(&o.m, j)) + carry; + set(&mut p, i + j, lo64(cur)); + carry = cur >> 64; + } + set(&mut p, i + 3, lo64(carry)); + } + let pair = |i: usize| (u128::from(limb(&p, i + 1)) << 64) | u128::from(limb(&p, i)); + let top = (pair(4), pair(2)); + let below = pair(0); + if top.0 >> 127 == 0 { + let shifted = wide_shl(top, 1); + return (top.0 >> 126 == 1).then(|| { + Self::round_wide( + neg, + e.saturating_sub(1), + (shifted.0, shifted.1 | below >> 127), + below << 1 != 0, + ) + }); + } + (top, below != 0) + }; + if acc.0 >> 127 == 0 { + return (acc.0 >> 126 == 1) + .then(|| Self::round_wide(neg, e.saturating_sub(1), wide_shl(acc, 1), sticky)); + } + Some(Self::round_wide(neg, e, acc, sticky)) + } + + fn mag_cmp(&self, o: &Self) -> Ordering { + match (self.is_zero(), o.is_zero()) { + (true, true) => return Ordering::Equal, + (true, false) => return Ordering::Less, + (false, true) => return Ordering::Greater, + (false, false) => {} + } + self.e + .cmp(&o.e) + .then_with(|| self.top().cmp(&o.top())) + .then_with(|| self.m.iter().rev().cmp(o.m.iter().rev())) + } + + fn recip(self) -> Self { + let top = self.top(); + if top == 0 { + return self; + } + let one = ::one(); + if N == 1 { + return one / self; + } + let b = self.abs(); + // At two limbs the exact 128/64 quotient beats the seed's chain of products. + let seed = if N == 2 { + u128::MAX / u128::from(top) + } else { + recip_seed(top) + }; + let mut y = Self::from_u128(false, seed, 64i32.saturating_sub(self.e)); + for _ in 0..2 { + y = y + y * (one - b * y); + } + y.with_sign(self.neg) + } +} + +impl PartialEq for Real { + fn eq(&self, o: &Self) -> bool { + self.cmp(o) == Ordering::Equal + } +} + +impl Eq for Real {} + +impl PartialOrd for Real { + fn partial_cmp(&self, o: &Self) -> Option { + Some(self.cmp(o)) + } +} + +impl Ord for Real { + fn cmp(&self, o: &Self) -> Ordering { + let sa = !self.is_zero() && self.neg; + let sb = !o.is_zero() && o.neg; + match (sa, sb) { + (false, true) => Ordering::Greater, + (true, false) => Ordering::Less, + (false, false) => self.mag_cmp(o), + (true, true) => o.mag_cmp(self), + } + } +} + +impl Neg for Real { + type Output = Self; + fn neg(self) -> Self { + if self.is_zero() { + return self; + } + Self { + neg: !self.neg, + ..self + } + } +} + +impl Add for Real { + type Output = Self; + fn add(self, other: Self) -> Self { + if N == 1 { + let (top_a, top_b) = (self.top(), other.top()); + if top_b == 0 { + return self; + } + if top_a == 0 { + return other; + } + return if (self.e, top_a) < (other.e, top_b) { + Self::add1(other, self) + } else { + Self::add1(self, other) + }; + } + if other.is_zero() { + return self; + } + if self.is_zero() { + return other; + } + let (big, small) = if self.mag_cmp(&other) == Ordering::Less { + (other, self) + } else { + (self, other) + }; + if N == 2 { + Self::add_two(big, small) + } else { + Self::add_wide(big, small) + } + } +} + +impl Sub for Real { + type Output = Self; + fn sub(self, o: Self) -> Self { + self + -o + } +} + +impl Mul for Real { + type Output = Self; + fn mul(self, o: Self) -> Self { + if self.is_zero() || o.is_zero() { + return ::zero(); + } + let neg = self.neg != o.neg; + let e = self.e.saturating_add(o.e); + if N == 1 { + let p = u128::from(self.top()) * u128::from(o.top()); + return if p >> 127 == 1 { + Self::round128(neg, e, p, false) + } else { + Self::round128(neg, e.saturating_sub(1), p << 1, false) + }; + } + if N <= 3 + && let Some(product) = self.mul_wide(o) + { + return product; + } + let mut p = self.product(&o); + Self::normalize(neg, e, &mut p, 2 * N, false) + } +} + +impl Div for Real { + type Output = Self; + /// To within a few units in the last place; correctly rounded for one limb. + fn div(self, o: Self) -> Self { + if self.is_zero() || o.is_zero() { + return ::zero(); + } + if N == 1 { + let num = u128::from(limb(&self.m, 0)) << 64; + let den = u128::from(limb(&o.m, 0)); + let q = num / den; + let mut ext = [0; SCRATCH]; + set(&mut ext, 0, lo64(q)); + set(&mut ext, 1, hi64(q)); + let e = self.e.saturating_sub(o.e).saturating_add(64); + return Self::normalize(self.neg != o.neg, e, &mut ext, 2, num % den != 0); + } + self * o.recip() + } +} + +impl Work for Real { + const PREC: u32 = match N { + 1 => 64, + 2 => 128, + _ => 192, + }; + + fn zero() -> Self { + Self { + neg: false, + e: 0, + m: [0; N], + } + } + + fn one() -> Self { + Self::from_int(1) + } + + fn from_int(i: i64) -> Self { + Self::from_raw(i128::from(i), 0) + } + + fn from_raw(raw: i128, frac: u32) -> Self { + Self::from_u128(raw < 0, raw.unsigned_abs(), 128 - frac.cast_signed()) + } + + fn constant(c: &C) -> Self { + let (neg, e, limbs) = *c; + let low = 6 - N; + let mut m = [0u64; N]; + for (i, x) in m.iter_mut().enumerate() { + *x = limb(&limbs, low + i); + } + let g = limb(&limbs, low.wrapping_sub(1)); + let rest = g << 1 != 0 || (0..low.saturating_sub(1)).any(|i| limb(&limbs, i) != 0); + if g >> 63 == 1 && (rest || m.first().is_some_and(|&x| x & 1 == 1)) { + let mut ext = [0; SCRATCH]; + for (i, &x) in m.iter().enumerate() { + set(&mut ext, i + 1, x); + } + set(&mut ext, 0, 1 << 63); + Self::pack(neg, e, &ext, N + 1, true) + } else { + Self { neg, e, m } + } + } + + fn to_raw(self, frac: u32, lo: i128, hi: i128, floor: bool) -> i128 { + if self.is_zero() { + return 0; + } + let saturated = if self.neg { lo } else { hi }; + let top = i64::from(self.e) + i64::from(frac); + if top > 128 { + return saturated; + } + let sh = top - i64::from(Self::PREC); + let mag = if N == 1 { + let m = u128::from(self.top()); + match u32::try_from(-sh) { + Err(_) => m << u32::try_from(sh).unwrap_or(0), + Ok(s) if s > 64 => u128::from(floor && self.neg), + Ok(s) => { + let dropped = m & ((1u128 << s) - 1); + let q = m >> s; + let half = s > 0 && dropped >> (s - 1) == 1; + let rest = s > 1 && dropped & ((1u128 << (s - 1)) - 1) != 0; + let up = if floor { + self.neg && dropped != 0 + } else { + half && (rest || q & 1 == 1) + }; + q + u128::from(up) + } + } + } else { + // |m·2^sh| from the mantissa at the top of 256 bits, shifted down + // to keep the half bit, the bits below it as `rest`. + let down = u32::try_from(256 - 64 * i64::from(u32_of(N)) - sh).unwrap_or(u32::MAX); + let ((high, low), rest) = wide_shr(self.wide(), down.saturating_sub(1)); + let (half, q) = (low & 1 == 1, (high << 127) | (low >> 1)); + let up = if floor { + self.neg && (half || rest) + } else { + half && (rest || q & 1 == 1) + }; + q.saturating_add(u128::from(up)) + }; + if self.neg { + 0i128.checked_sub_unsigned(mag).map_or(lo, |v| v.max(lo)) + } else { + i128::try_from(mag).map_or(hi, |v| v.min(hi)) + } + } + + fn is_zero(&self) -> bool { + // Normalized, so the top bit is set unless every bit is clear. + self.top() == 0 + } + + fn is_neg(&self) -> bool { + self.neg && !self.is_zero() + } + + fn exponent(&self) -> i32 { + self.e + } + + fn abs(self) -> Self { + Self { neg: false, ..self } + } + + fn with_sign(self, neg: bool) -> Self { + if neg { -self.abs() } else { self.abs() } + } + + fn ldexp(self, k: i32) -> Self { + if self.is_zero() { + return self; + } + Self { + e: self.e.saturating_add(k), + ..self + } + } + + fn square(self) -> Self { + self * self + } + + fn mul_int(self, k: i64) -> Self { + if self.is_zero() || k == 0 { + return Self::zero(); + } + let ku = k.unsigned_abs(); + let mut p = [0; SCRATCH]; + let mut carry: u128 = 0; + for i in 0..N { + let cur = u128::from(limb(&self.m, i)) * u128::from(ku) + carry; + set(&mut p, i, lo64(cur)); + carry = cur >> 64; + } + set(&mut p, N, lo64(carry)); + Self::normalize( + self.neg != (k < 0), + self.e.saturating_add(64), + &mut p, + N + 1, + false, + ) + } + + fn div_int(self, k: u64) -> Self { + if self.is_zero() || k == 0 { + return self; + } + let mut q = [0; SCRATCH]; + let mut rem: u128 = 0; + let small = u32::try_from(k).is_ok(); + for i in (0..=N).rev() { + let digit = if i == 0 { 0 } else { limb(&self.m, i - 1) }; + if small { + // rem < k < 2^32, so each half fits a 64-bit division. + let high = (lo64(rem) << 32) | (digit >> 32); + let low = ((high % k) << 32) | (digit & u64::from(u32::MAX)); + set(&mut q, i, ((high / k) << 32) | (low / k)); + rem = u128::from(low % k); + continue; + } + let cur = (rem << 64) | u128::from(digit); + set(&mut q, i, lo64(cur / u128::from(k))); + rem = cur % u128::from(k); + } + Self::normalize(self.neg, self.e, &mut q, N + 1, rem != 0) + } + + fn sqrt(self) -> Self { + if self.is_zero() { + return Self::zero(); + } + let top2 = if N == 1 { + u128::from(self.top()) << 64 + } else { + (u128::from(self.top()) << 64) | u128::from(limb(&self.m, N.saturating_sub(2))) + }; + let ee = i64::from(self.e) - 128; + let (x, ee) = if ee % 2 == 0 { + (top2, ee) + } else { + (top2 >> 1, ee + 1) + }; + let half_e = i32_of(ee / 2); + if N == 1 { + let r = isqrt_top(x); + let up = x - r * r > r; + return Self::from_u128(false, r + u128::from(up), half_e.saturating_add(128)); + } + let a = self.abs(); + let three = Self::from_int(3); + let mut y = Self::from_u128(false, rsqrt_seed(x), 0i32.saturating_sub(half_e)); + for _ in 0..2 { + y = (y * (three - a * y.square())).ldexp(-1); + } + a * y + } + + fn octant(mag: u128, frac: u32, radians: bool) -> (u32, Self) { + Self::octant_limbs(&[lo64(mag), hi64(mag)], frac, radians) + } + + fn octant_of(self) -> (u32, Self) { + let point = Self::PREC + .cast_signed() + .saturating_sub(self.e) + .max(0) + .cast_unsigned(); + Self::octant_limbs(&self.m, point, true) + } + + fn octant_of_product(a: Self, b: Self) -> (u32, Self) { + if a.is_zero() || b.is_zero() { + return (0, Self::zero()); + } + let prod = a.product(&b); + let mut turns = [0u64; 16]; + for (i, &cv) in FRAC_1_2PI_LONG.iter().enumerate() { + let mut carry: u128 = 0; + for j in 0..2 * N { + let cur = u128::from(limb(&turns, i + j)) + + u128::from(cv) * u128::from(limb(&prod, j)) + + carry; + set(&mut turns, i + j, lo64(cur)); + carry = cur >> 64; + } + set(&mut turns, i + 2 * N, lo64(carry)); + } + // |a·b| = prod·2^(e − 128N) and 1/2π = c·2^−578, so |a·b|/2π = turns·2^−point. + let point = i64::from(128 * u32_of(N) + 578) - i64::from(a.e) - i64::from(b.e); + let need = i64::from(64 * u32_of(N + 1) + 3); + let shift = u32::try_from((point - need).unsigned_abs()) + .unwrap_or(u32::MAX) + .min(1024); + if point < need { + shl(&mut turns, 16, shift); + } else { + shr(&mut turns, 16, shift); + } + Self::octant_from(&turns) + } + + fn dot2(a: Self, b: Self, c: Self, d: Self) -> Self { + if a.is_zero() || b.is_zero() { + return c * d; + } + if c.is_zero() || d.is_zero() { + return a * b; + } + let mut first = (a.neg != b.neg, a.e.saturating_add(b.e), a.product(&b)); + let mut second = (c.neg != d.neg, c.e.saturating_add(d.e), c.product(&d)); + if second.1 > first.1 { + core::mem::swap(&mut first, &mut second); + } + let low = 64 * u32_of(SCRATCH - 1 - 2 * N); + shl(&mut first.2, SCRATCH, low); + shl(&mut second.2, SCRATCH, low); + let sticky = shr(&mut second.2, SCRATCH, first.1.abs_diff(second.1).min(1024)); + let top = first.1.saturating_add(64); + if first.0 == second.0 { + add_into(&mut first.2, &second.2, SCRATCH); + return Self::normalize(first.0, top, &mut first.2, SCRATCH, sticky); + } + if first.2.iter().rev().cmp(second.2.iter().rev()) == Ordering::Less { + sub_into(&mut second.2, &first.2, SCRATCH); + return Self::normalize(second.0, top, &mut second.2, SCRATCH, false); + } + sub_into(&mut first.2, &second.2, SCRATCH); + if sticky { + // What fell off `second` takes one more unit, and the rest rounds. + let mut one = [0; SCRATCH]; + set(&mut one, 0, 1); + sub_into(&mut first.2, &one, SCRATCH); + } + Self::normalize(first.0, top, &mut first.2, SCRATCH, sticky) + } + + fn sixteenths(num: Self, den: Self) -> i64 { + if num.is_zero() { + return 0; + } + let gap = den.e.saturating_sub(num.e).clamp(0, 127).cast_unsigned(); + let ratio = ((u128::from(num.top()) << 5) >> gap) / u128::from(den.top().max(1)); + i64::try_from((ratio + 1) >> 1).unwrap_or(16) + } + + fn horner2( + u: Self, + table: &Series, + firsts: [u32; 2], + step: u32, + n: u32, + alternate: bool, + ) -> (Self, Self) { + if N != 2 { + let [a, b] = firsts; + return ( + Self::horner(u, table, a, step, n, alternate), + Self::horner(u, table, b, step, n, alternate), + ); + } + // Both chains in one loop, each hiding the other's latency. + let top = (u128::from(u.top()) << 64) | u128::from(limb(&u.m, 0)); + let uq = q126(u.neg, u.e, top); + let coef = |first: u32, k: u32| { + let c = table + .q126 + .get((first + step * k) as usize) + .copied() + .unwrap_or(0); + if alternate && k % 2 == 1 { -c } else { c } + }; + let (mut p, mut r): (i128, i128) = (0, 0); + for k in (0..=n).rev() { + p = coef(firsts[0], k) + mul_q126(uq, p); + r = coef(firsts[1], k) + mul_q126(uq, r); + } + (Self::from_raw(p, 126), Self::from_raw(r, 126)) + } + + fn horner(u: Self, table: &Series, first: u32, step: u32, n: u32, alternate: bool) -> Self { + let sign = |k: u32| alternate && k % 2 == 1; + let at = |k: u32| (first + step * k) as usize; + if N == 1 { + let uq = q62(u.neg, u.e, u.top()); + let mut p: i64 = 0; + for k in (0..=n).rev() { + let c = table.q62.get(at(k)).copied().unwrap_or(0); + let up = i64::try_from((i128::from(uq) * i128::from(p)) >> 62).unwrap_or(0); + p = (if sign(k) { -c } else { c }) + up; + } + return Self::from_raw(i128::from(p), 62); + } + if N == 2 { + let top = (u128::from(u.top()) << 64) | u128::from(limb(&u.m, 0)); + let uq = q126(u.neg, u.e, top); + let mut p: i128 = 0; + for k in (0..=n).rev() { + let c = table.q126.get(at(k)).copied().unwrap_or(0); + p = (if sign(k) { -c } else { c }) + mul_q126(uq, p); + } + return Self::from_raw(p, 126); + } + let uq = q190(u.neg, u.e, &u.m); + let mut p: Q190 = (false, [0; 3]); + for k in (0..=n).rev() { + let c = table.q190.get(at(k)).copied().unwrap_or([0; 3]); + p = add_q190((sign(k), c), mul_q190(uq, p)); + } + let mut ext = [0; SCRATCH]; + for (i, &v) in p.1.iter().enumerate() { + set(&mut ext, i, v); + } + Self::normalize(p.0, 2, &mut ext, 3, false) + } +} + +#[cfg(test)] +mod tests { + use super::{isqrt_top, recip_seed, rsqrt_seed}; + + #[test] + fn seeds_and_roots_are_exact_enough() { + let mut state = 0x9E37_79B9_7F4A_7C15u64; + let mut next = || { + state ^= state << 13; + state ^= state >> 7; + state ^= state << 17; + state + }; + for _ in 0..200_000 { + let x = (u128::from(next()) << 64 | u128::from(next())) >> (next() % 3); + let x = x.max(1 << 125); + assert_eq!(isqrt_top(x), x.isqrt(), "{x}"); + let t = next() | 1 << 63; + let inverse = u128::MAX / u128::from(t); + assert!(recip_seed(t).abs_diff(inverse) < inverse >> 55, "{t}"); + let root = u128::MAX / x.isqrt(); + assert!(rsqrt_seed(x).abs_diff(root) < root >> 55, "{x}"); + } + } +} + +#[cfg(test)] +mod two_limbs { + use super::*; + + /// The generic sum, `len = N + 1` limbs, as `add` did before `add_two`. + fn generic(big: Real<2>, small: Real<2>) -> Real<2> { + let len = 3; + let (mut acc, mut addend) = ([0; SCRATCH], [0; SCRATCH]); + for i in 0..2 { + set(&mut acc, i + 1, limb(&big.m, i)); + set(&mut addend, i + 1, limb(&small.m, i)); + } + let gap = big.e.abs_diff(small.e); + let sticky = if gap >= 192 { + addend = [0; SCRATCH]; + true + } else { + shr(&mut addend, len, gap) + }; + if big.neg == small.neg { + if add_into(&mut acc, &addend, len) { + let lost = shr(&mut acc, len, 1); + let top = limb(&acc, len - 1) | 1 << 63; + set(&mut acc, len - 1, top); + return Real::pack(big.neg, big.e.saturating_add(1), &acc, len, sticky || lost); + } + return Real::pack(big.neg, big.e, &acc, len, sticky); + } + sub_into(&mut acc, &addend, len); + if sticky { + let mut one = [0; SCRATCH]; + set(&mut one, 0, 1); + sub_into(&mut acc, &one, len); + } + Real::normalize(big.neg, big.e, &mut acc, len, sticky) + } + + #[test] + fn add_two_matches_the_generic_sum() { + let mut state = 0x2545_F491_4F6C_DD1Du64; + let mut next = move || { + state ^= state << 13; + state ^= state >> 7; + state ^= state << 17; + state + }; + for _ in 0..1_000_000 { + let limbs = |n: &mut dyn FnMut() -> u64| match n() % 4 { + 0 => u64::MAX, + 1 => 0, + _ => n(), + }; + let a = Real::<2> { + neg: next() % 2 == 1, + e: 0, + m: [limbs(&mut next), limbs(&mut next) | 1 << 63], + }; + let gap = i32::try_from(next() % 200).unwrap_or(0); + let b = Real::<2> { + neg: next() % 2 == 1, + e: -gap, + m: [limbs(&mut next), limbs(&mut next) | 1 << 63], + }; + let (big, small) = if a.mag_cmp(&b) == Ordering::Less { + (b, a) + } else { + (a, b) + }; + let (x, y) = (Real::add_two(big, small), generic(big, small)); + assert!( + x.neg == y.neg && x.e == y.e && x.m == y.m, + "{a:?} {b:?}: {x:?} {y:?}" + ); + } + } +} diff --git a/src/series.rs b/src/series.rs new file mode 100644 index 0000000..6982683 --- /dev/null +++ b/src/series.rs @@ -0,0 +1,131 @@ +//! Fixed-point coefficient tables, built at compile time from the 384-bit constants. + +use crate::consts::{C, INV_FACT, INV_ODD}; + +/// Coefficients in `[0, 1]` with 62, 126 and 190 fractional bits, one per working width. +pub struct Series { + pub q62: &'static [i64], + pub q126: &'static [i128], + pub q190: &'static [[u64; 3]], +} + +/// `c · 2^126` rounded, for `|c| ≤ 1`. +pub const fn q126(entry: &C) -> i128 { + let (neg, exp, mant) = *entry; + let top = ((mant[5] as u128) << 64) | mant[4] as u128; + let shift = 2 - exp; + let value = if shift <= 0 || shift >= 128 { + 0 + } else { + ((top >> shift) + ((top >> (shift - 1)) & 1)).cast_signed() + }; + if neg { -value } else { value } +} + +const fn q62(entry: &C) -> i64 { + let (neg, exp, mant) = *entry; + let shift = 2 - exp; + let value = if shift <= 0 || shift >= 64 { + 0 + } else { + ((mant[5] >> shift) + ((mant[5] >> (shift - 1)) & 1)).cast_signed() + }; + if neg { -value } else { value } +} + +/// `|c| · 2^190` rounded, as three limbs, least significant first. +const fn q190(entry: &C) -> [u64; 3] { + let (_, exp, mant) = *entry; + let shift = 2 - exp; + if shift < 0 || shift >= 190 { + return [0; 3]; + } + let shift = shift.cast_unsigned(); + let wide = [mant[2], mant[3], mant[4], mant[5]]; + let mut out = [0u64; 3]; + let mut rest: &mut [u64] = &mut out; + let mut index = 0; + while let [slot, tail @ ..] = rest { + let bit = 64 + 64 * index + shift; + *slot = word(&wide, bit); + rest = tail; + index += 1; + } + if (word(&wide, 63 + shift) & 1) == 1 { + let (low, c0) = out[0].overflowing_add(1); + let (mid, c1) = out[1].overflowing_add(c0 as u64); + out = [low, mid, out[2] + c1 as u64]; + } + out +} + +/// The 64 bits of `w` from bit `bit` up, reading zero past its end. +const fn word(w: &[u64; 4], bit: u32) -> u64 { + let lo = limb(w, bit / 64); + let hi = limb(w, bit / 64 + 1); + let shift = bit % 64; + if shift == 0 { + lo + } else { + (lo >> shift) | (hi << (64 - shift)) + } +} + +const fn limb(w: &[u64; 4], i: u32) -> u64 { + match i { + 0 => w[0], + 1 => w[1], + 2 => w[2], + 3 => w[3], + _ => 0, + } +} + +const fn tables(table: &[C; L]) -> ([i64; L], [i128; L], [[u64; 3]; L]) { + let mut q62s = [0; L]; + let mut q126s = [0; L]; + let mut q190s = [[0; 3]; L]; + let mut rest: &[C] = table; + let mut xs: &mut [i64] = &mut q62s; + let mut ys: &mut [i128] = &mut q126s; + let mut zs: &mut [[u64; 3]] = &mut q190s; + while let ([head, tail @ ..], [q62_slot, xt @ ..], [q126_slot, yt @ ..], [q190_slot, zt @ ..]) = + (rest, xs, ys, zs) + { + *q62_slot = q62(head); + *q126_slot = q126(head); + *q190_slot = q190(head); + rest = tail; + xs = xt; + ys = yt; + zs = zt; + } + (q62s, q126s, q190s) +} + +pub const FACT: Series = Series { + q62: &tables(&INV_FACT).0, + q126: &tables(&INV_FACT).1, + q190: &tables(&INV_FACT).2, +}; + +pub const ODD: Series = Series { + q62: &tables(&INV_ODD).0, + q126: &tables(&INV_ODD).1, + q190: &tables(&INV_ODD).2, +}; + +#[cfg(test)] +mod tests { + use super::{FACT, ODD, tables}; + use crate::consts::{INV_FACT, INV_ODD}; + + /// The tables built at compile time, built again at run time. + #[test] + fn tables_build_the_same_at_run_time() { + let fact = tables(&INV_FACT); + assert!(FACT.q62 == fact.0 && FACT.q126 == fact.1 && FACT.q190 == fact.2); + let odd = tables(&INV_ODD); + assert!(ODD.q62 == odd.0 && ODD.q126 == odd.1 && ODD.q190 == odd.2); + } +} diff --git a/src/stirling.rs b/src/stirling.rs new file mode 100644 index 0000000..bc32f12 --- /dev/null +++ b/src/stirling.rs @@ -0,0 +1,81 @@ +//! Constants of Stirling's series and of `ln Γ(1 + a)` to 384 bits, from mpmath. + +use crate::consts::C; + +/// `½·ln(2π)`. +#[rustfmt::skip] +pub const LN_SQRT_2PI: C = (false, 0, [0x93EA_BF90_5C55_69BC, 0x212F_9D7F_E00E_86BF, 0xDEC6_A313_3DAA_155D, 0xCFB0_8F8D_1345_8B4D, 0x94BC_9001_4419_2023, 0xEB3F_8E43_25F5_A534]); +/// Stirling's `B₂ₖ/(2k(2k − 1))` for k from 1 to 40. +#[rustfmt::skip] +pub const STIRLING: [C; 40] = [ + (false, -3, [0xAAAA_AAAA_AAAA_AAAB, 0xAAAA_AAAA_AAAA_AAAA, 0xAAAA_AAAA_AAAA_AAAA, 0xAAAA_AAAA_AAAA_AAAA, 0xAAAA_AAAA_AAAA_AAAA, 0xAAAA_AAAA_AAAA_AAAA]), + (true, -8, [0x0B60_B60B_60B6_0B61, 0x60B6_0B60_B60B_60B6, 0xB60B_60B6_0B60_B60B, 0x0B60_B60B_60B6_0B60, 0x60B6_0B60_B60B_60B6, 0xB60B_60B6_0B60_B60B]), + (false, -10, [0x0D00_D00D_00D0_0D01, 0x00D0_0D00_D00D_00D0, 0xD00D_00D0_0D00_D00D, 0x0D00_D00D_00D0_0D00, 0x00D0_0D00_D00D_00D0, 0xD00D_00D0_0D00_D00D]), + (true, -10, [0x09C0_9C09_C09C_09C1, 0xC09C_09C0_9C09_C09C, 0x9C09_C09C_09C0_9C09, 0x09C0_9C09_C09C_09C0, 0xC09C_09C0_9C09_C09C, 0x9C09_C09C_09C0_9C09]), + (false, -10, [0x6AE1_D4E7_00DC_A8F1, 0x9C03_72A3_C563_1FE4, 0x8F15_8C7F_91AB_8753, 0xFE46_AE1D_4E70_0DCA, 0x7539_C037_2A3C_5631, 0xDCA8_F158_C7F9_1AB8]), + (true, -9, [0x6CCC_9E3E_40FB_5587, 0x86CC_C9E3_E40F_B558, 0x586C_CC9E_3E40_FB55, 0x5586_CCC9_E3E4_0FB5, 0xB558_6CCC_9E3E_40FB, 0xFB55_86CC_C9E3_E40F]), + (false, -7, [0x0D20_D20D_20D2_0D21, 0x20D2_0D20_D20D_20D2, 0xD20D_20D2_0D20_D20D, 0x0D20_D20D_20D2_0D20, 0x20D2_0D20_D20D_20D2, 0xD20D_20D2_0D20_D20D]), + (true, -5, [0x4769_8BAD_CFF2_1436, 0x3658_7A9C_BEE1_0325, 0x2547_698B_ADCF_F214, 0x1436_587A_9CBE_E103, 0x0325_4769_8BAD_CFF2, 0xF214_3658_7A9C_BEE1]), + (false, -2, [0xF033_FFD0_C3B7_F4B2, 0x33FF_D0C3_B7F4_B1C0, 0xFFD0_C3B7_F4B1_C0F0, 0xD0C3_B7F4_B1C0_F033, 0xC3B7_F4B1_C0F0_33FF, 0xB7F4_B1C0_F033_FFD0]), + (true, 1, [0x14C1_F4AA_7022_3B38, 0xE731_2CC3_EA60_748B, 0xB380_8C0F_9CF6_DEDC, 0x48B1_4C1F_4AA7_0223, 0xEDCE_7312_CC3E_A607, 0xB23B_3808_C0F9_CF6D]), + (false, 4, [0x3A7A_9C88_6459_C00B, 0xE9EA_7221_9167_002D, 0xA7A9_C886_459C_00B4, 0x9EA7_2219_1670_02D3, 0x7A9C_8864_59C0_0B4E, 0xD672_2191_6700_2D3A]), + (true, 8, [0x5B3F_712E_B9E0_7CA4, 0xDB44_A929_2E66_60D5, 0xB3F7_12EB_9E07_CA39, 0xB44A_9292_E666_0D55, 0x3F71_2EB9_E07C_A39D, 0x9CD9_292E_6660_D55B]), + (false, 12, [0x740D_A740_DA74_0DA7, 0xA740_DA74_0DA7_40DA, 0xDA74_0DA7_40DA_740D, 0x0DA7_40DA_740D_A740, 0x40DA_740D_A740_DA74, 0x8911_A740_DA74_0DA7]), + (true, 16, [0xA199_6CF5_F3A9_C5E4, 0xEBD1_01B8_D006_A04A, 0x8D54_1CDB_CBC6_8B5D, 0x94D9_2E2F_D250_842C, 0xFF6E_EC24_B48F_F1B3, 0x8D0C_C570_E255_BF59]), + (false, 20, [0x6A57_DA7B_7468_3DCB, 0x6967_B29E_E369_D34F, 0x9609_3D5E_E562_C084, 0xDCB0_0698_234D_582B, 0x19EE_4FDC_4469_CCAE, 0xA8D1_044D_3708_D1C2]), + (true, 24, [0x9ABB_C406_169A_BBC4, 0xC406_169A_BBC4_0616, 0x169A_BBC4_0616_9ABB, 0xBBC4_0616_9ABB_C406, 0x0616_9ABB_C406_169A, 0xE884_4D8A_169A_BBC4]), + (false, 29, [0xF312_19DB_CC48_676F, 0x6F31_219D_BCC4_8676, 0x76F3_1219_DBCC_4867, 0x676F_3121_9DBC_C486, 0x8676_F312_19DB_CC48, 0xB694_D07B_219D_BCC4]), + (true, 34, [0xF9BE_6F5E_30FA_FCE7, 0xDBB1_324E_3CD1_0483, 0xF306_6B7E_A521_C5CD, 0xDA88_C079_3F07_A1F8, 0xA602_4D5C_4976_1634, 0xA228_8CEC_F233_76AE]), + (false, 39, [0xB881_2735_0B88_1273, 0x8127_350B_8812_7350, 0x2735_0B88_1273_50B8, 0x350B_8812_7350_B881, 0x0B88_1273_50B8_8127, 0xA1BB_CDE4_EA01_2735]), + (true, 44, [0x8171_4AF6_42A2_4358, 0x5817_14AF_642A_2435, 0x3581_714A_F642_A243, 0x4358_1714_AF64_2A24, 0x2435_8171_4AF6_42A2, 0xB400_5BDE_03D4_642A]), + (false, 49, [0xC3FC_1644_88D0_3738, 0x7D42_4E93_0F91_4416, 0xAAC0_952D_3B2D_CF5D, 0xEEEC_AC9E_8573_ED1E, 0xC3A9_E6DA_EAE4_6D98, 0xDE46_6B7C_78FB_AAE3]), + (true, 55, [0xABFE_D399_5BC0_B405, 0x877F_7942_E745_7DEB, 0xCBD5_6936_7A8E_3A5C, 0x7203_7C5E_1516_61B4, 0x4050_9F4F_D884_644B, 0x977D_7628_7772_9BCB]), + (false, 60, [0x1349_260C_05F8_A82A, 0x4E98_6E10_DD6C_4012, 0x46BA_523A_04C2_CB44, 0x83DA_6597_D322_6A6F, 0x0B6B_0A35_2D4F_335C, 0xE2E1_337F_5AF0_BED9]), + (true, 66, [0x6671_5B3A_DE4A_E3B8, 0x9209_871F_A77C_3AE3, 0x1F50_A7D3_0F4B_3A8C, 0x09D8_D881_AD45_7156, 0xB4F9_2FF9_86CD_EEA2, 0xB9E0_9405_8AD8_9016]), + (false, 72, [0xCFDF_11E3_D1C5_6C22, 0xD86B_A7F9_AAD3_322F, 0x0B3D_95ED_5188_A0D9, 0x478F_4715_B086_40E9, 0x0326_AB4C_C8BF_3F7C, 0xA5F7_EEF9_E71A_C7C8]), + (true, 78, [0x9E6F_BEA3_4445_E20F, 0x9B37_6479_67B1_C1DB, 0x0717_30F3_C513_F899, 0x694D_EEF0_8CB9_CEBD, 0xDA64_925C_6C86_491A, 0xA0EF_80E5_7954_084C]), + (false, 84, [0x7F8A_DDDF_7F1D_B5EB, 0x3250_2FFF_EBCC_7550, 0x66BA_7710_D482_31E8, 0xC758_797B_F482_6900, 0x9790_760B_0CE0_256E, 0xA8EB_FE48_DA17_DD99]), + (true, 90, [0x4782_409B_737C_796D, 0x2EF3_BA1B_5EF1_28B1, 0x2409_B737_C796_CBEC, 0x3BA1_B5EF_128B_1478, 0x24C9_5AB5_6CBE_C2EF, 0xBF58_2A43_3556_FB17]), + (false, 96, [0xD64F_6BD0_99DE_7492, 0x837D_2BB8_3DE9_B8B2, 0xEDFB_FDAA_6336_8772, 0x45C0_EB78_E86D_69D5, 0x7770_E671_0431_6DCB, 0xE940_B372_3E6C_7D0E]), + (true, 103, [0x7924_791F_CDBE_610E, 0xA1E0_619A_AA6B_15CB, 0xB5C7_3E81_CD5C_EE1B, 0x174B_18C9_BD60_A7D4, 0xF8CC_3B4F_4518_35E1, 0x989A_1506_8967_2663]), + (false, 109, [0xD4A2_45F2_02CC_3D8D, 0x8D4A_245F_202C_C3D8, 0xD8D4_A245_F202_CC3D, 0x3D8D_4A24_5F20_2CC3, 0x58FB_ED95_C390_02CC, 0xD5CE_C8C0_E5AC_CFED]), + (true, 116, [0xF775_6D4C_CAC2_A220, 0x756D_4CCA_C2A2_2017, 0x6D4C_CAC2_A220_17F7, 0x4CCA_C2A2_2017_F775, 0x12E5_B9C2_738F_0D8D, 0xA005_36CA_A55D_946E]), + (false, 122, [0x3400_E758_6295_F18E, 0xCC13_5E5D_9A18_C246, 0x400E_7586_295F_18D9, 0xC135_E5D9_A18C_2463, 0x0539_9005_5E27_2CEC, 0xFF5C_A480_718D_0478]), + (true, 129, [0x5618_F693_3B59_EB1D, 0x8B97_0D80_209A_DFA6, 0x618F_6933_B59E_B1CC, 0x8970_D802_09AD_FA65, 0x51CE_652D_A1AC_906F, 0xD8CA_CDE8_FD42_5243]), + (false, 136, [0x1BE1_0EF1_89B2_52A9, 0xD329_C3CE_04B2_58BE, 0x3609_112F_8D0C_E908, 0x7D64_C29C_C515_283C, 0x6D26_0A61_4FFD_D36D, 0xC375_BBF4_BD00_4BF4]), + (true, 143, [0x6F72_BBCE_D18E_9B83, 0x597E_8319_AB73_D9C5, 0x2F30_2BB2_A9EF_67B0, 0x5962_5477_74D0_4093, 0x39D8_E8F1_6B37_9936, 0xBAD3_3621_BFB2_52C9]), + (false, 150, [0x4AD0_12B4_04AD_012B, 0xD012_B404_AD01_2B40, 0x12B4_04AD_012B_404A, 0xBB0F_6601_2B40_4AD0, 0xF50F_D61D_72FE_906A, 0xBCFF_6120_E392_0B45]), + (true, 157, [0x452D_4897_18CD_B5D1, 0x3786_07BC_A4C0_0E90, 0xF6AB_93AE_FD7F_341C, 0xA3FE_41E5_230B_2674, 0xF8DC_79DC_8231_CF7E, 0xCA0A_AFB0_E0D2_AD9C]), + (false, 164, [0x47F8_9EAE_67A3_855F, 0xF5CC_D914_19DC_D2F4, 0xE327_1A69_E845_61C8, 0x767C_95D9_5FAA_C26F, 0x9781_34FC_5FBD_6DF4, 0xE3E6_A8D1_CECA_9D6A]), + (true, 172, [0x99E2_1870_3128_7729, 0x4C49_4E52_9790_5644, 0x959D_1FAA_42D3_2EF2, 0x87F8_B383_41B4_D93F, 0xEDBA_5116_DCFE_FF56, 0x876F_6FFD_5A2B_432D]), +]; + +/// `ln Γ(1 + a) = Σ dₖ·aᵏ`: `d₁ = −γ`, then `(−1)ᵏ·ζ(k)/k` for k from 2 to 25. +#[rustfmt::skip] +pub const LN_GAMMA_1P: [C; 25] = [ + (true, 0, [0xD064_9CCB_6210_57D1, 0x8E4B_59FA_03A9_F0EE, 0x0C03_DF34_709A_FFBD, 0xA1CE_CC3A_F65C_C019, 0xD1BE_3F81_0152_CB56, 0x93C4_67E3_7DB0_C7A4]), + (false, 0, [0x8B26_9F81_40C5_5A84, 0xEC0C_F375_B26E_C1C1, 0x3358_2C03_1A5E_F369, 0xB5C8_8DAA_8116_2D8B, 0x73D8_9122_00BA_CE5E, 0xD28D_3312_983E_9918]), + (true, -1, [0x6315_9D75_5BBB_0166, 0xE034_437C_A773_4414, 0x3631_2FC7_61E5_4F25, 0x1007_A920_F862_2AE9, 0xF64B_92E6_99B5_AA12, 0xCD26_AADF_5596_B7AC]), + (false, -1, [0x0AA4_833E_F806_7D2F, 0xF24E_74B6_38CD_6D13, 0xDE8F_EADB_673E_682D, 0xC586_B130_B802_8592, 0xF912_1119_6E52_35FB, 0x8A89_9156_3EC2_41B5]), + (true, -2, [0x6792_4211_F175_87B5, 0x1077_73BC_9CB7_044C, 0x2A64_FA75_D99B_DB83, 0x0998_6D37_23B2_DF88, 0x826A_4FDA_E199_04DC, 0xD45C_E0BD_530A_492E]), + (false, -2, [0x384E_19A2_DBDB_45B1, 0x555C_CBA4_32ED_070C, 0xEA7D_E6B9_1EE2_1F2E, 0x42AE_EE17_15CD_7EB1, 0x3CCA_078A_B4AD_8EFB, 0xADA0_6588_0618_30A5]), + (true, -2, [0x0F66_2CB4_DA4D_4A0D, 0x418E_9224_BD21_EC51, 0x7C9E_0C47_5036_19EF, 0xFC6D_D53D_60F9_9BCA, 0x0FFA_0F29_5870_1A43, 0x9381_D0EE_751D_72C6]), + (false, -2, [0x2716_5E99_C907_1F7F, 0x894B_647A_283C_5447, 0x2A44_BDD2_58B8_8BBC, 0xC379_CB84_EA4D_7EC1, 0xF2CE_526E_DAD3_266D, 0x8085_9B57_C31C_B745]), + (true, -3, [0xDF3E_5C53_6366_D621, 0x7824_D599_3E4B_D605, 0x722B_D732_CE4E_C97A, 0xE90D_ACEB_1301_1B58, 0x5500_DFBA_7DB2_2231, 0xE403_3836_ABED_A2AD]), + (false, -3, [0x523E_2E24_90F5_DDB0, 0x1499_EEEE_FD60_2978, 0x7BC2_BE6A_BD73_F152, 0xE3DE_34B6_DF9D_6FAC, 0x3418_F376_8CD6_6B0E, 0xCD00_F1C2_EAFC_7981]), + (true, -3, [0x53ED_8DE3_5B9D_4426, 0xA66F_B2C6_4077_2FBA, 0x0F4F_8BA4_5C52_0C29, 0xB33F_9590_CBD9_6894, 0x95D8_9020_1C7F_1587, 0xBA46_1988_A636_B07A]), + (false, -3, [0x4CE0_927E_7FB0_9E8C, 0xF872_71B1_0F1D_08DA, 0xC510_C5E7_3A17_DD32, 0xD6AB_E647_ED46_FA68, 0xBD0B_01F5_D03A_1CCC, 0xAAB5_6B19_21DE_202B]), + (true, -3, [0x35D9_D1C1_582A_BCDD, 0xB7F8_3F91_0889_22C1, 0xFD21_1CB0_C075_5778, 0xC896_E05D_44BB_2D86, 0x79F8_1B6C_2036_AFF1, 0x9D8E_CB8F_E2CC_261A]), + (false, -3, [0xB59D_3F5F_876B_929B, 0xBD20_323C_7268_DD09, 0x28B6_1D64_256D_9DCA, 0x6623_B174_2E8E_DEA5, 0x9695_C081_7D0A_6441, 0x924B_6FC1_9062_FB9E]), + (true, -3, [0xA900_66CD_3D55_70BE, 0x3590_3F58_0F68_5F66, 0x42B8_6108_50A5_89BC, 0x1069_EA2D_7185_1D53, 0x9A4F_F396_EE34_EC23, 0x8889_9A3B_73FF_01DC]), + (false, -3, [0x0D74_5ACD_0FA0_6210, 0x2F2E_757B_530E_42B9, 0x41CD_D75F_45C7_D7FF, 0x5834_005B_D89B_CE23, 0x79D0_156A_FFDB_C10B, 0x8000_8032_66F5_9178]), + (true, -4, [0xDB81_698B_D4CF_6AA0, 0x6E8A_9DD7_21EC_D0AE, 0x75F9_1189_52EB_490A, 0xABD3_9A90_3C73_E82A, 0xABDD_E62D_5294_E08B, 0xF0F1_6988_F455_E4E5]), + (false, -4, [0xCDF6_D390_1999_1E23, 0x6A71_611C_F41D_EAD3, 0x9099_D5CE_80D3_E668, 0x7507_9D8B_7118_25A4, 0xD3D0_5F94_9F43_1375, 0xE38E_71D1_05A0_C482]), + (true, -4, [0x6136_7BA1_063E_3CCA, 0x8D07_2910_5A0E_46D8, 0xDD53_5E24_5E2F_1B05, 0xE1AF_D617_6283_CCFF, 0xE83D_10BF_DF68_DA3B, 0xD794_50DA_B445_02E8]), + (false, -4, [0x50D5_8970_C998_0BE2, 0xC8DB_31A6_8FC5_0BF7, 0xD56B_1B53_C4B2_9A22, 0x1C26_0A54_2198_6C8B, 0x546D_31AF_95B3_951C, 0xCCCC_D99A_96AD_D052]), + (true, -4, [0x5D75_B711_DA03_8F6D, 0x8DA2_7FB2_1821_A737, 0x8FED_BF69_10FF_D0A6, 0x6ACD_6BBE_9FBA_8096, 0xDBDE_3EB0_EEF8_3E63, 0xC30C_36DB_BDFD_FADB]), + (false, -4, [0x2070_29C8_EAD8_B743, 0xD447_04EE_ABF3_6D36, 0xA28B_A3D2_8E70_0453, 0xED2D_D12F_0C0C_94AC, 0x5C7F_0544_C73B_8B65, 0xBA2E_8E8B_BC6F_C881]), + (true, -4, [0x4E3F_B2B7_333C_C9C9, 0xFAC7_D8E5_8338_5BDF, 0x0230_9B0E_2FB3_A061, 0xA4A5_CBB5_5BB2_5832, 0x1755_C6A0_7401_D738, 0xB216_442C_8DB3_65DF]), + (false, -4, [0x0194_999E_C8E8_27CC, 0x7875_26EA_1F3A_C7AB, 0x849E_0503_480E_C411, 0x485B_9BF2_38DC_EAED, 0x40BF_9C09_DEC5_D623, 0xAAAA_AB55_57EE_6AA9]), + (true, -4, [0xFDE0_0BFC_30B7_EABE, 0xE23B_A4A2_BB50_2FFA, 0x9F6E_1326_EF95_0C63, 0x1419_79AE_A51E_B00B, 0x71A9_A8CA_784B_F5F2, 0xA3D7_0A8F_5CFD_BBA0]), +]; diff --git a/src/traits.rs b/src/traits.rs index a31e668..26f5c05 100644 --- a/src/traits.rs +++ b/src/traits.rs @@ -5,13 +5,26 @@ use fixed::traits::{Fixed, FixedSigned}; use fixed::types::extra::{IsLessOrEqual, LeEqU128, True, Unsigned}; use fixed::{FixedI8, FixedI16, FixedI32, FixedI64, FixedI128}; +pub(crate) mod sealed { + /// The working arithmetic of a type, and its raw bits; this crate's types + /// alone implement it. + pub trait Sealed { + type Work: crate::real::Work; + type Narrow: crate::real::Work; + fn raw_i128(self) -> i128; + fn from_raw_i128(raw: i128) -> Self; + } +} + /// The operations and constants the algorithms need from a fixed-point type. /// /// Every signed `fixed` type with three or more integer bits implements it, /// from [`FixedI8`] to [`FixedI128`] with up -/// to 5, 13, 29, 61 or 125 fractional bits; `I16F16` and `I24F8` qualify. +/// to 5, 13, 29, 61 or 125 fractional bits; `I16F16` and `I24F8` qualify. It +/// is sealed: no other type can implement it. pub trait CordicNumber: - Copy + sealed::Sealed + + Copy + PartialEq + PartialOrd + Add @@ -218,12 +231,42 @@ macro_rules! impl_cordic_generic { $bits_type:ty, $wide_type:ty, $sqrt_impl:ident, + $work:ty, + $narrow:ty, $total_bits:expr, $max_frac:ty, $pi_frac:ty, $frac_pi_2:ty, $frac_pi_4:ty ) => { + impl sealed::Sealed for $fixed_type + where + Fract: Unsigned + + IsLessOrEqual<$max_frac, Output = True> + + IsLessOrEqual<$pi_frac, Output = True> + + IsLessOrEqual<$frac_pi_2, Output = True> + + IsLessOrEqual<$frac_pi_4, Output = True> + + LeEqU128, + { + type Work = $work; + type Narrow = $narrow; + + #[inline] + fn raw_i128(self) -> i128 { + i128::from(self.to_bits()) + } + + #[inline] + fn from_raw_i128(raw: i128) -> Self { + let saturated = if raw < 0 { + <$bits_type>::MIN + } else { + <$bits_type>::MAX + }; + Self::from_bits(<$bits_type>::try_from(raw).unwrap_or(saturated)) + } + } + impl CordicNumber for $fixed_type where Fract: Unsigned @@ -418,19 +461,34 @@ macro_rules! impl_cordic_generic { }; } +use crate::real::Real; use fixed::types::extra::{ U5, U6, U7, U8, U13, U14, U15, U16, U29, U30, U31, U32, U61, U62, U63, U64, U125, U126, U127, U128, }; // Each type's π bound is one bit stricter than π needs, for headroom. -impl_cordic_generic!(FixedI8, i8, u16, sqrt_round_widened, 8, U8, U5, U6, U7); +impl_cordic_generic!( + FixedI8, + i8, + u16, + sqrt_round_widened, + Real<1>, + Real<1>, + 8, + U8, + U5, + U6, + U7 +); impl_cordic_generic!( FixedI16, i16, u32, sqrt_round_widened, + Real<1>, + Real<1>, 16, U16, U13, @@ -443,6 +501,8 @@ impl_cordic_generic!( i32, u64, sqrt_round_widened, + Real<1>, + Real<1>, 32, U32, U29, @@ -455,6 +515,8 @@ impl_cordic_generic!( i64, u128, sqrt_round_widened, + Real<2>, + Real<1>, 64, U64, U61, @@ -467,6 +529,8 @@ impl_cordic_generic!( i128, u128, sqrt_round_i128, + Real<3>, + Real<2>, 128, U128, U125, diff --git a/tests/golden.txt b/tests/golden.txt new file mode 100644 index 0000000..31decdd --- /dev/null +++ b/tests/golden.txt @@ -0,0 +1,673 @@ +# function type hash changed_in +acos I16F16 5765d551e4743a50 4.0.0 +acos I24F8 2780e413d06ed50e 4.0.0 +acos I32F32 3fe7e281f41fd101 4.0.0 +acos I3F5 347f122d9ec5c0a1 - +acos I3F61 758531d5b86c61d6 - +acos I64F64 c71f327093f72e56 4.0.0 +acosh I16F16 5e1ce4d06f243c83 4.0.0 +acosh I24F8 2e7ede1d4bcfbf3a 4.0.0 +acosh I32F32 a93489408ccb0243 4.0.0 +acosh I3F5 23d874726d683999 - +acosh I3F61 c4379e4c3c960372 - +acosh I64F64 24f399fa88650a2e 4.0.0 +acoth I16F16 f04527b92be20343 4.0.0 +acoth I24F8 31bc8ea7058ecd58 4.0.0 +acoth I32F32 e563e0df423a59d0 4.0.0 +acoth I3F5 0609e81a282bc393 - +acoth I3F61 3eb6116e6485c45d - +acoth I64F64 fa1d99fb229f9ed1 4.0.0 +asin I16F16 f1a0dedf88aaa556 4.0.0 +asin I24F8 cc2f848964c8cfc2 4.0.0 +asin I32F32 fd490c891ceb5db2 4.0.0 +asin I3F5 93eaa82126829d2f - +asin I3F61 c3c4465d12b8b024 - +asin I64F64 cee58aea7f9bcf7f 4.0.0 +asin_from_gap I16F16 bc1ffe7934965668 - +asin_from_gap I24F8 fb30b05763bdc3f7 - +asin_from_gap I32F32 e9a99d0124708c08 - +asin_from_gap I3F5 388152efe04cc9f2 - +asin_from_gap I3F61 6d8997014b864c64 - +asin_from_gap I64F64 3b158b28daeed329 - +asinh I16F16 5fc9c869ed7052f6 4.0.0 +asinh I24F8 dd72e5a8460ab0d2 4.0.0 +asinh I32F32 4bde530bc4887dae 4.0.0 +asinh I3F5 df2fcc5860606036 - +asinh I3F61 2a2086cf2f5a43d9 - +asinh I64F64 2e726674405c55fb 4.0.0 +atan I16F16 cca08b1868932d38 4.0.0 +atan I24F8 d5d8dbdc3d54daec 4.0.0 +atan I32F32 0ee77fc2b36a0de6 4.0.0 +atan I3F5 fbe50431aa014635 - +atan I3F61 bd3af74303300f4b - +atan I64F64 a0a85b0d315e633d 4.0.0 +atan2 I16F16 b0cdf7115e7f6603 4.0.0 +atan2 I24F8 cb87e3fdc9c714bf 4.0.0 +atan2 I32F32 1fad4958dc984132 4.0.0 +atan2 I3F5 18721014d93c2569 - +atan2 I3F61 de229d78fbb4a6a8 - +atan2 I64F64 dceed18e29579ea7 4.0.0 +atan2_range I16F16 ea5afa4952e5ef50 - +atan2_range I24F8 58078745cb9ff422 - +atan2_range I32F32 a4d389a16075cf2d - +atan2_range I3F5 6454424829cd11f9 - +atan2_range I3F61 46da382c160479f3 - +atan2_range I64F64 c6b042dbc4322c0d - +atan2_turns I16F16 88d6ad0889ea7c17 - +atan2_turns I24F8 2f23d0fe753b3b86 - +atan2_turns I32F32 0a5776d697102181 - +atan2_turns I3F5 fff862c0b1dc4e7c - +atan2_turns I3F61 c254dbaf84846cd5 - +atan2_turns I64F64 3328beedebc385ec - +atan_range I16F16 98349bdb703e00a1 - +atan_range I24F8 efbee4a8c1b16ce5 - +atan_range I32F32 528c89b6813224a7 - +atan_range I3F5 2faa10e1727861ea - +atan_range I3F61 628cb621b4c6f9dd - +atan_range I64F64 b379f90974eaaf44 - +atan_ratio I16F16 cf0ed187224176e4 - +atan_ratio I24F8 bf2b438f0316b44a - +atan_ratio I32F32 5b3f135be4925119 - +atan_ratio I3F5 4cb8ce2cae393652 - +atan_ratio I3F61 09462dbd1de37718 - +atan_ratio I64F64 4bcacb4aa0778d2b - +atan_ratio_range I16F16 0dfbcc0d24b692ea - +atan_ratio_range I24F8 308e26b39bc46ac9 - +atan_ratio_range I32F32 99027b511aa83f48 - +atan_ratio_range I3F5 6c32e4b6de0dc42e - +atan_ratio_range I3F61 f34a48ce9298ec24 - +atan_ratio_range I64F64 7618afca9797b596 - +atanh I16F16 d05f43076e4a2247 4.0.0 +atanh I24F8 922fcf8289bf48ab 4.0.0 +atanh I32F32 fc3f6937917512eb 4.0.0 +atanh I3F5 7354bc32b3c5ca8d - +atanh I3F61 695979eceeb7b5b7 - +atanh I64F64 5cfca9e847278f40 4.0.0 +atanh_ratio I16F16 274257eeb7f42828 - +atanh_ratio I24F8 8a401e08024da8f6 - +atanh_ratio I32F32 8e7d018969304aad - +atanh_ratio I3F5 7d33e011f8b4a51a - +atanh_ratio I3F61 16070bec36c6e661 - +atanh_ratio I64F64 c72950f5dfe59547 - +atanh_ratio_range I16F16 f7aaf9118755d488 - +atanh_ratio_range I24F8 0674dd450dd0cdda - +atanh_ratio_range I32F32 c554352057ff7428 - +atanh_ratio_range I3F5 4110015df19bc7b5 - +atanh_ratio_range I3F61 24d4b425c4e95ba8 - +atanh_ratio_range I64F64 a8692933dcac1fa0 - +carlson_rc I16F16 898cc509327c010f - +carlson_rc I24F8 385f12c689001991 - +carlson_rc I32F32 be333dffb4e8db24 - +carlson_rc I3F5 62fa683456145b08 - +carlson_rc I3F61 230f74e3a1c2aadb - +carlson_rc I64F64 74e5cec776f017b0 - +carlson_rd I16F16 32ef56ce0dfdd5f4 - +carlson_rd I24F8 ebaf5547b507d029 - +carlson_rd I32F32 409b8397591e071f - +carlson_rd I3F5 3e9db819ef28cbc6 - +carlson_rd I3F61 3d1e2a1e781a9553 - +carlson_rd I64F64 89bbaa63d0d109ec - +carlson_rf I16F16 86438c0f107aa2d4 - +carlson_rf I24F8 c64808c2e100ac49 - +carlson_rf I32F32 ca5daf6baa2be721 - +carlson_rf I3F5 d553dfe9ea0f4b21 - +carlson_rf I3F61 31405041d0b934c2 - +carlson_rf I64F64 10822b72d6286a1e - +carlson_rf_roots I16F16 0d262c7ce2858911 - +carlson_rf_roots I24F8 a623e054a7f41497 - +carlson_rf_roots I32F32 1da4b1aa5bee5bf5 - +carlson_rf_roots I3F5 ac7a9c98ba7ed35d - +carlson_rf_roots I3F61 10df3aacedfbd4a7 - +carlson_rf_roots I64F64 ee4c7556833eed99 - +carlson_rj I16F16 3474175d2cc979ba - +carlson_rj I24F8 27f9382d66b73fcb - +carlson_rj I32F32 c278158a70738a69 - +carlson_rj I3F5 f5e9ff25e08279d4 - +carlson_rj I3F61 11dea54aa945ba61 - +carlson_rj I64F64 075b0d4ae92d54e8 - +carlson_rj_roots I16F16 293236ee5c40beb0 - +carlson_rj_roots I24F8 04148bd8219e49ee - +carlson_rj_roots I32F32 cd504c93bc6ef5ae - +carlson_rj_roots I3F5 09e30e9603e80e23 - +carlson_rj_roots I3F61 741a661ae40f5b26 - +carlson_rj_roots I64F64 9212549a1c8a46ae - +complex_div I16F16 2d0b233395134516 - +complex_div I24F8 c0e202a6f02ca071 - +complex_div I32F32 5bb7ab970f39035b - +complex_div I3F5 42b444923cecfe9c - +complex_div I3F61 7e79a84d764f90e1 - +complex_div I64F64 e94568a2e4d16452 - +complex_e1 I16F16 8dd2e3ca849a35a7 - +complex_e1 I24F8 1710085ec5b41696 - +complex_e1 I32F32 413384279b0a25b2 - +complex_e1 I3F5 9b30deff98ea4eba - +complex_e1 I3F61 0a94c84536020782 - +complex_e1 I64F64 95b6eefbbec74adc - +complex_erf I16F16 c4e9a99f69e1f6d6 - +complex_erf I24F8 6ae0ab513a564b33 - +complex_erf I32F32 f15e54ce0f91110a - +complex_erf I3F5 3a1c54341061099b - +complex_erf I3F61 4022d0c889bf17cd - +complex_erf I64F64 7376c2396c4a362b - +complex_exp I16F16 b20f152396893083 - +complex_exp I24F8 9bfa5f577043e80d - +complex_exp I32F32 9357426ac431d620 - +complex_exp I3F5 3bdd825ed4344fd1 - +complex_exp I3F61 5145cd2f132cfd56 - +complex_exp I64F64 45bb4accf80eb152 - +complex_exprel I16F16 73e6726bf75a677a - +complex_exprel I24F8 0585099090d06c7c - +complex_exprel I32F32 80812483a52d1d94 - +complex_exprel I3F5 164d92850bff5b47 - +complex_exprel I3F61 f9d5e876317a0980 - +complex_exprel I64F64 be5e53058c852e68 - +complex_exprel2 I16F16 ebd74235671aa7f6 - +complex_exprel2 I24F8 60b2c371521489d4 - +complex_exprel2 I32F32 77570b8e5a9f8edd - +complex_exprel2 I3F5 02937bdf751da12d - +complex_exprel2 I3F61 1da5e35dfeb15391 - +complex_exprel2 I64F64 351e57ae5934f184 - +complex_exprel_weighted I16F16 3c3e1b171a32ce71 - +complex_exprel_weighted I24F8 65f22cf1372de42e - +complex_exprel_weighted I32F32 911eefbcc0c05902 - +complex_exprel_weighted I3F5 72ce961aeaf4c4f5 - +complex_exprel_weighted I3F61 24dfd62651024a73 - +complex_exprel_weighted I64F64 8d3d189b7b1d7eb3 - +complex_faddeeva I16F16 daad96a649e0a316 - +complex_faddeeva I24F8 d9a382dd46bfc8b3 - +complex_faddeeva I32F32 1cafa9e39dff568c - +complex_faddeeva I3F5 01354c8ed382a371 - +complex_faddeeva I3F61 37280a59946d7c88 - +complex_faddeeva I64F64 e5accd4c81d47fac - +complex_gamma_upper I16F16 914ed7d6ecf24bda - +complex_gamma_upper I24F8 6c6e6b37cb093a51 - +complex_gamma_upper I32F32 1af98046d5b3ec1a - +complex_gamma_upper I3F5 2fa2831bb560e1ea - +complex_gamma_upper I3F61 45d087937002451a - +complex_gamma_upper I64F64 f405dfd7f9912851 - +complex_ln I16F16 47b4da7e1b25679e - +complex_ln I24F8 b8ebee29c3888cc4 - +complex_ln I32F32 11834fe02e0b3748 - +complex_ln I3F5 a143f9ca6232b618 - +complex_ln I3F61 8e090cc08a2030ee - +complex_ln I64F64 619de076a8d69a2f - +complex_mul I16F16 1b0f179f0e8502de - +complex_mul I24F8 6d78c364eb3a8e82 - +complex_mul I32F32 89345d0955aad768 - +complex_mul I3F5 56b60bdc73d3dfef - +complex_mul I3F61 74445b1ca553900e - +complex_mul I64F64 fbdc924a918b618b - +complex_pow I16F16 957613f5db4439a3 - +complex_pow I24F8 9412018d85ad05ea - +complex_pow I32F32 23df04610c75eec2 - +complex_pow I3F5 20982e5e5387dc12 - +complex_pow I3F61 83cc27942afee091 - +complex_pow I64F64 c4d971ded36505cc - +cos I16F16 babd45b73eb07245 4.0.0 +cos I24F8 b7cf07593356171a 4.0.0 +cos I32F32 1f39ae9f713bb358 4.0.0 +cos I3F5 769e95706a11dd29 - +cos I3F61 70bcebf2d3dafe9d - +cos I64F64 cdba1e5339a0f098 4.0.0 +cosh I16F16 eb8fcf2c7eb3dfdd 4.0.0 +cosh I24F8 0ffeebc3c988a0f9 4.0.0 +cosh I32F32 65cf97e9c9f4a277 4.0.0 +cosh I3F5 4d0fb4da724a813b - +cosh I3F61 f15036988e3a313d - +cosh I64F64 bc18a35eba2b6b3e 4.0.0 +coth I16F16 d88c9c6a6086d071 4.0.0 +coth I24F8 1e56e8abda14720b 4.0.0 +coth I32F32 66f0069efc4c9fd0 4.0.0 +coth I3F5 841f198f27bf913e - 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+norm I24F8 6232849c5ec0fa4e - +norm I32F32 9ed5e32f3f33a3f1 - +norm I3F5 d9110934a70e4b6d - +norm I3F61 b5818c833f5199f9 - +norm I64F64 b9df310d78ae65a1 - +normalize I16F16 bd3621d663699064 - +normalize I24F8 d9c99726bfd75094 - +normalize I32F32 f1cfad99c1e7c19c - +normalize I3F5 e06b55a25380bbd9 - +normalize I3F61 bd9d428eeeead3b6 - +normalize I64F64 95ddbb048973a2fe - +pow I16F16 a92e87a2227c53d9 4.0.0 +pow I24F8 a6c9db0d479f9a9b 4.0.0 +pow I32F32 1df4bf8be2db42f3 4.0.0 +pow I3F5 937fec6ce016f3f5 - +pow I3F61 a5ef200e825b6cec - +pow I64F64 ab2221213a378312 4.0.0 +pow2 I16F16 3bdbee77d70a1d09 4.0.0 +pow2 I24F8 aabf57385cfd78d6 4.0.0 +pow2 I32F32 107101298af99aae 4.0.0 +pow2 I3F5 ddc06e5fbac130ee - +pow2 I3F61 85c1f6e6541b5dd9 - +pow2 I64F64 29ff69b6682b6094 4.0.0 +root_in I16F16 f2470ef471da7612 - +root_in I24F8 991d2f09203c5f61 - +root_in I32F32 3561b8c49436a526 - +root_in I3F5 4f8f0803f0f10df9 - +root_in I3F61 9de39dff8350831f - +root_in I64F64 ea6279e9a3ce7f53 - +rsqrt I16F16 641340398291d90b - +rsqrt I24F8 428291274e9a12c5 - +rsqrt I32F32 23b63892453efaed - +rsqrt I3F5 6f0bbcb3dbf1cf7c - +rsqrt I3F61 b9dc899ac8d2595d - +rsqrt I64F64 4aef7f31b69663de - +scale_by I16F16 f086d86e4a62437b - +scale_by I24F8 5e61f6153d1a745d - +scale_by I32F32 89203467307a420f - +scale_by I3F5 c768e5df5df5a9c1 - +scale_by I3F61 b2008a65be70a27f - +scale_by I64F64 242c3ce73f826692 - +scale_exponent I16F16 339d3a8b16bd5240 - +scale_exponent I24F8 766eae20b413ff00 - +scale_exponent I32F32 c307a4f5e3e512ac - +scale_exponent I3F5 857c056993e76eca - +scale_exponent I3F61 12eebcf709875045 - +scale_exponent I64F64 876ba868deb6d47d - +sin I16F16 c3c8ac18b0526c06 4.0.0 +sin I24F8 273025ffd9f01bd5 4.0.0 +sin I32F32 f442835ada9fe6e1 4.0.0 +sin I3F5 5214f24e58adb576 - +sin I3F61 22394fba6eb6ed0b - +sin I64F64 809496eb1bfe9c30 4.0.0 +sin_cos I16F16 a0bd1332ce24a05e 4.0.0 +sin_cos I24F8 ee09705bae20e45a 4.0.0 +sin_cos I32F32 701f0eecb271c5a8 4.0.0 +sin_cos I3F5 db1baf1c78cb5ca3 - +sin_cos I3F61 2035d9f7fe45d5eb - +sin_cos I64F64 3febd2fa483ef4df 4.0.0 +sin_cos_range I16F16 cef0755342d051e9 - +sin_cos_range I24F8 38506efba6cfe761 - +sin_cos_range I32F32 3e560bda23cb8ffd - +sin_cos_range I3F5 f5edd1387833b7ef - +sin_cos_range I3F61 94bdeb07f26a11a4 - +sin_cos_range I64F64 bb09bd862b3d606f - +sin_cos_range_turns I16F16 1479465c829b7b69 - +sin_cos_range_turns I24F8 868dc83d29251401 - +sin_cos_range_turns I32F32 d4ab671a0769f2d0 - +sin_cos_range_turns I3F5 cd02d38793b9afde - +sin_cos_range_turns I3F61 5cc2344743442655 - +sin_cos_range_turns I64F64 ab9588d1ab4d41a4 - +sin_cos_turns I16F16 a02e6705098d93a4 - +sin_cos_turns I24F8 c8f0965acbb18896 - +sin_cos_turns I32F32 120c2a2a9d649b76 - +sin_cos_turns I3F5 46d754e32b0a6851 - +sin_cos_turns I3F61 ff62f63e18023638 - +sin_cos_turns I64F64 8956cd7917e24e28 - +sinh I16F16 4972ffd57255a07b 4.0.0 +sinh I24F8 1df393e08b8ea6af 4.0.0 +sinh I32F32 65b5dc51834ed56b 4.0.0 +sinh I3F5 e82df0b49c582739 - +sinh I3F61 2fa30d272d1a3057 - +sinh I64F64 011fe9b68486f006 4.0.0 +sinh_cosh I16F16 1b6b3147f0240393 4.0.0 +sinh_cosh I24F8 38c6fd4c4014b38f 4.0.0 +sinh_cosh I32F32 30fd20711294bc71 4.0.0 +sinh_cosh I3F5 918272d34654ab3c - +sinh_cosh I3F61 ca68c271771e1ee9 - +sinh_cosh I64F64 3e908edc4f27bf4a 4.0.0 +sqrt I16F16 2dc8349462170ec0 - +sqrt I24F8 d58006496d3544b3 - +sqrt I32F32 2f42d8f131ade225 - +sqrt I3F5 6e749920e258924f - +sqrt I3F61 7e6eed7a3d0f3f52 - +sqrt I64F64 194393ab41ff7071 - +sqrt_or_zero I16F16 25edcaba5bcd0e9c - +sqrt_or_zero I24F8 04ad05dbbddfe4dc - +sqrt_or_zero I32F32 089cb0ad8cb6b648 - +sqrt_or_zero I3F5 67f51ca4660ce8ea - +sqrt_or_zero I3F61 d645316dbdd4119f - +sqrt_or_zero I64F64 7a0f01a0d77177fa - +sqrt_product I16F16 521e3051da00dcbb - +sqrt_product I24F8 4d4fef8d23a1090b - +sqrt_product I32F32 8b3b954bd26b3312 - +sqrt_product I3F5 e0e57ca77d2ba867 - +sqrt_product I3F61 ab14275e94826ae2 - +sqrt_product I64F64 b58a481e4eb48099 - +sqrt_range I16F16 4db2dc303b0655f0 - +sqrt_range I24F8 b57528447c6a5a11 - +sqrt_range I32F32 a9cc02a74d5bed26 - +sqrt_range I3F5 5421c36a1ffd358a - +sqrt_range I3F61 ce37a1f023528c9d - +sqrt_range I64F64 1b106d8e487bd27a - +stumpff I16F16 c72f4c14b90dde23 - +stumpff I24F8 851b93e89641c505 - +stumpff I32F32 9df330ab2163a6f1 - +stumpff I3F5 4716072f3d00d6c3 - +stumpff I3F61 36e718ee17d3d537 - +stumpff I64F64 84cb65dfc5e47b13 - +stumpff_range I16F16 741912c1126c3f3c - +stumpff_range I24F8 797ae2a7e1ed38e4 - +stumpff_range I32F32 ef758c3c77cff80b - +stumpff_range I3F5 4505cc66aa67e3b9 - +stumpff_range I3F61 fde2b15357563b82 - +stumpff_range I64F64 106df84ad185b610 - +tan I16F16 9d46d2ee2b3f25f2 4.0.0 +tan I24F8 c63d148044eacc18 4.0.0 +tan I32F32 024d7195d813e5f6 4.0.0 +tan I3F5 13e70cbd51fb48d6 - +tan I3F61 10e4788ccf83ccd7 - +tan I64F64 b85cd185dd9fad94 4.0.0 +tan_ratio I16F16 041fc4380c90934b - +tan_ratio I24F8 94b5211235410772 - +tan_ratio I32F32 d62891e1fba35f91 - +tan_ratio I3F5 cf4a17eacee0f0f6 - +tan_ratio I3F61 e9703daa4c45dd4c - +tan_ratio I64F64 2eb74486b28ca413 - +tan_ratio_range I16F16 7ccb7cfbc0d05457 - +tan_ratio_range I24F8 900043332331f5a6 - +tan_ratio_range I32F32 ec6d8df3674a439c - +tan_ratio_range I3F5 8a1e568fdf904f97 - +tan_ratio_range I3F61 12b40ee7e36a6ee0 - +tan_ratio_range I64F64 ab53cd807171f868 - +tanh I16F16 7f06199525874f34 4.0.0 +tanh I24F8 6b1ec89061af8b95 4.0.0 +tanh I32F32 58050c3736ae54b8 4.0.0 +tanh I3F5 1b2c4e0f6bb4ea0f - +tanh I3F61 1126d238d81a2367 - +tanh I64F64 b8ee7bde68a6f6ad 4.0.0 +tanh_range I16F16 e22e7ea0d804a17b - +tanh_range I24F8 058c3bc9d99b7909 - +tanh_range I32F32 f2bd0cc04ee6be0e - +tanh_range I3F5 15413998f2bfb4ad - +tanh_range I3F61 b2b90ddd496465f9 - +tanh_range I64F64 debbdd69169871b5 - +tanh_ratio I16F16 fadd7efaaf18eabb - +tanh_ratio I24F8 b95caccb1a949979 - +tanh_ratio I32F32 2da0f45d96545fdd - +tanh_ratio I3F5 05cbf05cfb096911 - +tanh_ratio I3F61 2a41371d1eb619c6 - +tanh_ratio I64F64 fad3c26610d661f0 - +tanh_ratio_range I16F16 627b430293613857 - +tanh_ratio_range I24F8 b45db35c2305eeaf - +tanh_ratio_range I32F32 837705fc4179afd3 - +tanh_ratio_range I3F5 bdb71e441831a042 - +tanh_ratio_range I3F61 66ce98501214525d - +tanh_ratio_range I64F64 c8960a0731d53e80 - +wrap_principal I16F16 2cb0147852759f04 - +wrap_principal I24F8 95e23c61759e7c11 - +wrap_principal I32F32 2579e914d37d1e70 - +wrap_principal I3F5 d92ec3ac5c07ceac - +wrap_principal I3F61 32aa899c996522f8 - +wrap_principal I64F64 4a7ae9b23c3c73ef - +wrap_radians I16F16 b8abaa77fb06a93e - +wrap_radians I24F8 94ad51baa77fe9a1 - +wrap_radians I32F32 7310e74f3fda7782 - +wrap_radians I3F5 01f6435d86bc0c95 - +wrap_radians I3F61 bbf076139270ff37 - +wrap_radians I64F64 0ae59ec5bb4289c5 - diff --git a/tests/unit/golden.rs b/tests/unit/golden.rs new file mode 100644 index 0000000..7ca6213 --- /dev/null +++ b/tests/unit/golden.rs @@ -0,0 +1,597 @@ +//! Holds every function to the output bits pinned in tests/golden.txt. +//! CHANGELOG.md says how to pin a change and record it. + +#[cfg(test)] +mod tests { + use crate::unit::support::Lcg; + use fixed::types::extra::{U5, U61}; + use fixed::types::{I16F16, I24F8, I32F32, I64F64}; + use fixed::{FixedI8, FixedI64}; + use fixed_analytics::{Complex, erf, erfc, fresnel, gamma}; + use fixed_analytics::{ + CordicNumber, acos, acosh, acoth, asin, asin_from_gap, asinh, atan, atan_range, atan2, + atanh, cos, cosh, coth, div_or_saturate, dot, exp, exp_range, hypot, ln, log2, log10, + mul_add, mul_div, norm, normalize, pow, pow2, rsqrt, scale_by, scale_exponent, sin, + sin_cos, sin_cos_range, sinh, sinh_cosh, sqrt, sqrt_or_zero, sqrt_product, tan, tanh, + tanh_range, + }; + use fixed_analytics::{ + Modulus, carlson_rc, carlson_rd, carlson_rf, carlson_rf_roots, carlson_rj, + carlson_rj_roots, ellip_e, ellip_k, ellip_pi, ellip_pi_complete, jacobi_am, jacobi_inverse, + jacobi_sn_cn_dn, + }; + use fixed_analytics::{ + Rounding, atan_ratio_range, atan2_range, atanh_ratio_range, div_rounded, dot_rounded, + ellip_pi_increment, exp_moments_range, exprel_range, exprel2_range, hypot_range, + log1p_range, log1p_ratio_range, log1p_residual_ratio_range, mul_add_rounded, + mul_div_rounded, mul_rounded, sqrt_range, stumpff_range, tan_ratio_range, tanh_ratio_range, + }; + use fixed_analytics::{ + atan_ratio, atanh_ratio, exp_moments, expm1, exprel, exprel2, log1p, log1p_ratio, + log1p_residual, log1p_residual_ratio, stumpff, tan_ratio, tanh_ratio, + }; + use fixed_analytics::{ + atan2_turns, div_by_tau, kepler_universal, mul_by_tau, root_in, sin_cos_range_turns, + sin_cos_turns, wrap_principal, wrap_radians, + }; + use std::collections::BTreeMap; + use std::fmt::Write; + + const GOLDEN: &str = concat!(env!("CARGO_MANIFEST_DIR"), "/tests/golden.txt"); + const CHANGELOG: &str = concat!(env!("CARGO_MANIFEST_DIR"), "/CHANGELOG.md"); + const INPUTS: usize = 600; + + const FUNCTIONS: &[&str] = &[ + "sin", + "cos", + "tan", + "sin_cos", + "asin", + "acos", + "atan", + "atan2", + "sinh", + "cosh", + "tanh", + "sinh_cosh", + "coth", + "asinh", + "acosh", + "atanh", + "acoth", + "exp", + "pow2", + "ln", + "log2", + "log10", + "pow", + "sqrt", + "sqrt_or_zero", + "mul_add", + "mul_div", + "div_or_saturate", + "dot", + "hypot", + "norm", + "normalize", + "scale_exponent", + "scale_by", + "sqrt_product", + "rsqrt", + "asin_from_gap", + "sin_cos_range", + "exp_range", + "atan_range", + "tanh_range", + "expm1", + "exprel", + "exprel2", + "exp_moments", + "log1p", + "log1p_ratio", + "log1p_residual", + "log1p_residual_ratio", + "stumpff", + "tan_ratio", + "atan_ratio", + "atanh_ratio", + "tanh_ratio", + "ellip_k", + "ellip_e", + "ellip_pi_complete", + "ellip_pi", + "jacobi_sn_cn_dn", + "jacobi_am", + "jacobi_inverse", + "carlson_rf", + "carlson_rf_roots", + "carlson_rj", + "carlson_rj_roots", + "carlson_rc", + "carlson_rd", + "sin_cos_turns", + "atan2_turns", + "div_by_tau", + "wrap_radians", + "wrap_principal", + "mul_by_tau", + "sin_cos_range_turns", + "root_in", + "kepler_universal", + "complex_mul", + "complex_div", + "complex_exp", + "complex_exprel", + "complex_exprel2", + "complex_faddeeva", + "complex_erf", + "complex_e1", + "complex_gamma_upper", + "complex_ln", + "complex_pow", + "complex_exprel_weighted", + "erf", + "erfc", + "fresnel", + "gamma", + "mul_rounded", + "div_rounded", + "mul_div_rounded", + "mul_add_rounded", + "dot_rounded", + "exprel_range", + "exprel2_range", + "exp_moments_range", + "log1p_range", + "log1p_ratio_range", + "log1p_residual_ratio_range", + "tanh_ratio_range", + "atan_ratio_range", + "atanh_ratio_range", + "tan_ratio_range", + "sqrt_range", + "stumpff_range", + "hypot_range", + "atan2_range", + "ellip_pi_increment", + ]; + + trait Raw: CordicNumber { + const NAME: &'static str; + fn raw(self) -> i128; + fn from_raw(raw: i128) -> Self; + } + + macro_rules! raw { + ($($t:ty, $bits:ty, $name:literal);+ $(;)?) => {$( + impl Raw for $t { + const NAME: &'static str = $name; + fn raw(self) -> i128 { + i128::from(self.to_bits()) + } + fn from_raw(raw: i128) -> Self { + let clamped = raw.clamp(<$bits>::MIN.into(), <$bits>::MAX.into()); + Self::from_bits(<$bits>::try_from(clamped).unwrap_or(0)) + } + } + )+}; + } + + raw! { + FixedI8, i8, "I3F5"; + I16F16, i32, "I16F16"; + I24F8, i32, "I24F8"; + I32F32, i64, "I32F32"; + FixedI64, i64, "I3F61"; + I64F64, i128, "I64F64"; + } + + const fn pair(range: (T, T)) -> [Option; 2] { + [Some(range.0), Some(range.1)] + } + + fn eval(name: &str, x: T, y: T) -> [Option; 2] { + let one = |v: T| [Some(v), None]; + let res = |v: fixed_analytics::Result| [v.ok(), None]; + match name { + "sin" => one(sin(x)), + "cos" => one(cos(x)), + "tan" => one(tan(x)), + "sin_cos" => { + let (s, c) = sin_cos(x); + [Some(s), Some(c)] + } + "asin" => res(asin(x)), + "acos" => res(acos(x)), + "atan" => one(atan(x)), + "atan2" => one(atan2(x, y)), + "sinh" => one(sinh(x)), + "cosh" => one(cosh(x)), + "tanh" => one(tanh(x)), + "sinh_cosh" => { + let (s, c) = sinh_cosh(x); + [Some(s), Some(c)] + } + "coth" => res(coth(x)), + "asinh" => one(asinh(x)), + "acosh" => res(acosh(x)), + "atanh" => res(atanh(x)), + "acoth" => res(acoth(x)), + "exp" => one(exp(x)), + "pow2" => one(pow2(x)), + "ln" => res(ln(x)), + "log2" => res(log2(x)), + "log10" => res(log10(x)), + "pow" => res(pow(x, y)), + "sqrt" => res(sqrt(x)), + "sqrt_or_zero" => one(sqrt_or_zero(x)), + "mul_add" => one(mul_add(x, y, x)), + "mul_div" => one(mul_div(x, y, y)), + "div_or_saturate" => one(div_or_saturate(x, y)), + "dot" => one(dot(&[x, y, x], &[y, y, x])), + "hypot" => one(hypot(x, y)), + "norm" => one(norm(&[x, y, x])), + "normalize" => normalize([x, y]).map_or([None, None], |v| [Some(v[0]), Some(v[1])]), + "scale_exponent" => { + let e = scale_exponent(x); + [ + e.map(|v| T::from_num(v.rem_euclid(4))), + e.map(|v| T::from_num(v.div_euclid(16).clamp(-3, 3))), + ] + } + "scale_by" => one(scale_by(x, y.to_i32() % 40)), + "sqrt_product" => res(sqrt_product(x, y)), + "rsqrt" => res(rsqrt(x)), + "asin_from_gap" => res(asin_from_gap(x)), + "sin_cos_range" => { + let ((s_lo, s_hi), (c_lo, c_hi)) = sin_cos_range(x, y); + [ + Some(s_lo.saturating_add(s_hi)), + Some(c_lo.saturating_add(c_hi)), + ] + } + "exp_range" => pair(exp_range(x, y)), + "atan_range" => pair(atan_range(x, y)), + "tanh_range" => pair(tanh_range(x, y)), + "expm1" => one(expm1(x)), + "exprel" => one(exprel(x)), + "exprel2" => one(exprel2(x)), + "exp_moments" => moments(x), + "log1p" => res(log1p(x)), + "log1p_ratio" => res(log1p_ratio(x)), + "log1p_residual" => res(log1p_residual(x)), + "log1p_residual_ratio" => res(log1p_residual_ratio(x)), + "stumpff" => { + let [c0, c1, c2, c3] = stumpff(x); + [Some(c0.saturating_add(c2)), Some(c1.saturating_add(c3))] + } + "tan_ratio" => pair(tan_ratio(x)), + "atan_ratio" => one(atan_ratio(x)), + "atanh_ratio" => res(atanh_ratio(x)), + "tanh_ratio" => one(tanh_ratio(x)), + "carlson_rf" => res(carlson_rf(x.abs(), y.abs(), T::one())), + "carlson_rf_roots" => res(carlson_rf_roots(x, y, T::one())), + "carlson_rj" => res(carlson_rj(x.abs(), y.abs(), T::one(), T::two())), + "carlson_rj_roots" => res(carlson_rj_roots(x, y, T::one(), T::two())), + "carlson_rc" => res(carlson_rc(x.abs(), y.abs())), + "carlson_rd" => res(carlson_rd(x.abs(), y.abs(), T::one())), + "sin_cos_turns" => pair(sin_cos_turns(x)), + "atan2_turns" => one(atan2_turns(x, y)), + "div_by_tau" => one(div_by_tau(x)), + "wrap_radians" => one(wrap_radians(x)), + "wrap_principal" => one(wrap_principal(x)), + "mul_by_tau" | "sin_cos_range_turns" | "root_in" | "kepler_universal" => { + solving(name, x, y) + } + "erf" => one(erf(x)), + "erfc" => one(erfc(x)), + "fresnel" => pair(fresnel(x)), + "gamma" => res(gamma(x)), + n if n.starts_with("complex") => complex(n, x, y), + n if n.ends_with("_rounded") || n.ends_with("_range") => bounding(n, x, y), + _ => elliptic(name, x, y), + } + } + + fn solving(name: &str, x: T, y: T) -> [Option; 2] { + let one = |v: T| [Some(v), None]; + let res = |v: fixed_analytics::Result| [v.ok(), None]; + match name { + "mul_by_tau" => one(mul_by_tau(x)), + "sin_cos_range_turns" => { + let ((s_lo, s_hi), (c_lo, c_hi)) = sin_cos_range_turns(x, y); + [ + Some(s_lo.saturating_add(s_hi)), + Some(c_lo.saturating_add(c_hi)), + ] + } + "root_in" => { + let top = x.abs().saturating_add(T::one()); + let rise = |t: T| exp(t).saturating_sub(top); + root_in(rise, T::zero(), top, 64).map_or([None, None], pair) + } + "kepler_universal" => res(kepler_universal(T::one(), T::zero(), y, x)), + _ => [None, None], + } + } + + /// The directed forms, down then up, and the ranges' ends, several summed. + fn bounding(name: &str, x: T, y: T) -> [Option; 2] { + let directed = |f: &dyn Fn(Rounding) -> T| [Some(f(Rounding::Down)), Some(f(Rounding::Up))]; + let res = |v: fixed_analytics::Result<(T, T)>| v.map_or([None, None], pair); + let sums = |ends: &[(T, T)]| { + let add = |a: T, b: T| a.saturating_add(b); + let low = ends.iter().map(|e| e.0).fold(T::zero(), add); + [ + Some(low), + Some(ends.iter().map(|e| e.1).fold(T::zero(), add)), + ] + }; + let unit = |v: T| div_or_saturate(v.abs(), v.abs().saturating_add(T::one())); + match name { + "mul_rounded" => directed(&|m| mul_rounded(x, y, m)), + "div_rounded" => directed(&|m| div_rounded(x, y, m)), + "mul_div_rounded" => directed(&|m| mul_div_rounded(x, y, y.div_int(3), m)), + "mul_add_rounded" => directed(&|m| mul_add_rounded(x, y, x, m)), + "dot_rounded" => directed(&|m| dot_rounded(&[x, y, x], &[y, y, x], m)), + "exprel_range" => pair(exprel_range(x, y)), + "exprel2_range" => pair(exprel2_range(x, y)), + "exp_moments_range" => sums(&exp_moments_range(x, y)), + "log1p_range" => res(log1p_range(x.abs(), y.abs())), + "log1p_ratio_range" => res(log1p_ratio_range(x.abs(), y.abs())), + "log1p_residual_ratio_range" => res(log1p_residual_ratio_range(x.abs(), y.abs())), + "tanh_ratio_range" => pair(tanh_ratio_range(x, y)), + "atan_ratio_range" => pair(atan_ratio_range(x, y)), + "atanh_ratio_range" => res(atanh_ratio_range(unit(x), -unit(y))), + "tan_ratio_range" => pair(tan_ratio_range(x, y)), + "sqrt_range" => pair(sqrt_range(x, y)), + "stumpff_range" => sums(&stumpff_range(x, y)), + "hypot_range" => pair(hypot_range((x, y), (y, x.div_int(2)))), + _ => pair(atan2_range((x, y), (y, x.div_int(2)))), + } + } + + fn moments(x: T) -> [Option; 2] { + let [e1, e2, e3, e4, e5, decay] = exp_moments(x); + [ + Some(e1.saturating_add(e2).saturating_add(e4)), + Some(e3.saturating_add(e5).saturating_add(decay)), + ] + } + + /// The complex functions of `x + iy`, both parts. + fn complex(name: &str, x: T, y: T) -> [Option; 2] { + let z = Complex::new(x, y); + let w = Complex::new(y, x); + let parts = |c: Complex| [Some(c.re), Some(c.im)]; + match name { + "complex_mul" => parts(z * w), + "complex_div" => parts(z / w), + "complex_exp" => parts(z.exp()), + "complex_exprel" => parts(z.exprel()), + "complex_exprel2" => parts(z.exprel2()), + "complex_faddeeva" => parts(z.faddeeva()), + "complex_erf" => parts(z.erf()), + "complex_e1" => z.e1().map_or([None, None], parts), + "complex_gamma_upper" => z.gamma_upper(T::two()).map_or([None, None], parts), + "complex_ln" => z.ln().map_or([None, None], parts), + "complex_pow" => z.pow(T::two() + T::half()).map_or([None, None], parts), + "complex_exprel_weighted" => { + let eps = if x.abs() < T::one() { + x.abs() + } else { + T::half() + }; + z.exprel_weighted(eps, y.abs()) + .map_or([None, None], |(k1, k2)| parts(k1 + k2)) + } + _ => [None, None], + } + } + + /// The elliptic functions, their modulus `|y|` folded into `[0, 1]`. + fn elliptic(name: &str, x: T, y: T) -> [Option; 2] { + let one = |v: T| [Some(v), None]; + let res = |v: fixed_analytics::Result| [v.ok(), None]; + let fold = if y.abs() > T::one() { + T::one().div(y.abs()) + } else { + y.abs() + }; + let Some(m) = Modulus::from_complement(fold) else { + return [None, None]; + }; + match name { + "ellip_k" => res(ellip_k(m)), + "ellip_e" => one(ellip_e(m)), + "ellip_pi_complete" => res(ellip_pi_complete(x, m)), + "ellip_pi" => { + let (sn, cn, dn) = jacobi_sn_cn_dn(x, m); + res(ellip_pi(-T::one(), sn, cn, dn, m)) + } + "jacobi_sn_cn_dn" => { + let (sn, cn, dn) = jacobi_sn_cn_dn(x, m); + [Some(sn.saturating_add(dn)), Some(cn)] + } + "jacobi_am" => one(jacobi_am(x, m)), + "jacobi_inverse" => { + let (sn, cn, dn) = jacobi_sn_cn_dn(x, m); + res(jacobi_inverse(sn, cn, dn, m)) + } + "ellip_pi_increment" => res(ellip_pi_increment(T::half(), x, y, m)), + _ => [None, None], + } + } + + /// A few fixed values, then raw bits of every length in equal measure. + fn inputs(seed: u64) -> Vec { + let mut v: Vec = [0.0, 1.0, -1.0, 0.5, -0.5, 2.0, -2.0, 3.0, -3.0] + .iter() + .map(|&x| T::from_num(x)) + .collect(); + v.extend([ + T::min_value(), + T::max_value(), + T::from_raw(1), + T::from_raw(-1), + ]); + let mut rng = Lcg(seed); + let bits = T::total_bits(); + while v.len() < INPUTS { + let len = u32::try_from(rng.next_u64() % u64::from(bits)).unwrap_or(0); + let r = i128::from(rng.next_u64()) << 64 | i128::from(rng.next_u64()); + let mag = r.unsigned_abs().checked_shr(128 - len).unwrap_or(0); + let raw = i128::try_from(mag).unwrap_or(0); + v.push(T::from_raw(if rng.next_u64() & 1 == 0 { + raw + } else { + -raw + })); + } + v + } + + fn fnv(hash: &mut u64, bytes: &[u8]) { + for &b in bytes { + *hash = (*hash ^ u64::from(b)).wrapping_mul(0x0100_0000_01B3); + } + } + + fn digest(name: &str) -> u64 { + let seed = name.bytes().fold(0x9E37_79B9_7F4A_7C15, |h: u64, b| { + h.rotate_left(7) ^ u64::from(b) + }); + let xs = inputs::(seed); + let ys = inputs::(seed ^ 0x5555); + // Debug code runs the elliptic functions slowly. Fewer inputs do. + let count = if name.starts_with("ellip") + || name.starts_with("jacobi") + || name.starts_with("carlson") + { + 64 + } else { + INPUTS + }; + let mut hash = 0xCBF2_9CE4_8422_2325; + for (&x, &y) in xs.iter().zip(&ys).take(count) { + for out in eval(name, x, y) { + match out { + Some(v) => fnv(&mut hash, &v.raw().to_le_bytes()), + None => fnv(&mut hash, b"none"), + } + } + } + hash + } + + fn current() -> BTreeMap<(String, String), u64> { + let mut map = BTreeMap::new(); + for &name in FUNCTIONS { + for (ty, hash) in [ + ( as Raw>::NAME, digest::>(name)), + (::NAME, digest::(name)), + (::NAME, digest::(name)), + (::NAME, digest::(name)), + ( as Raw>::NAME, digest::>(name)), + (::NAME, digest::(name)), + ] { + map.insert((name.to_string(), ty.to_string()), hash); + } + } + map + } + + /// Rows keyed by function and type: the hash and the version that last moved it. + type Pinned = BTreeMap<(String, String), (u64, String)>; + + fn pinned() -> Result { + let text = std::fs::read_to_string(GOLDEN).map_err(|e| format!("{GOLDEN}: {e}"))?; + let mut map = BTreeMap::new(); + for line in text + .lines() + .filter(|l| !l.starts_with('#') && !l.is_empty()) + { + let cols: Vec<&str> = line.split_whitespace().collect(); + let [name, ty, hash, changed] = cols.as_slice() else { + return Err(format!("malformed golden row: {line}")); + }; + let hash = u64::from_str_radix(hash, 16).map_err(|e| format!("{line}: {e}"))?; + map.insert( + ((*name).to_string(), (*ty).to_string()), + (hash, (*changed).to_string()), + ); + } + Ok(map) + } + + #[test] + fn golden_outputs() -> Result<(), String> { + let now = current(); + let old = pinned().unwrap_or_default(); + let changed: Vec = now + .iter() + .filter(|(k, h)| old.get(*k).is_some_and(|(o, _)| o != *h)) + .map(|((name, ty), _)| format!("{name} on {ty}")) + .collect(); + let missing = now.keys().any(|k| !old.contains_key(k)); + if std::env::var_os("GOLDEN_UPDATE").is_some() { + let version = env!("CARGO_PKG_VERSION"); + let mut out = String::from("# function type hash changed_in\n"); + for ((name, ty), hash) in &now { + let stamp = match old.get(&(name.clone(), ty.clone())) { + Some((o, v)) if o == hash => v.clone(), + Some(_) => version.to_string(), + None => "-".to_string(), + }; + writeln!(out, "{name} {ty} {hash:016x} {stamp}").map_err(|e| e.to_string())?; + } + return std::fs::write(GOLDEN, out).map_err(|e| e.to_string()); + } + if changed.is_empty() && !missing { + return Ok(()); + } + Err(format!( + "outputs changed for {}{}; rerun with GOLDEN_UPDATE=1 and name each changed function in CHANGELOG.md", + changed.join(", "), + if missing { " (and new rows)" } else { "" } + )) + } + + #[test] + fn golden_changes_in_changelog() -> Result<(), String> { + let log = std::fs::read_to_string(CHANGELOG).map_err(|e| format!("{CHANGELOG}: {e}"))?; + let mut sections: BTreeMap = BTreeMap::new(); + let mut current: Option = None; + for line in log.lines() { + if let Some(head) = line.strip_prefix("## ") { + let version = head.trim_matches(|c: char| c == '[' || c == ']' || c == ' '); + let version = version.split_whitespace().next().unwrap_or(""); + current = Some(version.trim_end_matches(']').to_string()); + } else if let Some(v) = ¤t { + let body = sections.entry(v.clone()).or_default(); + body.push_str(line); + body.push('\n'); + } + } + let mut missing = Vec::new(); + for ((name, ty), (_, version)) in pinned()? { + if version == "-" { + continue; + } + let named = sections + .get(&version) + .is_some_and(|body| body.contains(&format!("`{name}`"))); + if !named { + missing.push(format!("{name} ({ty}) in {version}")); + } + } + if missing.is_empty() { + Ok(()) + } else { + Err(format!( + "CHANGELOG.md doesn't name these output changes: {}", + missing.join(", ") + )) + } + } +} diff --git a/tests/unit/mod.rs b/tests/unit/mod.rs index 723743d..82d2591 100644 --- a/tests/unit/mod.rs +++ b/tests/unit/mod.rs @@ -1,4 +1,5 @@ mod error; +mod golden; mod kernel; mod ops; mod smoke; diff --git a/tests/unit/ops/algebraic.rs b/tests/unit/ops/algebraic.rs index d99a47f..3064616 100644 --- a/tests/unit/ops/algebraic.rs +++ b/tests/unit/ops/algebraic.rs @@ -254,3 +254,68 @@ mod exact { Ok(()) } } + +#[cfg(test)] +mod scale_aware { + use crate::unit::support::TestResult; + use fixed::types::{I16F16, I64F64}; + use fixed_analytics::{ + asin, asin_from_gap, hypot, norm, normalize, rsqrt, scale_by, scale_exponent, sqrt_product, + }; + + #[test] + fn hypot_keeps_tiny_components() { + let tiny = I64F64::from_bits(3); + // 3² + 4² in ulps: the squares alone would underflow to zero. + assert_eq!(hypot(tiny, I64F64::from_bits(4)), I64F64::from_bits(5)); + let big = I16F16::from_num(30_000); + assert_eq!(hypot(big, big), I16F16::MAX); + assert_eq!( + norm(&[ + I16F16::from_num(2), + I16F16::from_num(3), + I16F16::from_num(6) + ]), + I16F16::from_num(7) + ); + } + + #[test] + fn normalize_gives_exact_directions() -> TestResult { + let v = normalize([I64F64::from_bits(3), I64F64::from_bits(-4)])?; + // Parsed decimals round correctly; `from_num(0.6)` would take the f64. + let want = ["0.6".parse::().ok(), "-0.8".parse::().ok()]; + assert_eq!([Some(v[0]), Some(v[1])], want); + assert!(normalize([I16F16::ZERO; 3]).is_err()); + Ok(()) + } + + #[test] + fn scaling_is_exact() { + assert_eq!(scale_exponent(I16F16::from_num(5)), Some(2)); + assert_eq!(scale_exponent(I16F16::from_bits(-1)), Some(-16)); + assert_eq!(scale_exponent(I16F16::ZERO), None); + assert_eq!(scale_by(I16F16::from_num(3), 4), I16F16::from_num(48)); + assert_eq!(scale_by(I16F16::from_num(3), -1), I16F16::from_num(1.5)); + assert_eq!(scale_by(I16F16::from_num(3), 20), I16F16::MAX); + } + + #[test] + fn roots_of_products_and_reciprocals() -> TestResult { + let tiny = I64F64::from_bits(1 << 20); + assert_eq!(sqrt_product(tiny, tiny)?, tiny); + assert!(sqrt_product(I16F16::ONE, -I16F16::ONE).is_err()); + assert_eq!(rsqrt(I16F16::from_num(4))?, I16F16::from_num(0.5)); + assert!(rsqrt(I16F16::ZERO).is_err()); + Ok(()) + } + + #[test] + fn asin_from_gap_matches_asin() -> TestResult { + assert_eq!(asin_from_gap(I16F16::ZERO)?, asin(I16F16::ONE)?); + assert_eq!(asin_from_gap(I16F16::from_num(2))?, asin(-I16F16::ONE)?); + assert_eq!(asin_from_gap(I16F16::ONE)?, I16F16::ZERO); + assert!(asin_from_gap(I16F16::from_num(-0.5)).is_err()); + Ok(()) + } +} diff --git a/tests/unit/ops/arith.rs b/tests/unit/ops/arith.rs new file mode 100644 index 0000000..55b38d8 --- /dev/null +++ b/tests/unit/ops/arith.rs @@ -0,0 +1,223 @@ +#[cfg(test)] +mod tests { + use crate::unit::support::Lcg; + use fixed::types::{I16F16, I32F32, I64F64}; + use fixed_analytics::Rounding::{Down, Nearest, Up}; + use fixed_analytics::{ + Rounding, div_or_saturate, div_rounded, dot, dot_rounded, mul_add, mul_add_rounded, + mul_div, mul_div_rounded, mul_rounded, sqrt_or_zero, + }; + + /// `num/den` rounded by `mode` to a raw `I32F32`, saturating, for `den > 0`. + fn exact(num: i128, den: i128, mode: Rounding) -> I32F32 { + let floor = num.div_euclid(den); + let rem = num.rem_euclid(den); + let raw = match mode { + Down => floor, + Up => floor + i128::from(rem != 0), + Nearest => floor + i128::from(2 * rem > den || (2 * rem == den && floor % 2 != 0)), + }; + I32F32::from_bits(i64::try_from(raw).unwrap_or(if raw < 0 { i64::MIN } else { i64::MAX })) + } + + #[test] + fn mul_div_keeps_the_product_whole() { + let big = I16F16::from_num(30_000); + // 30000² overflows I16F16 long before the division brings it back. + assert_eq!(mul_div(big, big, big), big); + let third = mul_div(I64F64::ONE, I64F64::ONE, I64F64::from_num(3)); + assert_eq!(third, I64F64::from_bits(0x5555_5555_5555_5555)); + } + + #[test] + fn mul_div_by_zero_saturates_by_sign() { + let (two, zero) = (I32F32::from_num(2), I32F32::ZERO); + assert_eq!(mul_div(two, two, zero), I32F32::MAX); + assert_eq!(mul_div(-two, two, zero), I32F32::MIN); + assert_eq!(mul_div(zero, two, zero), I32F32::ZERO); + assert_eq!(mul_div(I32F32::MAX, I32F32::MAX, I32F32::ONE), I32F32::MAX); + } + + #[test] + fn mul_add_rounds_once_to_nearest() { + let ulp = I16F16::from_bits(1); + let half = I16F16::from_bits(0x8000); + // ulp·½ is half an ulp: ties to the even neighbour, 0 then 2 ulps. + assert_eq!(mul_add(ulp, half, I16F16::ZERO), I16F16::ZERO); + assert_eq!(mul_add(ulp, half, ulp), I16F16::from_bits(2)); + // 182² = 33124 is past MAX; the sum is not. + let big = I16F16::from_num(182); + assert_eq!(mul_add(big, big, I16F16::MIN), I16F16::from_num(356)); + } + + #[test] + fn div_or_saturate_never_panics() { + let one = I16F16::ONE; + assert_eq!(div_or_saturate(one, I16F16::ZERO), I16F16::MAX); + assert_eq!(div_or_saturate(-one, I16F16::ZERO), I16F16::MIN); + assert_eq!(div_or_saturate(I16F16::ZERO, I16F16::ZERO), I16F16::ZERO); + assert_eq!( + div_or_saturate(one, I16F16::from_num(3)), + I16F16::from_bits(0x5555) + ); + assert_eq!( + div_or_saturate(I16F16::from_num(2), I16F16::from_num(3)), + I16F16::from_bits(0xAAAB) + ); + } + + #[test] + fn dot_sums_exactly_before_rounding() { + let big = I16F16::from_num(200); + let a = [big, big, big]; + let b = [big, -big, I16F16::from_num(0.25)]; + assert_eq!(dot(&a, &b), I16F16::from_num(50)); + assert_eq!(dot(&a, &b[..1]), I16F16::MAX); + assert_eq!(dot::(&[], &[]), I16F16::ZERO); + let m = [I64F64::MIN; 8]; + let n = [ + I64F64::MIN, + I64F64::MIN, + I64F64::MIN, + I64F64::MIN, + -I64F64::MAX, + -I64F64::MAX, + -I64F64::MAX, + -I64F64::MAX, + ]; + // Eight products of about 2^126 sum past 2^256 before cancelling. + assert_eq!(dot(&m[..4], &n[..4]), I64F64::MAX); + } + + #[test] + fn sqrt_or_zero_is_total() { + assert_eq!(sqrt_or_zero(I16F16::from_num(-4)), I16F16::ZERO); + assert_eq!(sqrt_or_zero(I16F16::ZERO), I16F16::ZERO); + assert_eq!(sqrt_or_zero(I16F16::from_num(9)), I16F16::from_num(3)); + } + + #[test] + fn directed_forms_round_the_exact_value_their_way() { + let mut rng = Lcg(0xD1); + let one = 1i128 << 32; + for _ in 0..20_000 { + let scale = |g: &mut Lcg| g.next_u64().cast_signed() >> (g.next_u64() % 40); + let (a, b, c) = (scale(&mut rng), scale(&mut rng), scale(&mut rng)); + let (fa, fb, fc) = ( + I32F32::from_bits(a), + I32F32::from_bits(b), + I32F32::from_bits(c), + ); + let (ra, rb, rc) = (i128::from(a), i128::from(b), i128::from(c)); + for mode in [Down, Nearest, Up] { + assert_eq!(mul_rounded(fa, fb, mode), exact(ra * rb, one, mode)); + assert_eq!( + mul_add_rounded(fa, fb, fc, mode), + exact(ra * rb + rc * one, one, mode) + ); + if c != 0 { + let (num, den) = if c < 0 { + (-ra * rb, -rc) + } else { + (ra * rb, rc) + }; + assert_eq!(mul_div_rounded(fa, fb, fc, mode), exact(num, den, mode)); + let (scaled, by) = if c < 0 { + (-ra * one, -rc) + } else { + (ra * one, rc) + }; + assert_eq!(div_rounded(fa, fc, mode), exact(scaled, by, mode)); + } + } + assert_eq!(mul_div_rounded(fa, fb, fc, Nearest), mul_div(fa, fb, fc)); + assert_eq!(div_rounded(fa, fc, Nearest), div_or_saturate(fa, fc)); + assert_eq!(mul_add_rounded(fa, fb, fc, Nearest), mul_add(fa, fb, fc)); + } + } + + #[test] + fn directed_dot_rounds_its_exact_sum() { + let mut rng = Lcg(0xD2); + for _ in 0..5000 { + let v: [i32; 6] = core::array::from_fn(|_| { + i32::try_from(rng.next_u64().cast_signed() >> 32).unwrap_or(0) + }); + let [a0, a1, a2, b0, b1, b2] = v; + let (fa, fb) = ( + [a0, a1, a2].map(I16F16::from_bits), + [b0, b1, b2].map(I16F16::from_bits), + ); + let wide = |x: i32, y: i32| i128::from(x) * i128::from(y); + let sum = wide(a0, b0) + wide(a1, b1) + wide(a2, b2); + let floor = sum.div_euclid(1 << 16); + let clamp = |r: i128| { + I16F16::from_bits(i32::try_from(r).unwrap_or(if r < 0 { + i32::MIN + } else { + i32::MAX + })) + }; + let (down, up) = (dot_rounded(&fa, &fb, Down), dot_rounded(&fa, &fb, Up)); + assert_eq!(down, clamp(floor)); + assert_eq!(up, clamp(floor + i128::from(sum.rem_euclid(1 << 16) != 0))); + assert_eq!(dot_rounded(&fa, &fb, Nearest), dot(&fa, &fb)); + } + } + + #[test] + fn directed_forms_bracket_on_128_bits() { + let ulp = I64F64::DELTA; + let half = I64F64::from_num(0.5); + assert_eq!( + (mul_rounded(ulp, half, Down), mul_rounded(ulp, half, Up)), + (I64F64::ZERO, ulp) + ); + assert_eq!( + (mul_rounded(-ulp, half, Down), mul_rounded(-ulp, half, Up)), + (-ulp, I64F64::ZERO) + ); + let three = I64F64::from_num(3); + assert_eq!( + div_rounded(I64F64::ONE, three, Down), + I64F64::from_bits(0x5555_5555_5555_5555) + ); + assert_eq!( + div_rounded(I64F64::ONE, three, Up), + I64F64::from_bits(0x5555_5555_5555_5556) + ); + assert_eq!( + div_rounded(-I64F64::ONE, three, Down), + -I64F64::from_bits(0x5555_5555_5555_5556) + ); + assert_eq!(mul_rounded(I64F64::MAX, three, Down), I64F64::MAX); + assert_eq!(div_rounded(-I64F64::ONE, I64F64::ZERO, Up), I64F64::MIN); + assert_eq!( + mul_div_rounded(I64F64::ZERO, three, I64F64::ZERO, Up), + I64F64::ZERO + ); + let mut rng = Lcg(0xD3); + for _ in 0..5000 { + let pick = |g: &mut Lcg| { + I64F64::from_bits( + (i128::from(g.next_u64().cast_signed()) << 64 | i128::from(g.next_u64())) + >> (g.next_u64() % 100), + ) + }; + let (a, b, c) = (pick(&mut rng), pick(&mut rng), pick(&mut rng)); + let forms: [&dyn Fn(Rounding) -> I64F64; 4] = [ + &|m| mul_rounded(a, b, m), + &|m| mul_div_rounded(a, b, c, m), + &|m| mul_add_rounded(a, b, c, m), + &|m| dot_rounded(&[a, b], &[c, a], m), + ]; + for f in forms { + let (down, near, up) = (f(Down), f(Nearest), f(Up)); + assert!( + down <= near && near <= up && up.saturating_sub(down) <= ulp, + "{a} {b} {c}" + ); + } + } + } +} diff --git a/tests/unit/ops/circular.rs b/tests/unit/ops/circular.rs index 27a240e..3c3d9c5 100644 --- a/tests/unit/ops/circular.rs +++ b/tests/unit/ops/circular.rs @@ -448,16 +448,17 @@ mod tests { use core::f64::consts::PI; let pole_3pi2 = 3.0 * PI / 2.0; - let far_from_pole = I16F16::from_num(pole_3pi2 - 0.0001); - let near_pole = I16F16::from_num(pole_3pi2 - 0.00005); + // tan(3π/2 − d) ≈ 1/d, which passes MAX = 32768 below d ≈ 3.05e-5. + let far_from_pole = I16F16::from_num(pole_3pi2 - 0.00005); + let near_pole = I16F16::from_num(pole_3pi2 - 0.00002); assert!( !is_max_16(tan(far_from_pole)), - "tan(3π/2 - 0.0001) should not saturate" + "tan(3π/2 - 0.00005) should not saturate" ); assert!( is_max_16(tan(near_pole)), - "tan(3π/2 - 0.00005) should saturate to MAX" + "tan(3π/2 - 0.00002) should saturate to MAX" ); } } @@ -478,35 +479,58 @@ mod reduction_and_accuracy { #[test] fn sin_cos_is_correct_within_pi_of_max() { - // n·2π exceeds MAX here; saturation used to return (0, 1). - for angle in [I32F32::MAX, I32F32::MIN, I32F32::MAX - I32F32::ONE] { + // Against mpmath at 400 bits; a rounded 2π would miss these by far. + let cases: [(I32F32, f64, f64); 3] = [ + ( + I32F32::MAX, + -0.971_310_175_848_310_1, + 0.237_816_194_346_652_58, + ), + ( + I32F32::MIN, + 0.971_310_175_792_939_2, + 0.237_816_194_572_803_37, + ), + ( + I32F32::MAX - I32F32::ONE, + -0.724_916_554_984_174_1, + -0.688_836_692_046_726_6, + ), + ]; + for (angle, want_s, want_c) in cases { let (s, c) = sin_cos(angle); - let r = reduced_f64(angle); let (s, c): (f64, f64) = (s.to_num(), c.to_num()); assert!( - (s - r.sin()).abs() < 1e-5, - "sin({angle}) = {s}, want {}", - r.sin() + (s - want_s).abs() < 2.4e-10, + "sin({angle}) = {s}, want {want_s}" ); assert!( - (c - r.cos()).abs() < 1e-5, - "cos({angle}) = {c}, want {}", - r.cos() + (c - want_c).abs() < 2.4e-10, + "cos({angle}) = {c}, want {want_c}" ); } - for angle in [I24F8::MAX, I24F8::MIN] { + let narrow: [(I24F8, f64, f64); 2] = [ + ( + I24F8::MAX, + 0.435_767_374_705_169_85, + -0.900_059_328_679_262_2, + ), + ( + I24F8::MIN, + -0.432_248_202_256_797_8, + -0.901_754_673_758_759_3, + ), + ]; + for (angle, want_s, want_c) in narrow { let (s, c) = sin_cos(angle); - let r = reduced_f64(angle); let (s, c): (f64, f64) = (s.to_num(), c.to_num()); assert!( - (s - r.sin()).abs() < 0.02, - "sin({angle}) = {s}, want {}", - r.sin() + (s - want_s).abs() < 0.002, + "sin({angle}) = {s}, want {want_s}" ); assert!( - (c - r.cos()).abs() < 0.02, - "cos({angle}) = {c}, want {}", - r.cos() + (c - want_c).abs() < 0.002, + "cos({angle}) = {c}, want {want_c}" ); } } diff --git a/tests/unit/ops/complex.rs b/tests/unit/ops/complex.rs new file mode 100644 index 0000000..81f2d94 --- /dev/null +++ b/tests/unit/ops/complex.rs @@ -0,0 +1,274 @@ +#[cfg(test)] +mod tests { + use crate::unit::support::TestResult; + use fixed::types::{I8F0, I8F24, I10F118, I16F0, I16F16, I28F100, I32F0, I64F64, I128F0}; + use fixed_analytics::{Complex, erf, erfc, fresnel, gamma}; + + fn z(re: f64, im: f64) -> Complex { + Complex::new(I64F64::from_num(re), I64F64::from_num(im)) + } + + fn near(got: Complex, re: f64, im: f64) -> bool { + let tol = 4e-16 * re.hypot(im).max(1.0); + (got.re.to_num::() - re).abs() <= tol && (got.im.to_num::() - im).abs() <= tol + } + + #[test] + fn arithmetic_rounds_each_part_once() { + let big = Complex::new(I16F16::from_num(200), I16F16::from_num(150)); + let q = big / big; + assert_eq!(q, Complex::new(I16F16::ONE, I16F16::ZERO)); + let p = big * Complex::new(I16F16::from_num(0.25), I16F16::from_num(0.5)); + assert_eq!( + p, + Complex::new(I16F16::from_num(-25), I16F16::from_num(137.5)) + ); + let zero = Complex::new(I16F16::ZERO, I16F16::ZERO); + assert_eq!(big / zero, Complex::new(I16F16::MAX, I16F16::MAX)); + assert_eq!(big.conj().im, I16F16::from_num(-150)); + assert_eq!( + Complex::new(I16F16::from_num(3), I16F16::from_num(4)).norm(), + I16F16::from_num(5) + ); + } + + #[test] + fn named_methods_are_the_operators() { + let (a, b) = (z(1.5, -2.25), z(-0.75, 0.5)); + assert_eq!(a.saturating_add(b), a + b); + assert_eq!(a.saturating_sub(b), a - b); + assert_eq!(a.saturating_mul(b), a * b); + assert_eq!(a.saturating_div(b), a / b); + assert_eq!(a.saturating_neg(), -a); + } + + #[test] + fn ln_and_pow_match_mpmath() -> TestResult { + let w = z(-3.0, 4.0); + assert!(near( + w.ln()?, + 1.609_437_912_434_100_4, + 2.214_297_435_588_181 + )); + assert!(near(w.pow(I64F64::from_num(0.5))?, 1.0, 2.0)); + assert!(near( + w.pow(I64F64::from_num(-1.25))?, + -0.124_516_151_201_356_84, + -0.048_829_006_850_428_146 + )); + assert!(z(0.0, 0.0).ln().is_err()); + assert!(z(0.0, 0.0).pow(-I64F64::ONE).is_err()); + assert_eq!(z(0.0, 0.0).pow(I64F64::ZERO)?, z(1.0, 0.0)); + assert_eq!(z(0.0, 2.0).pow(I64F64::from_num(3))?, z(0.0, -8.0)); + assert_eq!(z(-1.0, 1.0).pow(I64F64::from_num(4))?, z(-4.0, 0.0)); + let big = Complex::new(I16F0::from_num(0), I16F0::from_num(30)); + assert_eq!(big.pow(I16F0::from_num(7))?.re, I16F0::ZERO); + Ok(()) + } + + #[test] + fn weighted_exprel_matches_mpmath() -> TestResult { + let t = |x: f64| I64F64::from_num(x); + let check = |w: Complex, eps: f64, p: f64, k: [f64; 4]| -> TestResult { + let (k1, k2) = w.exprel_weighted(t(eps), t(p))?; + assert!(near(k1, k[0], k[1]), "{w:?} {eps} {p}: {k1:?}"); + assert!(near(k2, k[2], k[3]), "{w:?} {eps} {p}: {k2:?}"); + Ok(()) + }; + check( + z(0.3, 2.0), + 0.25, + 1.5, + [ + 0.359_227_728_201_932_8, + 0.870_528_713_966_726_7, + 0.345_972_961_403_165_2, + 0.321_314_056_573_191_9, + ], + )?; + check( + z(0.0, 40.0), + 0.062_5, + 3.0, + [ + 0.019_621_287_045_008_31, + 0.038_514_840_070_235_416, + 0.000_931_489_629_153_719_7, + 0.022_263_155_901_843_38, + ], + )?; + check( + z(-1.0, 0.5), + 0.875, + 0.0, + [ + 1.210_862_569_552_848, + 0.359_836_283_700_871, + 0.526_759_808_612_434_7, + 0.096_898_276_774_117, + ], + )?; + check( + z(0.0, 1.0), + 2f64.powi(-40), + 2f64.powi(30), + [ + 0.841_022_212_426_340_8, + 0.459_542_919_068_431_9, + 0.459_542_919_068_287_8, + 0.158_489_665_230_998_7, + ], + )?; + let w = z(0.75, -1.5); + assert_eq!( + w.exprel_weighted(t(0.0), t(4.0))?, + (w.exprel(), w.exprel2()) + ); + assert!(w.exprel_weighted(t(1.0), t(0.0)).is_err()); + assert!(w.exprel_weighted(t(0.5), t(-1.0)).is_err()); + Ok(()) + } + + #[test] + fn exponentials_match_mpmath() { + let one = z(1.0, 0.0); + assert!(near(one.exp(), core::f64::consts::E, 0.0)); + assert!(near(z(0.0, 0.0).exprel(), 1.0, 0.0)); + assert!(near(z(0.0, 0.0).exprel2(), 0.5, 0.0)); + let w = z(0.0, core::f64::consts::PI); + assert!(near(w.exp(), -1.0, 0.0)); + } + + #[test] + fn faddeeva_and_erf_match_mpmath() { + assert!(near( + z(0.5, 0.3).faddeeva(), + 0.614_851_539_146_991, + 0.303_124_349_647_351_1 + )); + assert!(near( + z(5.0, 0.2).faddeeva(), + 0.004_807_037_335_964_243, + 0.115_040_120_131_140_57 + )); + assert!(near( + z(-2.0, 1.0).faddeeva(), + 0.140_239_581_366_277_94, + -0.222_213_440_179_899_1 + )); + assert!(near( + z(0.5, 0.3).erf(), + 0.561_565_188_524_213_2, + 0.267_605_864_957_603_6 + )); + assert!(near( + z(-2.0, 1.0).erf(), + -1.003_606_342_725_651_8, + -0.011_259_006_028_815_025 + )); + } + + #[test] + fn exponential_integral_matches_mpmath() -> TestResult { + assert!(near( + z(0.5, 0.3).e1()?, + 0.422_211_324_225_017_94, + -0.305_371_136_174_266_65 + )); + assert!(near( + z(5.0, 0.2).e1()?, + 0.001_116_122_642_478_627, + -0.000_266_867_640_371_248_9 + )); + assert!(near( + z(0.0, 6.0).e1()?, + 0.068_057_243_893_247_13, + -0.146_108_775_514_390_08 + )); + // On the negative axis, the value from above it. + assert!(near( + z(-3.0, 0.0).e1()?, + -9.933_832_570_625_416, + -core::f64::consts::PI + )); + assert!(z(0.0, 0.0).e1().is_err()); + Ok(()) + } + + #[test] + fn incomplete_gamma_matches_mpmath() -> TestResult { + let a = I64F64::from_num(2.5); + assert!(near( + z(0.5, 0.3).gamma_upper(a)?, + 1.299_139_215_932_174_5, + -0.066_377_001_160_412_6 + )); + assert!(near( + z(-2.0, 1.0).gamma_upper(a)?, + -11.043_544_259_962_623, + 4.945_042_286_802_596 + )); + assert!(near( + z(0.0, 6.0).gamma_upper(a)?, + -10.838_494_867_793_583, + 10.170_960_665_022_385 + )); + assert!(near( + z(0.0, 0.0).gamma_upper(a)?, + 1.329_340_388_179_137, + 0.0 + )); + assert!(z(1.0, 1.0).gamma_upper(I64F64::ZERO).is_err()); + Ok(()) + } + + #[test] + fn wide_values_keep_exact_parts_and_phases() -> TestResult { + let base: i128 = (1 << 100) + 12_345; + let wide = |re: i128, im: i128| Complex::new(I128F0::from_num(re), I128F0::from_num(im)); + assert_eq!( + (wide(base, base + 1) * wide(base + 2, base + 3)).re, + I128F0::from_num(-2 * base - 3) + ); + let far = wide( + 16_669_777_780_427_424 << 60, + -(167_377_028_183_139_440 << 60), + ); + assert_eq!(far.faddeeva(), wide(i128::MAX, i128::MAX)); + let axis = Complex::new(I8F0::ZERO, I8F0::from_num(-7)).erf(); + assert_eq!(axis, Complex::new(I8F0::ZERO, I8F0::MIN)); + let on_axis = Complex::new(I8F0::from_num(-50), I8F0::ZERO).gamma_upper(I8F0::ONE)?; + assert_eq!(on_axis, Complex::new(I8F0::MAX, I8F0::ZERO)); + let far_left = Complex::new(I16F0::from_num(-12_570), I16F0::from_num(-2)).e1()?; + assert_eq!(far_left, Complex::new(I16F0::MAX, I16F0::MIN)); + assert_eq!(erfc(I32F0::MIN), I32F0::from_num(2)); + let part = Complex::new(I8F24::from_bits(16), I8F24::from_num(4)) + .erf() + .re; + assert!((part.to_num::() - 9.562_398_898_839_027).abs() < 6e-8); + let tiny = I28F100::from_bits(1 << 10); + let near_axis = + Complex::new(I28F100::from_num(-40), tiny).gamma_upper(I28F100::from_num(3))?; + assert!((near_axis.im.to_num::() + 3.042_283_268_878_188_5e-7).abs() < 1e-22); + let one = I10F118::ONE; + let small = Complex::new(one + one, one).gamma_upper(I10F118::from_bits(1 << 8))?; + assert!((small.re.to_bits() - 3_119_751_710_710_189_250_821_825_353_125_381).abs() <= 1); + assert!((small.im.to_bits() + 14_775_364_147_287_538_117_270_599_311_667_794).abs() <= 1); + Ok(()) + } + + #[test] + fn real_special_functions() -> TestResult { + let f = |x: f64| I64F64::from_num(x); + assert!((erf(f(1.0)).to_num::() - 0.842_700_792_949_714_9).abs() < 3e-16); + assert!((erfc(f(2.0)).to_num::() - 0.004_677_734_981_047_266).abs() < 1e-17); + assert_eq!(erfc(f(-30.0)), f(2.0)); + let (c, s) = fresnel(f(1.0)); + assert!((c.to_num::() - 0.779_893_400_376_822_8).abs() < 3e-16); + assert!((s.to_num::() - 0.438_259_147_390_354_77).abs() < 3e-16); + assert!((gamma(f(4.5))?.to_num::() - 11.631_728_396_567_45).abs() < 1e-14); + assert!((gamma(f(0.5))?.to_num::() - 1.772_453_850_905_516).abs() < 5e-16); + assert!(gamma(f(0.0)).is_err()); + Ok(()) + } +} diff --git a/tests/unit/ops/edges.rs b/tests/unit/ops/edges.rs new file mode 100644 index 0000000..9374c1f --- /dev/null +++ b/tests/unit/ops/edges.rs @@ -0,0 +1,122 @@ +#[cfg(test)] +mod tests { + use crate::unit::support::TestResult; + use fixed::types::{I16F16, I32F32, I64F64}; + use fixed_analytics::{ + Complex, Error, Modulus, carlson_rc, carlson_rd, carlson_rj, carlson_rj_roots, ellip_e, + ellip_k, ellip_pi, ellip_pi_complete, jacobi_am, jacobi_inverse, jacobi_sn_cn_dn, + kepler_universal, root_in, wrap_principal, wrap_radians, + }; + + #[test] + fn wraps_that_round_onto_an_end_step_back_inside() { + // Just below 2π, where the rounding first lands on 2π itself. + let tau = core::f64::consts::TAU; + let narrow = wrap_radians(-I16F16::DELTA).to_num::(); + assert!(narrow < tau && narrow > tau - 1e-3); + let wide = wrap_radians(-I64F64::DELTA).to_num::(); + assert!(wide <= tau && wide > tau - 1e-9); + // Exact wraps that round onto 2π, nearest 0 modulo 2π, and below −π. + assert_eq!(wrap_radians(I16F16::from_bits(1_647_099)), I16F16::ZERO); + let below = wrap_principal(I32F32::from_bits(1_524_713_260_611)).to_num::(); + let pi = core::f64::consts::PI; + assert!(below > -pi && below < -pi + 1e-9); + for x in [ + I32F32::from_num(pi), + I32F32::from_num(-pi), + I32F32::from_num(3.0 * pi), + ] { + let w = wrap_principal(x).to_num::(); + assert!(w > -pi && w <= pi + 1e-9, "{x}: {w}"); + } + } + + #[test] + fn solvers_take_their_edge_cases() -> TestResult { + let line = |x: I32F32| x - I32F32::from_num(2); + let two = I32F32::from_num(2); + assert_eq!(root_in(line, I32F32::ZERO, two, 8)?, (two, two)); + assert_eq!(root_in(line, two, I32F32::from_num(5), 8)?, (two, two)); + let (a, b) = root_in(line, I32F32::from_num(9), I32F32::ONE, 100)?; + assert!(a <= two && two <= b); + let t = |x: f64| I64F64::from_num(x); + assert!(kepler_universal(t(-1.0), t(0.0), t(1.0), t(1.0)).is_err()); + let far = kepler_universal(t(1.0), t(0.0), t(-1.0), t(1e6))?; + assert!(far > t(10.0) && far < t(20.0)); + let back = kepler_universal(t(1.0), t(0.5), t(0.5), t(-3.0))?; + assert!(back < t(0.0)); + Ok(()) + } + + #[test] + fn elliptic_functions_take_their_edge_cases() -> TestResult { + let m = |x: f64| { + Modulus::from_parameter(I32F32::from_num(x)).ok_or(Error::domain("test", "modulus")) + }; + let one = m(1.0)?; + assert!(ellip_k(one).is_err()); + assert_eq!(ellip_e(one), I32F32::ONE); + let (sn, cn, dn) = jacobi_sn_cn_dn(I32F32::from_num(0.5), one); + assert!((sn.to_num::() - 0.5f64.tanh()).abs() < 1e-9 && cn == dn); + assert!(jacobi_am(I32F32::from_num(5000), one) > I32F32::ONE); + assert!(jacobi_inverse(I32F32::ONE, I32F32::ZERO, I32F32::ZERO, m(0.5)?).is_err()); + let half = m(0.5)?; + assert_eq!(half.parameter(), I32F32::from_num(0.5)); + assert!((half.complement().to_num::() - 0.5f64.sqrt()).abs() < 1e-9); + assert!(Modulus::from_parameter(I32F32::from_num(1.5)).is_none()); + assert!(Modulus::::from_complement(-I32F32::ONE).is_none()); + let kc = Modulus::from_complement(I32F32::from_num(0.6)) + .ok_or(Error::domain("test", "modulus"))?; + assert!((kc.parameter().to_num::() - 0.64).abs() < 1e-9); + assert!(carlson_rc(I32F32::ZERO, I32F32::ONE)? > I32F32::ONE); + assert!(carlson_rc(-I32F32::ONE, I32F32::ONE).is_err()); + assert!(carlson_rd(I32F32::ZERO, I32F32::ZERO, I32F32::ONE).is_err()); + assert!(carlson_rj(I32F32::ONE, I32F32::ONE, I32F32::ONE, I32F32::ZERO).is_err()); + assert!(carlson_rj_roots(I32F32::ONE, I32F32::ONE, I32F32::ONE, I32F32::ZERO).is_err()); + assert!(carlson_rj_roots(I32F32::ZERO, I32F32::ZERO, I32F32::ONE, I32F32::ONE).is_err()); + let far = carlson_rj( + I32F32::from_num(0.01), + I32F32::from_num(0.5), + I32F32::ONE, + I32F32::from_num(400), + )?; + assert!(far > I32F32::ZERO && far < I32F32::from_num(0.05)); + let rj0 = carlson_rj( + I32F32::ZERO, + I32F32::from_num(0.5), + I32F32::ONE, + I32F32::from_num(400), + )?; + assert!(rj0 > I32F32::ZERO); + assert!(ellip_pi_complete(I32F32::ONE, half).is_err()); + assert!( + ellip_pi( + I32F32::from_num(2), + I32F32::ONE, + I32F32::ZERO, + I32F32::ONE, + half + ) + .is_err() + ); + let wide = ellip_pi( + I32F32::from_num(0.5), + I32F32::from_num(0.6), + I32F32::from_num(-0.8), + I32F32::ONE, + one, + )?; + assert_eq!(wide, I32F32::MAX); + Ok(()) + } + + #[test] + fn complex_edges() -> TestResult { + let zero = Complex::new(I16F16::ZERO, I16F16::ZERO); + assert_eq!(zero.arg(), I16F16::ZERO); + assert_eq!(zero.pow(I16F16::ONE)?, zero); + let up = Complex::new(I16F16::ZERO, I16F16::ONE); + assert_eq!(up.arg(), fixed_analytics::atan2(I16F16::ONE, I16F16::ZERO)); + Ok(()) + } +} diff --git a/tests/unit/ops/elliptic.rs b/tests/unit/ops/elliptic.rs new file mode 100644 index 0000000..62bb279 --- /dev/null +++ b/tests/unit/ops/elliptic.rs @@ -0,0 +1,234 @@ +#[cfg(test)] +mod tests { + use crate::unit::support::{Lcg, TestResult}; + use fixed::types::{I32F32, I64F64}; + use fixed_analytics::{ + Error, Modulus, carlson_rc, carlson_rd, carlson_rf, carlson_rf_roots, carlson_rj, ellip_e, + ellip_k, ellip_pi, ellip_pi_complete, ellip_pi_increment, jacobi_am, jacobi_inverse, + jacobi_sn_cn_dn, + }; + + fn half() -> Result, Error> { + Modulus::from_parameter(I64F64::from_num(0.5)).ok_or(Error::domain("test", "m")) + } + + fn near(got: I64F64, want: f64, tol: f64) -> bool { + (got.to_num::() - want).abs() <= tol + } + + #[test] + fn complete_integrals_match_mpmath() -> TestResult { + let m = half()?; + assert!(near(ellip_k(m)?, 1.854_074_677_301_372, 1e-15)); + assert!(near(ellip_e(m), 1.350_643_881_047_675_5, 1e-15)); + assert!(near( + ellip_pi_complete(I64F64::from_num(-0.5), m)?, + 1.487_846_992_668_798_4, + 1e-15 + )); + assert!(near( + ellip_pi_complete(I64F64::from_num(-1000), m)?, + 0.050_138_952_552_005_41, + 1e-15 + )); + let one = Modulus::from_complement(I64F64::ZERO).ok_or(Error::domain("test", "kc"))?; + assert!(ellip_k(one).is_err()); + assert_eq!(ellip_e(one), I64F64::ONE); + Ok(()) + } + + #[test] + fn jacobi_functions_match_mpmath() -> TestResult { + let (sn, cn, dn) = jacobi_sn_cn_dn(I64F64::from_num(0.5), half()?); + assert!(near(sn, 0.470_750_473_655_657_3, 1e-15)); + assert!(near(cn, 0.882_266_394_890_440_3, 1e-15)); + assert!(near(dn, 0.942_972_425_777_385_7, 1e-15)); + let m9 = + Modulus::from_parameter(I64F64::from_num(0.9)).ok_or(Error::domain("test", "m"))?; + let far = jacobi_sn_cn_dn(I64F64::from_num(3), m9); + assert!(near(far.0, 0.990_630_599_937_832_6, 1e-15)); + assert!(near(far.1, -0.136_568_717_013_853_34, 1e-15)); + assert!(near(far.2, 0.341_739_539_737_691_07, 1e-15)); + Ok(()) + } + + #[test] + fn jacobi_identities_and_inverse() -> TestResult { + let mut rng = Lcg(0xE11); + for _ in 0..300 { + let kc = I32F32::from_num(rng.range(0.0, 1.0)); + let m = Modulus::from_complement(kc).ok_or(Error::domain("test", "kc"))?; + let k: f64 = ellip_k(m)?.to_num(); + let u = I32F32::from_num(rng.range(-1.9, 1.9) * k); + let (sn, cn, dn) = jacobi_sn_cn_dn(u, m); + let (sf, cf, df): (f64, f64, f64) = (sn.to_num(), cn.to_num(), dn.to_num()); + let kf = kc.to_num::(); + let mm = kf.mul_add(-kf, 1.0); + assert!(sf.mul_add(sf, cf * cf) - 1.0 < 1e-8); + assert!((mm * sf).mul_add(sf, df * df) - 1.0 < 1e-8); + let back: f64 = jacobi_inverse(sn, cn, dn, m)?.to_num(); + assert!( + (back - u.to_num::()).abs() < 1e-6 * k, + "u = {u}, back = {back}" + ); + let am = jacobi_am(u, m).to_num::(); + assert!((am.sin() - sf).abs() < 1e-8 && (am.cos() - cf).abs() < 1e-8); + } + Ok(()) + } + + #[test] + fn separatrix_closed_forms() -> TestResult { + let one = Modulus::from_complement(I64F64::ZERO).ok_or(Error::domain("test", "kc"))?; + let (sn, cn, dn) = jacobi_sn_cn_dn(I64F64::from_num(2), one); + assert!(near(sn, 2f64.tanh(), 1e-15)); + assert!(near(cn, 1.0 / 2f64.cosh(), 1e-15) && cn == dn); + assert_eq!( + jacobi_sn_cn_dn(I64F64::MAX, one), + (I64F64::ONE, I64F64::ZERO, I64F64::ZERO) + ); + Ok(()) + } + + #[test] + fn carlson_integrals_match_mpmath() -> TestResult { + let f = I64F64::from_num::; + assert!(near( + carlson_rf(f(1.0), f(2.0), f(3.0))?, + 0.726_945_935_468_908_2, + 1e-15 + )); + assert!(near( + carlson_rj(f(2.0), f(3.0), f(4.0), f(5.0))?, + 0.142_975_796_671_567_54, + 1e-15 + )); + let big_p = carlson_rj(f(0.001), f(0.002), f(0.004), f(1000.0))?; + assert!(near(big_p, 0.064_844_213_471_528_12, 1e-15)); + assert!(near( + carlson_rc(f(2.25), f(2.0))?, + core::f64::consts::LN_2, + 1e-15 + )); + assert!(near( + carlson_rd(f(0.0), f(2.0), f(1.0))?, + 1.797_210_352_103_388_3, + 1e-15 + )); + // R_F(t², t², t²) = 1/t, though t² = 2^-108 underflows I64F64. + let tiny = I64F64::from_bits(1 << 10); + assert_eq!( + carlson_rf_roots(tiny, tiny, tiny)?, + I64F64::from_num(1u64 << 54) + ); + assert!(carlson_rf(f(0.0), f(0.0), f(1.0)).is_err()); + assert!(carlson_rj(f(1.0), f(1.0), f(1.0), f(0.0)).is_err()); + Ok(()) + } + + #[test] + fn incomplete_third_kind() -> TestResult { + let (sn, cn, dn) = ( + I64F64::from_num(0.644_217_687_237_691), + I64F64::from_num(0.764_842_187_284_488_4), + I64F64::from_num(0.890_220_077_129_841_1), + ); + let p = ellip_pi(I64F64::from_num(-0.5), sn, cn, dn, half()?)?; + assert!(near(p, 0.679_470_265_670_900_9, 1e-14)); + assert!( + ellip_pi( + I64F64::from_num(2), + I64F64::ONE, + I64F64::ZERO, + I64F64::from_num(0.7), + half()? + ) + .is_err() + ); + Ok(()) + } + + #[test] + fn pi_increments_match_mpmath() -> TestResult { + let t = |x: f64| I64F64::from_num(x); + let rise = |n: f64, u: f64, s: I64F64| ellip_pi_increment(t(n), t(u), s, half()?); + assert!(near(rise(-0.5, 0.3, t(0.2))?, 0.186_114_348_208_225, 1e-15)); + assert!(near( + rise(-0.5, 1.8, t(0.5))?, + 0.336_687_301_019_070_5, + 1e-15 + )); + assert!(near(rise(-0.5, 5.0, t(3.0))?, 2.425_522_980_217_884, 4e-15)); + assert!(near( + rise(-0.5, -2.0, t(7.5))?, + 6.007_295_236_841_554, + 8e-15 + )); + assert!(near( + rise(-0.5, 1.854, t(0.001))?, + 6.666_666_960_258_615e-4, + 1e-17 + )); + assert!(near( + rise(0.75, 100.0, t(0.25))?, + 0.250_983_515_421_914_3, + 1e-14 + )); + assert!(near( + rise(0.75, -3.5, t(-20.0))?, + -41.369_096_674_879_01, + 4e-14 + )); + let tiny = rise(-0.5, 1.0, I64F64::from_bits(1 << 34))?; + assert!(near( + tiny, + 7.042_637_426_469_899e-10, + I64F64::DELTA.to_num() + )); + assert_eq!(rise(-0.5, 1.0, I64F64::ZERO)?, I64F64::ZERO); + Ok(()) + } + + #[test] + fn pi_increments_add_and_agree_with_pi() -> TestResult { + let modulus = + Modulus::from_parameter(I32F32::from_num(0.8)).ok_or(Error::domain("test", "m"))?; + let n = I32F32::from_num(0.3); + let ulp = I32F32::DELTA; + let mut rng = Lcg(0x91); + for _ in 0..300 { + let u = I32F32::from_num(rng.range(-6.0, 6.0)); + let (first, second) = ( + I32F32::from_num(rng.range(-3.0, 3.0)), + I32F32::from_num(rng.range(-3.0, 3.0)), + ); + let whole = ellip_pi_increment(n, u, first + second, modulus)?; + let parts = ellip_pi_increment(n, u, first, modulus)? + + ellip_pi_increment(n, u + first, second, modulus)?; + assert!( + (whole - parts).abs() <= ulp * 2, + "{u} {first} {second}: {whole} {parts}" + ); + } + let u = I32F32::from_num(0.9); + let (sn, cn, dn) = jacobi_sn_cn_dn(u, modulus); + let from_zero = ellip_pi_increment(n, I32F32::ZERO, u, modulus)?; + assert!((from_zero - ellip_pi(n, sn, cn, dn, modulus)?).abs() <= ulp * 4); + Ok(()) + } + + #[test] + fn pi_increments_on_the_separatrix_and_past_n_of_1() -> TestResult { + let t = |x: f64| I64F64::from_num(x); + let sep = Modulus::from_complement(I64F64::ZERO).ok_or(Error::domain("test", "m"))?; + let ahead = ellip_pi_increment(t(-0.5), t(3.0), t(2.0), sep)?; + assert!(near(ahead, 1.334_413_912_023_380_5, 1e-15)); + let across = ellip_pi_increment(t(-0.5), t(-1.0), t(1.5), sep)?; + assert!(near(across, 1.381_753_614_509_052_7, 1e-15)); + let far = ellip_pi_increment(t(-0.5), t(5000.0), t(3.0), sep)?; + assert_eq!(far, t(2.0)); + assert!(ellip_pi_increment(t(1.0), t(0.2), t(0.3), half()?).is_err()); + assert!(ellip_pi_increment(t(1.5), t(0.2), t(0.3), sep).is_err()); + Ok(()) + } +} diff --git a/tests/unit/ops/exponential.rs b/tests/unit/ops/exponential.rs index 7946e5b..dcbee71 100644 --- a/tests/unit/ops/exponential.rs +++ b/tests/unit/ops/exponential.rs @@ -506,7 +506,7 @@ mod reduction { ); assert_eq!( pow(-I16F16::from_num(200), I16F16::from_num(3))?, - -I16F16::MAX + I16F16::MIN ); assert_eq!(pow(-I16F16::from_num(3), I16F16::ZERO)?, I16F16::ONE); assert_eq!(pow(-I16F16::ONE, I16F16::from_num(7))?, -I16F16::ONE); diff --git a/tests/unit/ops/interval.rs b/tests/unit/ops/interval.rs new file mode 100644 index 0000000..dae36d9 --- /dev/null +++ b/tests/unit/ops/interval.rs @@ -0,0 +1,284 @@ +#[cfg(test)] +mod tests { + use crate::unit::support::Lcg; + use fixed::types::{I16F16, I32F32}; + use fixed_analytics::{ + atan_range, atan_ratio, atan_ratio_range, atan2, atan2_range, atanh_ratio, + atanh_ratio_range, enclose, exp, exp_moments, exp_moments_range, exp_range, exprel, + exprel_range, exprel2, exprel2_range, hypot, hypot_range, log1p, log1p_range, log1p_ratio, + log1p_ratio_range, log1p_residual_ratio, log1p_residual_ratio_range, sin, sin_cos_range, + sin_cos_range_turns, sqrt_or_zero, sqrt_range, stumpff, stumpff_range, tan_ratio, + tan_ratio_range, tanh_range, tanh_ratio, tanh_ratio_range, + }; + + fn across(lo: I32F32, hi: I32F32, n: i64) -> impl Iterator { + let span = i128::from(hi.to_bits()) - i128::from(lo.to_bits()); + (0..=n).map(move |k| { + let step = span * i128::from(k) / i128::from(n); + I32F32::from_bits(i64::try_from(i128::from(lo.to_bits()) + step).unwrap_or(0)) + }) + } + + fn draw(rng: &mut Lcg, options: &[f64]) -> f64 { + let at = usize::try_from(rng.next_u64() % options.len() as u64).unwrap_or(0); + options.get(at).copied().unwrap_or(0.0) + } + + fn holds(range: (I32F32, I32F32), v: I32F32) -> bool { + range.0 <= v && v <= range.1 + } + + #[test] + fn enclose_spans_an_ulp_each_way() { + let y = exp(I32F32::ONE); + let (lo, hi) = enclose(y); + let e = core::f64::consts::E; + assert!(lo.to_num::() < e && e < hi.to_num::()); + assert_eq!(enclose(I16F16::MAX).1, I16F16::MAX); + } + + #[test] + fn sin_cos_range_finds_interior_extremes() { + let ((smin, smax), (cmin, cmax)) = sin_cos_range(I32F32::ZERO, I32F32::PI); + assert_eq!(smax, I32F32::ONE); + assert!(smin <= I32F32::ZERO); + assert_eq!((cmin, cmax), (-I32F32::ONE, I32F32::ONE)); + let wide = sin_cos_range(I32F32::from_num(-10), I32F32::from_num(10)); + assert_eq!( + wide, + ((-I32F32::ONE, I32F32::ONE), (-I32F32::ONE, I32F32::ONE)) + ); + let (s, _) = sin_cos_range(I32F32::from_num(0.25), I32F32::from_num(0.5)); + assert!(s.0 <= sin(I32F32::from_num(0.25)) && sin(I32F32::from_num(0.5)) <= s.1); + assert!(s.1 < I32F32::ONE); + } + + #[test] + fn turn_ranges_hold_every_sampled_value() { + let quarter = I32F32::from_num(0.25); + let below = -I32F32::DELTA; + assert_eq!( + sin_cos_range_turns(I32F32::ZERO, quarter), + ((below, I32F32::ONE), (below, I32F32::ONE)) + ); + assert_eq!( + sin_cos_range_turns(I32F32::from_num(-3), I32F32::from_num(-2)), + ((-I32F32::ONE, I32F32::ONE), (-I32F32::ONE, I32F32::ONE)) + ); + let mut rng = Lcg(0x7E57); + for _ in 0..2000 { + let a = rng.range(-8.0, 8.0); + let b = a + rng.range(0.0, 0.6); + let (lo, hi) = (I32F32::from_num(a), I32F32::from_num(b)); + let ((smin, smax), (cmin, cmax)) = sin_cos_range_turns(hi, lo); + for k in 0..=16 { + let t = lo.to_num::().mul_add( + 1.0 - f64::from(k) / 16.0, + hi.to_num::() * f64::from(k) / 16.0, + ); + let x = t * core::f64::consts::TAU; + let inside = |v: f64, l: I32F32, h: I32F32| { + l.to_num::() - 1e-12 <= v && v <= h.to_num::() + 1e-12 + }; + assert!( + inside(x.sin(), smin, smax), + "sin at {t} turns outside [{smin}, {smax}]" + ); + assert!( + inside(x.cos(), cmin, cmax), + "cos at {t} turns outside [{cmin}, {cmax}]" + ); + } + } + } + + #[test] + fn ranges_hold_every_sampled_value() { + let mut rng = Lcg(0x5EED); + for _ in 0..2000 { + let a = rng.range(-50.0, 50.0); + let b = a + rng.range(0.0, 3.0); + let (lo, hi) = (I32F32::from_num(a), I32F32::from_num(b)); + let ((smin, smax), (cmin, cmax)) = sin_cos_range(lo, hi); + let (lo4, hi4) = (lo / 4, hi / 4); + let (emin, emax) = exp_range(lo4, hi4); + let (amin, amax) = atan_range(lo, hi); + let (tmin, tmax) = tanh_range(lo, hi); + for k in 0..=16 { + let x = (lo.to_num::()).mul_add( + 1.0 - f64::from(k) / 16.0, + hi.to_num::() * f64::from(k) / 16.0, + ); + let inside = |v: f64, l: I32F32, h: I32F32| { + l.to_num::() - 1e-12 <= v && v <= h.to_num::() + 1e-12 + }; + assert!( + inside(x.sin(), smin, smax), + "sin({x}) outside [{smin}, {smax}]" + ); + assert!( + inside(x.cos(), cmin, cmax), + "cos({x}) outside [{cmin}, {cmax}]" + ); + let x4 = (lo4.to_num::()).mul_add( + 1.0 - f64::from(k) / 16.0, + hi4.to_num::() * f64::from(k) / 16.0, + ); + assert!(inside(x4.exp(), emin, emax), "exp outside at {x4}"); + assert!(inside(x.atan(), amin, amax), "atan outside at {x}"); + assert!(inside(x.tanh(), tmin, tmax), "tanh outside at {x}"); + } + } + } + + #[test] + fn kernel_ranges_hold_every_value_between() -> crate::unit::support::TestResult { + let mut rng = Lcg(0x4A7E); + for _ in 0..1500 { + let a = rng.range(-30.0, 30.0); + let width = draw(&mut rng, &[1e-6, 1e-3, 0.1, 2.0, 20.0]); + let (lo, hi) = ( + I32F32::from_num(a), + I32F32::from_num(a + rng.range(0.0, width)), + ); + let (ex, ex2, th, at, tn, sq) = ( + exprel_range(hi, lo), + exprel2_range(lo, hi), + tanh_ratio_range(lo, hi), + atan_ratio_range(lo, hi), + tan_ratio_range(lo, hi), + sqrt_range(lo, hi), + ); + let moments = exp_moments_range(lo, hi); + for x in across(lo, hi, 64) { + assert!( + holds(ex, exprel(x)) && holds(ex2, exprel2(x)), + "exprel at {x}" + ); + assert!( + holds(th, tanh_ratio(x)) && holds(at, atan_ratio(x)), + "ratio at {x}" + ); + assert!( + holds(tn, tan_ratio(x).0), + "tan_ratio at {x} in [{lo}, {hi}]" + ); + assert!(holds(sq, sqrt_or_zero(x)), "sqrt at {x}"); + for (range, v) in moments.iter().zip(exp_moments(x)) { + assert!(holds(*range, v), "moment at {x}"); + } + } + let (ul, uh) = (lo / 32, hi / 32); + if ul > -I32F32::ONE { + let (l1, lr, lq) = ( + log1p_range(ul, uh)?, + log1p_ratio_range(ul, uh)?, + log1p_residual_ratio_range(ul, uh)?, + ); + for x in across(ul, uh, 64) { + assert!( + holds(l1, log1p(x)?) && holds(lr, log1p_ratio(x)?), + "log1p at {x}" + ); + assert!(holds(lq, log1p_residual_ratio(x)?), "residual at {x}"); + } + } + if ul > -I32F32::ONE && uh < I32F32::ONE { + let range = atanh_ratio_range(ul, uh)?; + assert!(across(ul, uh, 64).all(|x| atanh_ratio(x).is_ok_and(|v| holds(range, v)))); + } + } + assert!(log1p_range(-I32F32::ONE, I32F32::ZERO).is_err()); + assert!(atanh_ratio_range(I32F32::ZERO, I32F32::ONE).is_err()); + Ok(()) + } + + #[test] + fn stumpff_ranges_hold_through_their_extremes() { + let mut rng = Lcg(0x57F); + // Extremes of c₀ at (jπ)², of c₁ near 20.19 and 59.68, of c₂ at 4π² and near 80.8. + let hot = [9.8696, 20.19, 39.478, 59.68, 80.76, 88.83, 157.9, 4.0, 0.0]; + for i in 0..3000 { + let centre = if i % 3 == 0 { + rng.range(-40.0, 400.0) + } else { + draw(&mut rng, &hot) + }; + let width = draw(&mut rng, &[1e-7, 1e-4, 0.01, 1.0, 30.0]); + let lo = I32F32::from_num(centre - rng.range(0.0, width)); + let hi = I32F32::from_num(centre + rng.range(0.0, width)); + let ranges = stumpff_range(lo, hi); + for x in across(lo, hi, 200) { + for (k, (range, v)) in ranges.iter().zip(stumpff(x)).enumerate() { + assert!( + holds(*range, v), + "c{k}({x}) = {v} outside {range:?} over [{lo}, {hi}]" + ); + } + } + } + let (c1lo, c1hi) = stumpff_range(I32F32::from_num(20.18), I32F32::from_num(20.2))[1]; + assert!(c1lo.to_num::() > -0.21724 && c1hi.to_num::() < -0.21723); + let (lo, hi) = (I32F32::from_num(-50), I32F32::MAX); + let whole = stumpff_range(lo, hi); + for x in across(lo, hi, 4000).chain(across(I32F32::ZERO, I32F32::from_num(500), 4000)) { + for (range, v) in whole.iter().zip(stumpff(x)) { + assert!(holds(*range, v), "{v} at {x} outside {range:?}"); + } + } + let wide = stumpff_range(I32F32::from_num(4), I32F32::from_num(300))[1]; + assert!(wide.0.to_num::() > -0.21731 && wide.1.to_num::() < 0.455); + let (c0lo, c0hi) = stumpff_range(I32F32::from_num(-3), I32F32::from_num(2))[0]; + assert!( + c0lo == stumpff(I32F32::from_num(2))[0] - I32F32::DELTA + && c0hi == stumpff(I32F32::from_num(-3))[0] + I32F32::DELTA + ); + } + + #[test] + fn tan_ratio_range_spans_its_poles() { + let (lo, hi) = (I32F32::from_num(1.5), I32F32::from_num(1.6)); + assert_eq!(tan_ratio_range(lo, hi), (I32F32::MIN, I32F32::MAX)); + assert_eq!( + tan_ratio_range(I32F32::from_num(-2), I32F32::from_num(2)), + (I32F32::MIN, I32F32::MAX) + ); + let (one, _) = tan_ratio_range(I32F32::from_num(-1), I32F32::from_num(0.5)); + assert_eq!(one, I32F32::ONE); + let far = tan_ratio_range(I32F32::from_num(3.5), I32F32::from_num(4.5)); + assert!(far.0 > I32F32::ZERO && far.1 < I32F32::from_num(1.1)); + } + + #[test] + fn box_ranges_hold_every_point() { + let mut rng = Lcg(0xB0C5); + for _ in 0..1500 { + let corner = |g: &mut Lcg| { + let a = g.range(-4.0, 4.0); + (I32F32::from_num(a), I32F32::from_num(a + g.range(0.0, 2.0))) + }; + let (x, y) = (corner(&mut rng), corner(&mut rng)); + let (h, angle) = (hypot_range(x, y), atan2_range(y, x)); + for px in across(x.0, x.1, 16) { + for py in across(y.0, y.1, 16) { + assert!(holds(h, hypot(px, py)), "hypot({px}, {py})"); + assert!( + holds(angle, atan2(py, px)), + "atan2({py}, {px}) outside {angle:?}" + ); + } + } + } + let pi = I32F32::PI + I32F32::DELTA; + let across_cut = atan2_range( + (I32F32::from_num(-1), I32F32::ONE), + (I32F32::from_num(-2), I32F32::from_num(-1)), + ); + assert!(across_cut.0 <= -pi + I32F32::DELTA && across_cut.1 >= pi - I32F32::DELTA); + let quadrant = atan2_range( + (I32F32::ONE, I32F32::from_num(2)), + (I32F32::ONE, I32F32::from_num(2)), + ); + assert!(quadrant.0 > I32F32::from_num(0.46) && quadrant.1 < I32F32::from_num(1.11)); + } +} diff --git a/tests/unit/ops/mod.rs b/tests/unit/ops/mod.rs index f8f5bc8..17fa5a1 100644 --- a/tests/unit/ops/mod.rs +++ b/tests/unit/ops/mod.rs @@ -1,4 +1,12 @@ mod algebraic; +mod arith; mod circular; +mod complex; +mod edges; +mod elliptic; mod exponential; mod hyperbolic; +mod interval; +mod ratio; +mod solve; +mod turns; diff --git a/tests/unit/ops/ratio.rs b/tests/unit/ops/ratio.rs new file mode 100644 index 0000000..545fd02 --- /dev/null +++ b/tests/unit/ops/ratio.rs @@ -0,0 +1,88 @@ +#[cfg(test)] +mod tests { + use crate::unit::support::{Lcg, TestResult}; + use fixed::types::{I16F16, I32F32, I64F64}; + use fixed_analytics::{ + atan_ratio, atanh_ratio, exp_moments, expm1, exprel, exprel2, log1p, log1p_ratio, + log1p_residual, log1p_residual_ratio, mul_add, stumpff, tan_ratio, tanh_ratio, + }; + + #[test] + fn values_at_zero() -> TestResult { + let z = I32F32::ZERO; + assert_eq!(exprel(z), I32F32::ONE); + assert_eq!(exprel2(z), I32F32::from_num(0.5)); + assert_eq!( + exp_moments(z), + [1.0, 0.5, 1.0 / 3.0, 0.25, 0.2, 1.0].map(I32F32::from_num) + ); + assert_eq!(log1p_ratio(z)?, I32F32::ONE); + assert_eq!(log1p_residual_ratio(z)?, I32F32::from_num(-0.5)); + assert_eq!( + stumpff(z)[..3], + [I32F32::ONE, I32F32::ONE, I32F32::from_num(0.5)] + ); + assert_eq!(tan_ratio(z), (I32F32::ONE, I32F32::ONE)); + assert_eq!(atan_ratio(z), I32F32::ONE); + assert_eq!(atanh_ratio(z)?, I32F32::ONE); + assert_eq!(tanh_ratio(z), I32F32::ONE); + Ok(()) + } + + #[test] + fn tiny_arguments_keep_their_bits() -> TestResult { + let tiny = I64F64::from_bits(1 << 20); + assert_eq!(expm1(tiny), tiny); + assert_eq!(log1p(tiny)?, tiny); + // ln(1 + v) − v ≈ −v²/2, far below an ulp here: it rounds to 0 or −0. + assert!(log1p_residual(tiny)?.abs() <= I64F64::from_bits(1)); + let v = I64F64::from_num(1e-6); + let r: f64 = log1p_residual(v)?.to_num(); + assert!((r - (-5e-13 + 1e-18 / 3.0)).abs() < 1e-19); + Ok(()) + } + + #[test] + fn stumpff_identities_hold() { + let mut rng = Lcg(0x5717); + for _ in 0..500 { + let z = I32F32::from_num(rng.range(-30.0, 30.0)); + let [c0, c1, c2, c3] = stumpff(z); + // c₀ + z·c₂ = 1 and c₁ + z·c₃ = 1, each to a few ulps. + let e0 = mul_add(z, c2, c0) - I32F32::ONE; + let e1 = mul_add(z, c3, c1) - I32F32::ONE; + let tol = I32F32::from_bits(40); + assert!(e0.abs() <= tol && e1.abs() <= tol, "z = {z}: {e0}, {e1}"); + } + } + + #[test] + fn moments_and_relatives_agree() { + let x = I32F32::from_num(2); + let [e1, e2, e3, e4, e5, decay] = exp_moments(x); + let ex = core::f64::consts::E.powi(-2); + let close = |a: I32F32, b: f64| (a.to_num::() - b).abs() < 1e-9; + assert!(close(e1, (1.0 - ex) / 2.0)); + assert!(close(e2, ex.mul_add(-3.0, 1.0) / 4.0)); + assert!(close(e3, ex.mul_add(-10.0, 2.0) / 8.0)); + assert!(close(e4, ex.mul_add(-38.0, 6.0) / 16.0)); + assert!(close(e5, ex.mul_add(-168.0, 24.0) / 32.0)); + assert!(close(decay, ex)); + assert!(close( + exprel(x), + core::f64::consts::E.mul_add(core::f64::consts::E, -1.0) / 2.0 + )); + } + + #[test] + fn domains_and_poles() { + assert!(log1p(-I16F16::ONE).is_err()); + assert!(log1p_ratio(I16F16::from_num(-2)).is_err()); + assert!(atanh_ratio(I16F16::ONE).is_err()); + // fixed's FRAC_PI_2 truncates, so it sits just below the pole. + assert_eq!(tan_ratio(I16F16::FRAC_PI_2).0, I16F16::MAX); + assert_eq!(tan_ratio(I16F16::FRAC_PI_2 + I16F16::DELTA).0, I16F16::MIN); + assert_eq!(exprel(I16F16::from_num(30)), I16F16::MAX); + assert_eq!(exp_moments(I16F16::from_num(-40)), [I16F16::MAX; 6]); + } +} diff --git a/tests/unit/ops/solve.rs b/tests/unit/ops/solve.rs new file mode 100644 index 0000000..532427d --- /dev/null +++ b/tests/unit/ops/solve.rs @@ -0,0 +1,71 @@ +#[cfg(test)] +mod tests { + use crate::unit::support::TestResult; + use fixed::types::{I16F16, I32F32, I64F64}; + use fixed_analytics::{kepler_universal, root_in, sin}; + + #[test] + fn root_in_closes_to_adjacent_values() -> TestResult { + let (a, b) = root_in(sin, I32F32::from_num(3), I32F32::from_num(3.5), 64)?; + assert_eq!((a, b), (I32F32::from_num(core::f64::consts::PI), a)); + assert_eq!(sin(a), I32F32::ZERO); + let cube = |x: I64F64| x * x * x - I64F64::from_num(2); + let (low, high) = root_in(cube, I64F64::from_num(2), I64F64::ZERO, 200)?; + assert_eq!(high - low, I64F64::DELTA); + let root = 2f64.cbrt(); + assert!((low.to_num::() - root).abs() < 1e-15); + Ok(()) + } + + #[test] + fn root_in_keeps_to_its_budget() -> TestResult { + let mut calls = 0; + let f = |x: I16F16| { + calls += 1; + x * I16F16::from_num(3) - I16F16::ONE + }; + let (a, b) = root_in(f, I16F16::from_num(-100), I16F16::from_num(100), 5)?; + assert_eq!(calls, 7); + let third = 1.0 / 3.0; + assert!(a.to_num::() < third && third < b.to_num::()); + let line = |x: I16F16| x - I16F16::from_num(0.5); + assert_eq!( + root_in(line, I16F16::ZERO, I16F16::ONE, 40)?, + (I16F16::from_num(0.5), I16F16::from_num(0.5)) + ); + assert!(root_in(line, I16F16::ONE, I16F16::from_num(2), 40).is_err()); + Ok(()) + } + + #[test] + fn kepler_matches_mpmath() -> TestResult { + let t = |x: f64| I64F64::from_num(x); + let close = |got: I64F64, want: f64| { + (got.to_num::() - want).abs() < 4e-16 * want.abs().max(1.0) + }; + // A circle, whose anomaly is its time, a parabola, a hyperbola and an ellipse. + assert_eq!(kepler_universal(t(1.0), t(0.0), t(1.0), t(2.5))?, t(2.5)); + assert!(close( + kepler_universal(t(1.0), t(0.5), t(0.0), t(2.0))?, + 1.263_997_823_528_725_5 + )); + assert!(close( + kepler_universal(t(1.0), t(0.25), t(-0.5), t(3.0))?, + 1.586_900_711_280_019 + )); + assert!(close( + kepler_universal(t(1.0), t(-0.2), t(0.75), t(10.0))?, + 7.591_239_334_586_656 + )); + assert_eq!(kepler_universal(t(1.0), t(0.5), t(0.0), t(0.0))?, t(0.0)); + Ok(()) + } + + #[test] + fn kepler_needs_a_real_orbit() { + let t = |x: f64| I32F32::from_num(x); + assert!(kepler_universal(t(0.0), t(0.0), t(1.0), t(1.0)).is_err()); + assert!(kepler_universal(t(1.0), t(2.0), t(1.0), t(1.0)).is_err()); + assert!(kepler_universal(t(1.0), t(1.0), t(1.0), t(1.0)).is_ok()); + } +} diff --git a/tests/unit/ops/turns.rs b/tests/unit/ops/turns.rs new file mode 100644 index 0000000..3892563 --- /dev/null +++ b/tests/unit/ops/turns.rs @@ -0,0 +1,82 @@ +#[cfg(test)] +mod tests { + use fixed::FixedU128; + use fixed::types::extra::U64; + use fixed::types::{I16F16, I32F32, I64F64}; + use fixed_analytics::{ + atan2_turns, div_by_tau, mul_by_tau, sin_cos, sin_cos_turns, wrap_principal, wrap_radians, + }; + + #[test] + fn mul_by_tau_rounds_once() { + // fixed's PI and FRAC_PI_2 truncate; these round to nearest. + let pi = I32F32::from_num(core::f64::consts::PI); + assert_eq!(mul_by_tau(I32F32::from_num(0.5)), pi); + let quarter = fixed_analytics::atan2(I64F64::ONE, I64F64::ZERO); + assert_eq!(mul_by_tau(I64F64::from_num(-0.25)), -quarter); + assert_eq!(mul_by_tau(I16F16::from_num(10_000)), I16F16::MAX); + let t = I64F64::from_num(0.123_456_789); + assert_eq!(div_by_tau(mul_by_tau(t)), t); + } + + #[test] + fn quarter_turns_are_exact() { + let quarter = I32F32::from_num(0.25); + assert_eq!(sin_cos_turns(quarter), (I32F32::ONE, I32F32::ZERO)); + assert_eq!(sin_cos_turns(-quarter), (-I32F32::ONE, I32F32::ZERO)); + assert_eq!( + sin_cos_turns(I32F32::from_num(1_000_000.5)), + (I32F32::ZERO, -I32F32::ONE) + ); + } + + #[test] + fn turns_agree_with_radians() { + let t = I64F64::from_num(0.1); + let (s, c) = sin_cos_turns(t); + let (sr, cr) = sin_cos(t * I64F64::TAU); + assert!( + (s - sr).abs() < I64F64::from_num(1e-17) && (c - cr).abs() < I64F64::from_num(1e-17) + ); + assert_eq!( + atan2_turns(I32F32::ONE, I32F32::ZERO), + I32F32::from_num(0.25) + ); + assert_eq!( + atan2_turns(I32F32::ZERO, -I32F32::ONE), + I32F32::from_num(0.5) + ); + assert_eq!( + atan2_turns(-I32F32::ONE, -I32F32::ONE), + I32F32::from_num(-0.375) + ); + } + + #[test] + fn turns_from_an_unsigned_fraction() { + // A U64F64 fraction of a turn is the same bits as an I64F64 one below ½. + let frac = FixedU128::::from_num(0.125); + let t = I64F64::from_bits(frac.to_bits().cast_signed()); + let (s, c) = sin_cos_turns(t); + assert!((s - c).abs() <= I64F64::from_bits(1)); + } + + #[test] + fn wraps_stay_in_range() { + for k in -2000..2000 { + let x = I16F16::from_num(k) / 7; + let w = wrap_radians(x); + assert!( + w >= I16F16::ZERO && w.to_num::() < core::f64::consts::TAU, + "{x} -> {w}" + ); + let p = wrap_principal(x); + let pf = p.to_num::(); + assert!( + pf > -core::f64::consts::PI && pf <= core::f64::consts::PI, + "{x} -> {p}" + ); + } + assert_eq!(div_by_tau(I32F32::TAU), I32F32::ONE); + } +} diff --git a/tools/accuracy-bench/Cargo.toml b/tools/accuracy-bench/Cargo.toml index ae69b35..484917a 100644 --- a/tools/accuracy-bench/Cargo.toml +++ b/tools/accuracy-bench/Cargo.toml @@ -3,6 +3,7 @@ name = "accuracy-bench" version = "0.1.0" edition = "2024" publish = false +default-run = "accuracy-bench" [dependencies] fixed_analytics = { path = "../.." } diff --git a/tools/accuracy-bench/baseline.json b/tools/accuracy-bench/baseline.json index 9f57369..2331fa0 100644 --- a/tools/accuracy-bench/baseline.json +++ b/tools/accuracy-bench/baseline.json @@ -1,33 +1,33 @@ { - "timestamp": 1788426873, + "timestamp": 1793186481, "results": [ { "name": "sin", "i16f16": { "count": 59007, - "abs_max": 0.0002151700847870322, - "abs_mean": 0.00006812612370136238, - "abs_p50": 0.000055933053498491425, - "abs_p95": 0.00017059742871838868, - "abs_p99": 0.00019168937678995523, - "rel_max": 1.116598003152961, - "rel_mean": 0.0006057467007882896, - "rel_p50": 0.00008775899233342654, - "rel_p95": 0.0012777646003236927, - "rel_p99": 0.006502970657507115 + "abs_max": 0.000015149291003387945, + "abs_mean": 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7.026546882116236e-7, "rel_max": 8.171585162535583e-7, - "rel_mean": 1.0902569528034319e-8, - "rel_p50": 1.230148501795134e-9, - "rel_p95": 4.8596052359287584e-8, + "rel_mean": 1.0065793896804114e-8, + "rel_p50": 2.0193712715291326e-10, + "rel_p95": 4.850229537057499e-8, "rel_p99": 2.313153839052801e-7 }, "samples_tested": 59003 diff --git a/tools/accuracy-bench/src/bin/consts.rs b/tools/accuracy-bench/src/bin/consts.rs new file mode 100644 index 0000000..1cffa9d --- /dev/null +++ b/tools/accuracy-bench/src/bin/consts.rs @@ -0,0 +1,102 @@ +//! Prints the engine's constants as 384 normalized bits, for src/consts.rs. + +use rug::float::Constant; +use rug::{Float, Integer}; + +const BITS: u32 = 384; + +fn hex(v: u64) -> String { + format!( + "0x{:04X}_{:04X}_{:04X}_{:04X}", + v >> 48, + (v >> 32) & 0xFFFF, + (v >> 16) & 0xFFFF, + v & 0xFFFF + ) +} + +fn parts(x: &Float) -> String { + if x.is_zero() { + return "(false, 0, [0; 6])".to_string(); + } + let neg = x.is_sign_negative(); + let x = &Float::with_val(x.prec(), x.abs_ref()); + let e = x.get_exp().unwrap_or(0); + let m: Integer = (Float::with_val(BITS + 64, x) << (BITS as i32 - e)) + .to_integer() + .unwrap(); + let limbs: Vec = (0..6) + .map(|i| hex((m.clone() >> (64 * i) as u32).to_u64_wrapping())) + .collect(); + format!("({neg}, {e}, [{}])", limbs.join(", ")) +} + +fn table(name: &str, xs: &[Float]) { + println!("pub const {name}: [C; {}] = [", xs.len()); + for x in xs { + println!(" {},", parts(x)); + } + println!("];"); +} + +fn emit(name: &str, x: Float) { + let e = x.get_exp().unwrap_or(0); + let m: Integer = (Float::with_val(BITS + 64, &x) << (BITS as i32 - e)) + .to_integer() + .unwrap(); + let limbs: Vec = (0..6) + .map(|i| hex((m.clone() >> (64 * i) as u32).to_u64_wrapping())) + .collect(); + println!("pub const {name}: C = (false, {e}, [{}]);", limbs.join(", ")); +} + +/// `x` in [2^−3, 2^−2) as `limbs` normalized limbs, for products reduced +/// past what 384 bits cover. +fn emit_long(name: &str, x: Float, limbs: u32) { + let m: Integer = (Float::with_val(64 * limbs + 128, &x) << (64 * limbs as i32 + 2)) + .to_integer() + .unwrap(); + let parts: Vec = (0..limbs) + .map(|i| hex((m.clone() >> (64 * i)).to_u64_wrapping())) + .collect(); + println!("pub const {name}: [u64; {limbs}] = [{}];", parts.join(", ")); +} + +fn main() { + let p = BITS + 128; + let long = Float::with_val(768, Constant::Pi) * 2u32; + emit_long("FRAC_1_2PI_LONG", Float::with_val(768, 1) / long, 9); + let pi = Float::with_val(p, Constant::Pi); + let ln2 = Float::with_val(p, Constant::Log2); + let ln10 = Float::with_val(p, 10).ln(); + let one = Float::with_val(p, 1); + emit("PI", pi.clone()); + emit("LN_2", ln2.clone()); + emit("LN_10", ln10.clone()); + emit("FRAC_1_LN_2", one.clone() / &ln2); + emit("FRAC_1_LN_10", one.clone() / &ln10); + emit("FRAC_1_2PI", one.clone() / (pi.clone() * 2u32)); + emit("EULER_GAMMA", Float::with_val(p, Constant::Euler)); + emit("FRAC_2_SQRT_PI", Float::with_val(p, 2) / pi.clone().sqrt()); + emit("SQRT_PI", pi.clone().sqrt()); + emit("SQRT_2", Float::with_val(p, 2).sqrt()); + let mut fact = Float::with_val(p, 1); + let mut inv_fact = vec![]; + for n in 0..49u32 { + if n > 0 { + fact *= n; + } + inv_fact.push(Float::with_val(p, 1) / &fact); + } + table("INV_FACT", &inv_fact); + let inv_odd: Vec = (0..49u32).map(|k| Float::with_val(p, 1) / (2 * k + 1)).collect(); + table("INV_ODD", &inv_odd); + let atan16: Vec = (0..17u32).map(|j| Float::with_val(p, j) / 16u32).map(|x| x.atan()).collect(); + table("ATAN_SIXTEENTHS", &atan16); + let ln16: Vec = (-5..8i32).map(|j| (Float::with_val(p, j) / 16u32 + 1u32).ln()).collect(); + table("LN_SIXTEENTHS", &ln16); + let angle = |j: u32| pi.clone() * j / 128u32; + table("SIN_PI_128THS", &(0..33u32).map(|j| angle(j).sin()).collect::>()); + table("COS_PI_128THS", &(0..33u32).map(|j| angle(j).cos()).collect::>()); + table("EXP2_32NDS", &(0..32u32).map(|j| (Float::with_val(p, j) / 32u32).exp2()).collect::>()); +} diff --git a/tools/accuracy-bench/src/bin/sweep_some.rs b/tools/accuracy-bench/src/bin/sweep_some.rs new file mode 100644 index 0000000..d704f64 --- /dev/null +++ b/tools/accuracy-bench/src/bin/sweep_some.rs @@ -0,0 +1,24 @@ +//! The worst-case sweep for the named functions only, on every type: +//! `cargo run --release --bin sweep_some -- [--quick] name…`. + +use accuracy_bench::{sut::types, sweep, worst}; +use rayon::prelude::*; + +fn main() { + let args: Vec = std::env::args().skip(1).collect(); + let quick = args.iter().any(|a| a == "--quick"); + sweep::QUICK.store(quick, std::sync::atomic::Ordering::Relaxed); + let funs: Vec = sweep::ALL + .iter() + .copied() + .filter(|f| args.iter().any(|a| a == f.name())) + .collect(); + let types = types(); + let jobs: Vec<(sweep::Fun, usize)> = funs + .iter() + .flat_map(|&f| (0..types.len()).map(move |t| (f, t))) + .collect(); + let stats: Vec<_> = jobs.par_iter().map(|&(f, t)| (types[t].run)(f)).collect(); + print!("{}", worst::to_text(&stats)); + worst::check_bounds(&stats); +} diff --git a/tools/accuracy-bench/src/complex_ref.rs b/tools/accuracy-bench/src/complex_ref.rs new file mode 100644 index 0000000..facbe68 --- /dev/null +++ b/tools/accuracy-bench/src/complex_ref.rs @@ -0,0 +1,442 @@ +//! Reference complex special functions in MPC, at precisions that cover each +//! series' cancellation, by methods unlike the crate's where they can be. + +use rug::float::Constant; +use rug::{Complex, Float}; + + +fn cx(re: &Float, im: &Float, p: u32) -> Complex { + Complex::with_val(p, (re, im)) +} + +/// `e^z` with its real part capped at 10⁴, so that MPFR, whose exponents stop +/// near 2^(2^30), keeps the phase of values every type saturates from. +pub fn exp_capped(z: &Complex) -> Complex { + let p = z.prec().0; + if *z.real() > 10_000 { + return Complex::with_val(p, (10_000, z.imag())).exp(); + } + z.clone().exp() +} + +/// `z^s·e^(−z)`, capped like [`exp_capped`]; on the negative axis its phase +/// `πs` is exact, where a rounded one times the huge modulus would swamp a +/// zero part. +fn power_exp(s: &Float, z: &Complex, p: u32) -> Complex { + if z.imag().is_zero() && z.real().is_sign_negative() { + let x = Float::with_val(p, z.real().abs_ref()); + let size = (Float::with_val(p, x.ln_ref()) * s + &x).min(&Float::with_val(p, 10_000)).exp(); + return Complex::with_val(p, (s.cos_pi_ref(), s.sin_pi_ref())) * size; + } + exp_capped(&(Complex::with_val(p, z.ln_ref()) * s - z)) +} + +/// `Γ(a, z) ~ z^(a−1)·e^(−z)·Σ (a − 1)…(a − k)/z^k`, for `|z|` past 300 near +/// the negative axis, where the continued fraction converges too slowly. +fn gamma_asymptotic(a: &Float, z: &Complex, p: u32) -> Complex { + let inv = Complex::with_val(p, z.recip_ref()); + let mut term = Complex::with_val(p, (1, 0)); + let mut sum = term.clone(); + for k in 1..5000u32 { + let next = Complex::with_val(p, &term * &inv) * Float::with_val(p, a - k); + if next.is_zero() || Float::with_val(64, next.abs_ref()) > Float::with_val(64, term.abs_ref()) { + break; + } + sum += &next; + term = next; + if settled(&term, &sum, 300) { + break; + } + } + power_exp(&Float::with_val(p, a - 1u32), z, p) * sum +} + +fn settled(term: &Complex, sum: &Complex, bits: i32) -> bool { + let t = Float::with_val(64, term.abs_ref()); + let s = Float::with_val(64, sum.abs_ref()); + t.is_zero() || (s.get_exp().unwrap_or(0) - t.get_exp().unwrap_or(0) > bits) +} + +/// `erf u` by its Maclaurin series `(2/√π)·Σ (−1)ⁿu^(2n+1)/(n!(2n + 1))`. +fn erf_series(u: &Complex, p: u32) -> Complex { + let u2 = Complex::with_val(p, u * u); + let mut power = u.clone(); + let mut sum = u.clone(); + let mut n: u32 = 0; + loop { + n += 1; + power = -(power * &u2) / n; + let term = Complex::with_val(p, &power / (2 * n + 1)); + sum += &term; + if n as f64 > Float::with_val(64, u2.abs_ref()).to_f64() && settled(&term, &sum, 300) { + break; + } + } + sum * (Float::with_val(p, 2) / Float::with_val(p, Constant::Pi).sqrt()) +} + +/// `w(z)` from `(i/√π)/z·Σ (2n − 1)!!/(2z²)ⁿ`, for `|z| ≥ 20`. +fn w_asymptotic(z: &Complex, p: u32) -> Complex { + let inv = Complex::with_val(p, Complex::with_val(p, z * z) * 2u32).recip(); + let mut term = Complex::with_val(p, (1, 0)); + let mut sum = term.clone(); + for n in 1..2000u32 { + let next = Complex::with_val(p, &term * &inv) * (2 * n - 1); + if Float::with_val(64, next.abs_ref()) > Float::with_val(64, term.abs_ref()) { + break; + } + sum += &next; + term = next; + if settled(&term, &sum, 300) { + break; + } + } + let root_pi = Float::with_val(p, Constant::Pi).sqrt(); + (sum / z) * Complex::with_val(p, (0, 1)) / root_pi +} + +/// Past this `|z|` the asymptotic series for `w`, whose smallest term is +/// about `e^(−|z|²)`, is good to 2^-174. +const FAR: u32 = 11; + +/// `|z|`, roughly. +fn modulus(re: &Float, im: &Float) -> f64 { + (Float::with_val(64, re.square_ref()) + Float::with_val(64, im.square_ref())).sqrt().to_f64() +} + +/// Bits that cover `e^(|z|²)` of cancellation with 320 to spare. +fn bits_for(re: &Float, im: &Float) -> u32 { + let r = modulus(re, im).min(f64::from(FAR)); + (320.0 + 1.5 * r * r) as u32 +} + +pub fn faddeeva(re: &Float, im: &Float) -> Complex { + let p = bits_for(re, im); + let z = cx(re, im, p); + if Float::with_val(64, z.abs_ref()) >= FAR { + if im.is_sign_negative() && !im.is_zero() { + let e = exp_capped(&Complex::with_val(p, -Complex::with_val(p, &z * &z))) * 2u32; + return e - w_asymptotic(&(-z), p); + } + return w_asymptotic(&z, p); + } + // w = e^(−z²)·erfc(−iz) = e^(−z²)·(1 − erf(−iz)). + let u = Complex::with_val(p, &z * Complex::with_val(p, (0, -1))); + let one = Complex::with_val(p, (1, 0)); + Complex::with_val(p, -Complex::with_val(p, &z * &z)).exp() * (one - erf_series(&u, p)) +} + +/// `erf z`, imaginary on the imaginary axis, where the asymptotic `w` leaves +/// out the real part `e^(−x²)` that `1 − e^(−z²)·w` would cancel. +pub fn erf(re: &Float, im: &Float) -> Complex { + let p = bits_for(re, im); + let z = cx(re, im, p); + if Float::with_val(64, z.abs_ref()) < FAR { + return erf_series(&z, p); + } + let flip = re.is_sign_negative(); + let z = if flip { -z } else { z }; + let iz = Complex::with_val(p, &z * Complex::with_val(p, (0, 1))); + let w = faddeeva(iz.real(), iz.imag()); + let one = Complex::with_val(p, (1, 0)); + let v = one - exp_capped(&Complex::with_val(p, -Complex::with_val(p, &z * &z))) * w; + let v = if flip { -v } else { v }; + if re.is_zero() { Complex::with_val(p, (0, v.imag())) } else { v } +} + +/// A continued fraction `b₀ + a₁/(b₁ + …)` by Lentz's method at `p` bits. +fn lentz(b0: Complex, p: u32, term: impl Fn(u32) -> (Complex, Complex)) -> Complex { + let tiny = Complex::with_val(p, (Float::with_val(p, Float::i_exp(1, -3000)), 0)); + let mut f = if b0.is_zero() { tiny.clone() } else { b0 }; + let mut c = f.clone(); + let mut d = Complex::with_val(p, (0, 0)); + for k in 1..100_000u32 { + let (a, b) = term(k); + d = Complex::with_val(p, &b + Complex::with_val(p, &a * &d)); + if d.is_zero() { + d = tiny.clone(); + } + c = Complex::with_val(p, &b + Complex::with_val(p, &a / &c)); + if c.is_zero() { + c = tiny.clone(); + } + d = d.recip(); + let delta = Complex::with_val(p, &c * &d); + f *= δ + let diff = Complex::with_val(p, &delta - 1u32); + let off = Float::with_val(64, diff.abs_ref()); + if off.is_zero() || off.get_exp().unwrap_or(0) < -300 { + break; + } + } + f +} + +pub fn e1(re: &Float, im: &Float) -> Complex { + let z = cx(re, im, (400.0 + 2.0 * modulus(re, im).min(1000.0)) as u32); + let r = Float::with_val(64, z.abs_ref()).to_f64(); + let near_neg = re.is_sign_negative() && Float::with_val(64, im.abs_ref()) <= Float::with_val(64, re.abs_ref()); + if near_neg && r > 300.0 { + let v = gamma_asymptotic(&Float::with_val(512, 0), &cx(re, im, 512), 512); + // On the axis, the value from above it: −Ei(−z) − iπ. + let pi = Float::with_val(512, Constant::Pi); + return if im.is_zero() { Complex::with_val(512, (v.real(), -pi)) } else { v }; + } + if r <= 60.0 || (near_neg && r <= 300.0) { + let p = z.prec().0; + let mut term = Complex::with_val(p, (1, 0)); + let mut sum = Complex::with_val(p, (0, 0)); + let mut n: u32 = 0; + loop { + n += 1; + term = -(term * &z) / n; + let part = Complex::with_val(p, &term / n); + sum += ∂ + if n as f64 > r && settled(&part, &sum, 300) { + break; + } + } + let gamma = Float::with_val(p, Constant::Euler); + return -(Complex::with_val(p, (gamma, 0)) + z.ln() + sum); + } + let p = 512; + let z = cx(re, im, p); + let one = Complex::with_val(p, (1, 0)); + let f = lentz(Complex::with_val(p, &z + &one), p, |k| { + let i = Float::with_val(p, k); + (Complex::with_val(p, (-(i.clone() * &i), 0)), Complex::with_val(p, &z + &one) + Complex::with_val(p, (i * 2u32, 0))) + }); + exp_capped(&Complex::with_val(p, -z)) / f +} + +pub fn gamma_upper(a: &Float, re: &Float, im: &Float) -> Complex { + let p = (512.0 + 2.0 * modulus(re, im).min(1000.0)) as u32; + let z = cx(re, im, p); + if z.is_zero() { + return Complex::with_val(p, (Float::with_val(p, a).gamma(), 0)); + } + if im.is_zero() && re.is_sign_positive() && *re <= 100 { + let v = Float::with_val(p, a).gamma_inc(&Float::with_val(p, re)); + return Complex::with_val(p, (v, 0)); + } + let r = Float::with_val(64, z.abs_ref()).to_f64(); + let near_neg = re.is_sign_negative() && Float::with_val(64, im.abs_ref()) <= Float::with_val(64, re.abs_ref()); + if near_neg && r > 300.0 { + let v = gamma_asymptotic(a, &z, p); + // On the axis Γ(a, z) = Γ(a) − e^(iπa)·G with G real; the series gives + // the second part. + if im.is_zero() && !a.is_integer() { + return Complex::with_val(p, (Float::with_val(p, v.real() + Float::with_val(p, a).gamma()), v.imag())); + } + return v; + } + let lead = power_exp(&Float::with_val(p, a), &z, p); + if r <= (a.to_f64() + 1.0).max(8.0) || (near_neg && r <= 300.0) { + let mut denom = Float::with_val(p, a); + let mut term = Complex::with_val(p, (Float::with_val(p, 1) / &denom, 0)); + let mut sum = term.clone(); + loop { + denom += 1u32; + term = Complex::with_val(p, &term * &z) / &denom; + sum += &term; + if denom.to_f64() > r && settled(&term, &sum, 300) { + break; + } + } + return Complex::with_val(p, (Float::with_val(p, a).gamma(), 0)) - lead * sum; + } + let one_minus_a = Float::with_val(p, 1) - a; + let f = lentz(Complex::with_val(p, &z + &one_minus_a), p, |k| { + let i = Float::with_val(p, k); + let ai = -(i.clone() * (i.clone() - a)); + let b = Complex::with_val(p, &z + &one_minus_a) + Float::with_val(p, i * 2u32); + (Complex::with_val(p, (ai, 0)), b) + }); + lead / f +} + +/// `z^a` for real `a`: on the axes and diagonals, `|z|^a` turned by an exact +/// `a·k·π/4`, else `e^(a·ln z)`, both capped like [`exp_capped`]. +pub fn pow(re: &Float, im: &Float, a: &Float) -> Complex { + let p = 960; + let z = cx(re, im, p); + if z.is_zero() { + return Complex::with_val(p, (u32::from(a.is_zero()), 0)); + } + let eighths = if im.is_zero() { + Some(if re.is_sign_negative() { 4 } else { 0 }) + } else if re.is_zero() || re.clone().abs() == im.clone().abs() { + let k = if re.is_zero() { 2 } else if re.is_sign_positive() { 1 } else { 3 }; + Some(if im.is_sign_negative() { -k } else { k }) + } else { + None + }; + let Some(k) = eighths else { + return exp_capped(&(Complex::with_val(p, z.ln_ref()) * a)); + }; + let ln_abs = Float::with_val(p, z.abs_ref()).ln(); + let size = (ln_abs * a).min(&Float::with_val(p, 10_000)).exp(); + let phase = Float::with_val(p, a * k) / 4u32; + Complex::with_val(p, (phase.cos_pi_ref(), phase.sin_pi_ref())) * size +} + +/// Bits for [`exprel_weighted`]'s quadrature. +const WP: u32 = 320; + +/// Gauss–Legendre nodes and weights on `[0, 1]`, 40 of them, by Newton's method. +fn legendre() -> &'static [(Float, Float)] { + static NODES: std::sync::OnceLock> = std::sync::OnceLock::new(); + NODES.get_or_init(|| { + let n = 40u32; + let p = WP + 64; + let pi = Float::with_val(p, Constant::Pi); + (1..=n) + .map(|i| { + let mut x = Float::with_val(p, (Float::with_val(p, i) - 0.25) * &pi / (f64::from(n) + 0.5)).cos(); + let mut deriv = Float::with_val(p, 0); + for _ in 0..100 { + let (mut p0, mut p1) = (Float::with_val(p, 1), x.clone()); + for k in 2..=n { + let p2 = (Float::with_val(p, &x * &p1) * (2 * k - 1) - Float::with_val(p, &p0 * (k - 1))) / k; + p0 = p1; + p1 = p2; + } + deriv = Float::with_val(p, Float::with_val(p, &x * &p1) - &p0) * n / (Float::with_val(p, &x * &x) - 1u32); + let dx = Float::with_val(p, &p1 / &deriv); + x -= &dx; + if dx.is_zero() || dx.get_exp().unwrap_or(0) < -(p as i32) { + break; + } + } + let w = Float::with_val(p, 2) / ((Float::with_val(p, 1) - Float::with_val(p, &x * &x)) * deriv.square()); + ((x + 1u32) / 2u32, w / 2u32) + }) + .collect() + }) +} + +/// `(K₁, K₂)` of `Complex::exprel_weighted`: past `|z| = 8(a + 170)·δ` by +/// parts, else Gauss–Legendre panels on `e^(z(1 − u))·{W, V}(u)/(1 − ε)` from +/// each end toward the least of the integrand; NaN where that would take more +/// than 800 panels. +pub fn exprel_weighted(re: &Float, im: &Float, eps: &Float, pw: &Float) -> (Complex, Complex) { + let p = WP; + let z = cx(re, im, p); + let one = Float::with_val(p, 1); + if eps.is_zero() { + // exprel and exprel2, at the bits their cancellation near 0 needs. + let p = 1024; + let z = cx(re, im, p); + if z.is_zero() { + return (Complex::with_val(p, (1, 0)), Complex::with_val(p, (0.5, 0))); + } + let em1 = exp_capped(&z) - Complex::with_val(p, (1, 0)); + let k1 = Complex::with_val(p, &em1 / &z); + let k2 = (em1 - &z) / Complex::with_val(p, &z * &z); + return (k1, k2); + } + let a = Float::with_val(p, pw + 1u32); + let keep = Float::with_val(p, &one - eps); + let delta = Float::with_val(p, eps / &keep); + let log_keep = Float::with_val(p, -eps).ln_1p(); + let r = Float::with_val(64, z.abs_ref()).to_f64(); + let nan = || (Complex::with_val(p, (f64::NAN, f64::NAN)), Complex::with_val(p, (f64::NAN, f64::NAN))); + let a64 = a.to_f64(); + let d64 = delta.to_f64(); + if r >= (8.0 * (a64 + 170.0) * d64).max(1.0) { + let ez = exp_capped(&z); + let sum = |shift: &Float, q: &Float, first: Complex| { + let step = Complex::with_val(p, z.recip_ref()) * q; + let mut term = first; + let mut total = term.clone(); + for k in 0..2000u32 { + term = -(term * &step) * Float::with_val(p, shift + k); + total += &term; + if settled(&term, &total, p as i32 + 20) { + break; + } + } + total + }; + let inv = Complex::with_val(p, z.recip_ref()); + let inv2 = Complex::with_val(p, &inv * &inv); + let pow_p = (Float::with_val(p, pw * &log_keep)).exp(); + let pow_a = (Float::with_val(p, &a * &log_keep)).exp(); + let k1 = Complex::with_val(p, &ez * sum(&a, &delta, inv.clone())) / &keep - sum(&a, eps, inv.clone()) * pow_p; + let a1 = Float::with_val(p, &a + 1u32); + let tail = -(Float::with_val(p, &a * &log_keep)).exp_m1() / Float::with_val(p, &a * eps); + let k2 = -(inv * tail) + Complex::with_val(p, &ez * sum(&a1, &delta, inv2.clone())) / &keep + - sum(&a1, eps, inv2) * pow_a; + return (k1, k2); + } + // Integrand parts at u: e^(z(from))·W(u) and e^(z(from))·V(u), from = −u + // forward and 1 − u backward, so neither grows; e^z multiplies the first. + let at = |u: &Float, forward: bool| { + let l = Float::with_val(p, Float::with_val(p, &delta * u).ln_1p() * &a); + let from = if forward { Float::with_val(p, -u) } else { Float::with_val(p, &one - u) }; + let lead = Complex::with_val(p, (Float::with_val(p, &from * re), Float::with_val(p, &from * im))); + let base = lead.exp(); + let w = Float::with_val(p, -&l).exp(); + let v = -Float::with_val(p, -&l).exp_m1() / Float::with_val(p, &a * &delta); + (Complex::with_val(p, &base * &w), base * v, Float::with_val(p, &l + Float::with_val(p, re * u))) + }; + let meet = if re.is_sign_negative() && !re.is_zero() { + Float::with_val(p, -(Float::with_val(p, &a / re)) - Float::with_val(p, &one / &delta)).clamp(&0, &1) + } else { + one.clone() + }; + let nodes = legendre(); + let (mut k1, mut k2) = (Complex::with_val(p, (0, 0)), Complex::with_val(p, (0, 0))); + let (mut k1f, mut k2f) = (k1.clone(), k2.clone()); + let mut panels = 0; + for forward in [true, false] { + let mut u = if forward { Float::with_val(p, 0) } else { one.clone() }; + loop { + let rest = if forward { Float::with_val(p, &meet - &u) } else { Float::with_val(p, &u - &meet) }; + if rest <= 0 { + break; + } + // The integrand's size only falls toward the meeting point, so its + // size here times what is left bounds the rest. + let (f1, f2, _) = at(&u, forward); + let sum = if forward { (&k1f, &k2f) } else { (&k1, &k2) }; + let small = |f: &Complex, s: &Complex| { + let left = Float::with_val(64, f.abs_ref()) * rest.to_f64(); + let got = Float::with_val(64, s.abs_ref()); + !got.is_zero() && left < got * Float::with_val(64, Float::i_exp(1, -(p as i32) - 20)) + }; + if small(&f1, sum.0) && small(&f2, sum.1) { + break; + } + let rr = d64 / (1.0 + d64 * u.to_f64()); + let slope = (-re.to_f64() - a64 * rr).abs(); + let h = (5.0 / (im.to_f64().abs() + slope + 1.0)).min(0.25 / rr).min(1.5 / (rr * a64.sqrt())); + let h = Float::with_val(p, h).min(&rest); + panels += 1; + if panels > 800 { + return nan(); + } + let lo = if forward { u.clone() } else { Float::with_val(p, &u - &h) }; + for (x, w) in nodes { + let (f1, f2, _) = at(&Float::with_val(p, &lo + Float::with_val(p, x * &h)), forward); + let wh = Float::with_val(p, w * &h); + let (s1, s2) = if forward { (&mut k1f, &mut k2f) } else { (&mut k1, &mut k2) }; + *s1 += f1 * &wh; + *s2 += f2 * &wh; + } + if forward { u += &h } else { u -= &h } + } + } + let ez = exp_capped(&z); + let k1 = (Complex::with_val(p, &ez * &k1f) + k1) / &keep; + // V's quadrature stops where W's does, so past a small aδ, K₂ from the + // identity K₂ = (φ₁(z) − (1 − ε)K₁)/(aε) at this precision. + let ad = Float::with_val(p, &a * &delta); + let k2 = if ad > Float::with_val(p, Float::i_exp(1, -8)) { + let phi1 = if z.is_zero() { Complex::with_val(p, (1, 0)) } else { (Complex::with_val(p, &ez - 1u32)) / &z }; + (phi1 - Complex::with_val(p, &k1 * &keep)) / Float::with_val(p, &a * eps) + } else { + (ez * k2f + k2) / &keep + }; + (k1, k2) +} diff --git a/tools/accuracy-bench/src/elliptic_ref.rs b/tools/accuracy-bench/src/elliptic_ref.rs new file mode 100644 index 0000000..347f4e1 --- /dev/null +++ b/tools/accuracy-bench/src/elliptic_ref.rs @@ -0,0 +1,206 @@ +//! Reference elliptic functions at 512 bits, independent of the crate's: the +//! AGM from MPFR, Carlson's duplication run until its deviations are below +//! 2^-70, and the Jacobi functions by the AGM with plain arcsines. + +use rug::Float; +use rug::float::Constant; + +pub const P: u32 = 512; + +fn f(x: f64) -> Float { + Float::with_val(P, x) +} + +fn big(x: &Float) -> Float { + Float::with_val(P, x) +} + +/// Duplication until the largest deviation `|1 − v/A|` is below 2^-70. +fn duplicate(mut v: Vec, weights: &[u32], mut each: impl FnMut(&[Float], &[Float], &Float)) -> (Vec, Float) { + let total: u32 = weights.iter().sum(); + let mut a = v.iter().zip(weights).fold(f(0.0), |acc, (x, &w)| acc + Float::with_val(P, x * w)) / total; + let tiny = Float::with_val(P, Float::i_exp(1, -70)); + for _ in 0..400 { + let dev = v.iter().map(|x| (Float::with_val(P, &a - x) / &a).abs()).fold(f(0.0), |m, d| if d > m { d } else { m }); + if dev < tiny { + break; + } + let roots: Vec = v.iter().map(|x| x.clone().sqrt()).collect(); + let lam = Float::with_val(P, &roots[0] * &roots[1]) + Float::with_val(P, &roots[0] * &roots[2]) + Float::with_val(P, &roots[1] * &roots[2]); + each(&v, &roots, &lam); + for x in v.iter_mut() { + *x = (x.clone() + &lam) / 4u32; + } + a = (a + &lam) / 4u32; + } + (v, a) +} + +pub fn rf(x: &Float, y: &Float, z: &Float) -> Float { + let (v, a) = duplicate(vec![big(x), big(y), big(z)], &[1, 1, 1], |_, _, _| {}); + let dx = Float::with_val(P, &a - &v[0]) / &a; + let dy = Float::with_val(P, &a - &v[1]) / &a; + let dz = -(dx.clone() + &dy); + let e2 = Float::with_val(P, &dx * &dy) - dz.clone().square(); + let e3 = dx * dy * dz; + let s = f(1.0) - Float::with_val(P, &e2 / 10u32) + Float::with_val(P, &e3 / 14u32) + + e2.clone().square() / 24u32 + - Float::with_val(P, &e2 * &e3) * 3u32 / 44u32; + s / a.sqrt() +} + +pub fn rc(x: &Float, y: &Float) -> Float { + if x.is_zero() { + return Float::with_val(P, Constant::Pi) / 2u32 / y.clone().sqrt(); + } + if x == y { + return f(1.0) / x.clone().sqrt(); + } + let d = Float::with_val(P, y - x); + if d.is_sign_positive() { + let t = (d.clone() / x).sqrt(); + t.atan() / d.sqrt() + } else { + let d = -d; + let t = (d.clone() / x).sqrt(); + t.atanh() / d.sqrt() + } +} + +pub fn rj(x: &Float, y: &Float, z: &Float, p: &Float) -> Float { + let mut sum = f(0.0); + let mut scale = f(1.0); + let (v, a) = duplicate(vec![big(x), big(y), big(z), big(p)], &[1, 1, 1, 2], |v, r, lam| { + let alpha = (Float::with_val(P, &v[3] * (r[0].clone() + &r[1] + &r[2])) + Float::with_val(P, &r[0] * &r[1]) * &r[2]).square(); + let beta = Float::with_val(P, &v[3] * (v[3].clone() + lam).square()); + sum += Float::with_val(P, &scale * rc(&alpha, &beta)); + scale /= 4u32; + }); + let dx = Float::with_val(P, &a - &v[0]) / &a; + let dy = Float::with_val(P, &a - &v[1]) / &a; + let dz = Float::with_val(P, &a - &v[2]) / &a; + let dp = -(dx.clone() + &dy + &dz) / 2u32; + let xyz = Float::with_val(P, &dx * &dy) * &dz; + let e2 = Float::with_val(P, &dx * &dy) + Float::with_val(P, &dx * &dz) + Float::with_val(P, &dy * &dz) - dp.clone().square() * 3u32; + let p3 = Float::with_val(P, dp.clone().square() * &dp); + let e3 = xyz.clone() + Float::with_val(P, &e2 * &dp) * 2u32 + p3.clone() * 4u32; + let e4 = (xyz.clone() * 2u32 + Float::with_val(P, &e2 * &dp) + p3 * 3u32) * &dp; + let e5 = xyz * dp.square(); + let series = f(1.0) - Float::with_val(P, &e2 * 3u32) / 14u32 + Float::with_val(P, &e3 / 6u32) + + e2.clone().square() * 9u32 / 88u32 + - e4 * 3u32 / 22u32 + - Float::with_val(P, &e2 * &e3) * 9u32 / 52u32 + + e5 * 3u32 / 26u32; + let a32 = Float::with_val(P, &a * a.clone().sqrt()); + scale * series / a32 + sum * 3u32 +} + +pub fn rd(x: &Float, y: &Float, z: &Float) -> Float { + rj(x, y, z, z) +} + +pub fn k(kc: &Float) -> Float { + Float::with_val(P, Constant::Pi) / 2u32 / f(1.0).agm(kc) +} + +pub fn e(kc: &Float) -> Float { + if kc.is_zero() { + return f(1.0); + } + let kc2 = kc.clone().square(); + let m = f(1.0) - &kc2; + rf(&f(0.0), &kc2, &f(1.0)) - m * rd(&f(0.0), &kc2, &f(1.0)) / 3u32 +} + +/// `(sn, cn, dn, am)` of `u`. +pub fn jacobi(u: &Float, kc: &Float) -> (Float, Float, Float, Float) { + let one = f(1.0); + if kc.is_zero() { + let s = u.clone().tanh(); + let c = one.clone() / u.clone().cosh(); + let am = u.clone().sinh().atan(); + return (s, c.clone(), c, am); + } + let m = one.clone() - kc.clone().square(); + let (mut a, mut b, mut c) = (vec![one.clone()], vec![big(kc)], vec![m.clone().sqrt()]); + let tiny = Float::with_val(P, Float::i_exp(1, -500)); + while c.last().map_or(false, |x| x.clone().abs() > tiny) && a.len() < 64 { + let (an, bn) = (a.last().unwrap().clone(), b.last().unwrap().clone()); + a.push((an.clone() + &bn) / 2u32); + b.push((an.clone() * &bn).sqrt()); + c.push((an - bn) / 2u32); + } + let n = a.len() - 1; + let mut phi = Float::with_val(P, &a[n] * u) * Float::with_val(P, Float::i_exp(1, n as i32)); + for i in (1..=n).rev() { + let x = Float::with_val(P, &c[i] / &a[i]) * phi.clone().sin(); + phi = (phi + x.asin()) / 2u32; + } + let (s, cn) = (phi.clone().sin(), phi.clone().cos()); + let dn = (one - m * s.clone().square()).sqrt(); + (s, cn, dn, phi) +} + +/// `Π(n; φ|m)` from `sn`, `cn`, `dn` of `φ`, over `(−π, π]`. +pub fn pi(n: &Float, sn: &Float, cn: &Float, dn: &Float, kc: &Float) -> Float { + let one = f(1.0); + let s2 = sn.clone().square(); + let pole = one.clone() - Float::with_val(P, n * &s2); + let (c2, d2) = (cn.clone().square(), dn.clone().square()); + let part = sn.clone().abs() * (rf(&c2, &d2, &one) + Float::with_val(P, n * &s2) * rj(&c2, &d2, &one, &pole) / 3u32); + let total = if cn.is_sign_negative() && !cn.is_zero() { pi_complete(n, kc) * 2u32 - part } else { part }; + if sn.is_sign_negative() { -total } else { total } +} + +pub fn pi_complete(n: &Float, kc: &Float) -> Float { + let one = f(1.0); + let kc2 = kc.clone().square(); + let pole = one.clone() - n; + rf(&f(0.0), &kc2, &one) + Float::with_val(P, n * rj(&f(0.0), &kc2, &one, &pole)) / 3u32 +} + +/// `u` with these `(sn, cn, dn)`, in `(−2K, 2K]`. +pub fn inverse(sn: &Float, cn: &Float, dn: &Float, kc: &Float) -> Float { + let part = sn.clone().abs() * rf(&cn.clone().square(), &dn.clone().square(), &f(1.0)); + let u = if cn.is_sign_negative() && !cn.is_zero() { k(kc) * 2u32 - part } else { part }; + if sn.is_sign_negative() { -u } else { u } +} + +/// `u` less the nearest multiple of `2K`, or `u` itself on the separatrix. +fn reduced(u: &Float, kc: &Float) -> Float { + if kc.is_zero() { + return big(u); + } + let period = k(kc) * 2u32; + let turns = Float::with_val(P, u / &period).round(); + Float::with_val(P, u - turns * period) +} + +/// `Π(n; am v|m)` for `v` within a period of 0, continued on the separatrix past 10⁴ +/// by `1/(1 − n)`, below which `sech²` has vanished. +fn pi_of_u(n: &Float, v: &Float, kc: &Float) -> Float { + let cap = f(1e4); + if kc.is_zero() && Float::with_val(P, v.clone().abs()) > cap { + let edge = if v.is_sign_negative() { -cap } else { cap }; + let rest = Float::with_val(P, v - &edge) / (f(1.0) - n); + return pi_of_u(n, &edge, kc) + rest; + } + if kc.is_zero() { + let (sn, cn, dn, _) = jacobi(v, kc); + return pi(n, &sn, &cn, &dn, kc); + } + // Π(v) = 2Π(n|m)·round(v/2K) + Π of the rest, within (−K, K]. + let period = k(kc) * 2u32; + let turns = Float::with_val(P, v / &period).round(); + let rest = Float::with_val(P, v - Float::with_val(P, &turns * &period)); + let (sr, cr, dr, _) = jacobi(&rest, kc); + let local = pi(n, &sr, &cr, &dr, kc); + if turns.is_zero() { local } else { local + turns * pi_complete(n, kc) * 2u32 } +} + +/// `∫ dv/(1 − n·sn²v)` from `u` to `u + s`. +pub fn pi_increment(n: &Float, u: &Float, s: &Float, kc: &Float) -> Float { + let from = reduced(u, kc); + let to = Float::with_val(P, &from + s); + pi_of_u(n, &to, kc) - pi_of_u(n, &from, kc) +} diff --git a/tools/accuracy-bench/src/lib.rs b/tools/accuracy-bench/src/lib.rs index 6f79c8a..d21412f 100644 --- a/tools/accuracy-bench/src/lib.rs +++ b/tools/accuracy-bench/src/lib.rs @@ -2,10 +2,15 @@ pub mod functions; pub mod metrics; +pub mod complex_ref; +pub mod elliptic_ref; pub mod readme; pub mod reference; pub mod report; pub mod sampling; +pub mod sut; +pub mod sweep; +pub mod worst; use fixed::traits::Fixed; use metrics::ErrorStats; diff --git a/tools/accuracy-bench/src/main.rs b/tools/accuracy-bench/src/main.rs index 4a16eb4..fe25b6d 100644 --- a/tools/accuracy-bench/src/main.rs +++ b/tools/accuracy-bench/src/main.rs @@ -2,8 +2,9 @@ //! //! Run with: cargo run --release //! Compare: cargo run --release -- --baseline path/to/baseline.json +//! [--worst-baseline path/to/worst.txt] -use accuracy_bench::{build_registry, readme, report::Report, sampling::SampleStrategy}; +use accuracy_bench::{build_registry, readme, report::Report, sampling::SampleStrategy, worst}; use rayon::prelude::*; use std::{env, fs, path::Path, process}; @@ -16,6 +17,14 @@ fn main() { .position(|a| a == "--baseline") .and_then(|i| args.get(i + 1)) .map(String::as_str); + let worst_baseline_path = args + .iter() + .position(|a| a == "--worst-baseline") + .and_then(|i| args.get(i + 1)) + .map(String::as_str); + + let quick = args.iter().any(|a| a == "--quick"); + accuracy_bench::sweep::QUICK.store(quick, std::sync::atomic::Ordering::Relaxed); let strategy = SampleStrategy::thorough(); @@ -38,11 +47,18 @@ fn main() { let report = Report::new(results); + eprintln!("Sweeping worst cases..."); + let worst_stats = worst::sweep(); + let worst_text = worst::to_text(&worst_stats); + // Save JSON report fs::create_dir_all("reports").ok(); let json_path = format!("reports/accuracy-{}.json", report.timestamp); fs::write(&json_path, report.to_json()).expect("Failed to write report"); eprintln!("Report saved: {json_path}"); + let worst_report = format!("reports/worst-{}.txt", report.timestamp); + fs::write(&worst_report, &worst_text).expect("Failed to write worst-case report"); + eprintln!("Worst cases saved: {worst_report}"); // Determine README path (handle running from different directories) let readme_path = find_readme_path(); @@ -65,11 +81,40 @@ fn main() { eprintln!("\nWarning: Could not find README.md to verify"); } + if let Some(ref path) = readme_path.as_ref().filter(|_| !quick) { + match worst::verify_readme(path, &worst_stats) { + Ok(()) => eprintln!("README worst case: OK"), + Err(e) => { + eprintln!("README worst case: FAILED\n{e}"); + all_passed = false; + } + } + } + if !worst::check_bounds(&worst_stats) { + all_passed = false; + } + if !quick && fs::read_to_string(worst::WORST_PATH).ok().as_deref() != Some(worst_text.as_str()) { + eprintln!("{} is out of date; run `cargo run --release` here", worst::WORST_PATH); + all_passed = false; + } + // Compare to baseline let baseline_passed = compare_and_report(&report, baseline_path); if !baseline_passed { all_passed = false; } + // Quick mode samples other points than the full run that wrote the + // baseline, so its maxima are held to the bounds alone. + if let Some(path) = worst_baseline_path.filter(|_| !quick) { + match fs::read_to_string(path) { + Ok(text) => { + if !worst::compare(&worst_stats, &text) { + all_passed = false; + } + } + Err(e) => eprintln!("No worst-case baseline at {path} ({e}); skipping"), + } + } process::exit(if all_passed { 0 } else { 1 }); } else { @@ -80,9 +125,16 @@ fn main() { Ok(false) => eprintln!("README.md already up-to-date"), Err(e) => eprintln!("Warning: Could not update README: {e}"), } + match worst::update_readme(path, &worst_stats) { + Ok(true) => eprintln!("README.md updated with worst-case data"), + Ok(false) => {} + Err(e) => eprintln!("Warning: Could not update README worst case: {e}"), + } } + fs::write(worst::WORST_PATH, &worst_text).expect("Failed to write worst.txt"); report.print_table(); + worst::check_bounds(&worst_stats); } } diff --git a/tools/accuracy-bench/src/sut.rs b/tools/accuracy-bench/src/sut.rs new file mode 100644 index 0000000..686d7aa --- /dev/null +++ b/tools/accuracy-bench/src/sut.rs @@ -0,0 +1,111 @@ +//! The fixed-point types the worst-case sweep runs on, with exact conversion +//! of their values to MPFR. + +use fixed::traits::Fixed; +use fixed_analytics::CordicNumber; +use rug::{Float, Integer}; + +/// Working precision of every reference value, enough for 128-bit inputs. +pub const PREC: u32 = 320; + +/// A type under test: any signed `fixed` type the crate implements. +pub trait Sut: CordicNumber + Fixed + Send + Sync + core::fmt::Debug + 'static { + fn rb(self) -> i128; + fn from_rb(raw: i128) -> Self; + + fn bits() -> u32 { + ::total_bits() + } + + fn frac_n() -> u32 { + ::frac_bits() + } + + fn min_raw() -> i128 { + ::min_value().rb() + } + + fn max_raw() -> i128 { + ::max_value().rb() + } + + fn name() -> String { + format!("I{}F{}", Self::bits() - Self::frac_n(), Self::frac_n()) + } + + fn to_float(self) -> Float { + Float::with_val(PREC, Integer::from(self.rb())) >> Self::frac_n() + } + + /// The nearest value to `x`, saturating at the ends of the range. + fn from_float(x: &Float) -> Self { + let scaled = Float::with_val(PREC, x << Self::frac_n()); + let raw = match scaled.to_integer() { + Some(i) => i.to_i128().unwrap_or(if x.is_sign_negative() { + i128::MIN + } else { + i128::MAX + }), + None => 0, + }; + Self::from_rb(raw.clamp(Self::min_raw(), Self::max_raw())) + } + + fn neg_sat(self) -> Self { + CordicNumber::saturating_sub(::zero(), self) + } +} + +impl Sut for F +where + F: CordicNumber + Fixed + Send + Sync + core::fmt::Debug + 'static, + F::Bits: TryInto + TryFrom, +{ + fn rb(self) -> i128 { + self.to_bits().try_into().ok().unwrap_or(0) + } + + fn from_rb(raw: i128) -> Self { + let clamped = raw.clamp(F::MIN.rb(), F::MAX.rb()); + F::from_bits(clamped.try_into().ok().unwrap_or_else(|| F::ZERO.to_bits())) + } +} + +/// One tested type: its name, width, and the sweep instantiated for it. +pub struct TypeEntry { + pub name: String, + pub bits: u32, + pub run: fn(crate::sweep::Fun) -> crate::sweep::FunStats, +} + +macro_rules! entries { + ($($ty:ident < $($frac:ident),+ >);+ $(;)?) => {{ + use fixed::types::extra::*; + use fixed::{FixedI8, FixedI16, FixedI32, FixedI64, FixedI128}; + let mut v: Vec = Vec::new(); + $($( + v.push(TypeEntry { + name: <$ty<$frac> as Sut>::name(), + bits: <$ty<$frac> as Sut>::bits(), + run: crate::sweep::run::<$ty<$frac>>, + }); + )+)+ + v + }}; +} + +/// Every 8-, 16- and 32-bit layout, and a spread of 64- and 128-bit ones +/// that covers both ends and the 63-bit edge of the I1F63 tables. +pub fn types() -> Vec { + entries! { + FixedI8; + FixedI16; + FixedI32; + FixedI64; + FixedI128; + } +} diff --git a/tools/accuracy-bench/src/sweep.rs b/tools/accuracy-bench/src/sweep.rs new file mode 100644 index 0000000..e43a852 --- /dev/null +++ b/tools/accuracy-bench/src/sweep.rs @@ -0,0 +1,1431 @@ +//! The worst-case sweep: each function's largest error in ulps on every +//! tested type, from exact inputs against MPFR, with symmetry and +//! monotonicity checks on the same inputs. +//! +//! Types of 16 bits or fewer are swept exhaustively (8 bits for functions of +//! two arguments); wider ones on a seeded sample that crowds round the points +//! where each function is hardest. + +use crate::sut::{PREC, Sut}; +use fixed_analytics::CordicNumber; +use rug::Float; +use rug::float::Constant; +use rug::ops::Pow; + +#[derive(Clone, Copy, Debug, PartialEq, Eq, PartialOrd, Ord, Hash)] +pub enum Fun { + Sin, + Cos, + Tan, + Asin, + Acos, + Atan, + Atan2, + Sinh, + Cosh, + Tanh, + Coth, + Asinh, + Acosh, + Atanh, + Acoth, + Exp, + Pow2, + Ln, + Log2, + Log10, + Pow, + Sqrt, + SqrtOrZero, + MulAdd, + MulDiv, + DivOrSaturate, + Dot3, + Hypot, + Norm3, + Normalize3, + ScaleBy, + SqrtProduct, + AsinFromGap, + Rsqrt, + Expm1, + Exprel, + Exprel2, + MomentE1, + MomentE2, + MomentE3, + MomentE4, + MomentE5, + Log1p, + Log1pRatio, + Log1pResidual, + Log1pResidualRatio, + StumpffC0, + StumpffC1, + StumpffC2, + StumpffC3, + TanRatio, + AtanRatio, + AtanhRatio, + TanhRatio, + EllipK, + EllipKm, + EllipE, + JacobiSn, + JacobiCn, + JacobiDn, + JacobiAm, + JacobiInverse, + CarlsonRf, + CarlsonRfRoots, + CarlsonRj, + CarlsonRc, + CarlsonRd, + EllipPi, + EllipPiComplete, + SinTurns, + CosTurns, + Atan2Turns, + DivByTau, + WrapRadians, + WrapPrincipal, + MulByTau, + CMulRe, + CMulIm, + CDivRe, + CDivIm, + CExpRe, + CExpIm, + CExprelRe, + CExprelIm, + CExprel2Re, + CExprel2Im, + FaddeevaRe, + FaddeevaIm, + CErfRe, + CErfIm, + E1Re, + E1Im, + GammaUpperRe, + GammaUpperIm, + CLnRe, + CLnIm, + CPowRe, + CPowIm, + WeightedK1Re, + WeightedK1Im, + WeightedK2Re, + WeightedK2Im, + KeplerUniversal, + Erf, + Erfc, + FresnelC, + FresnelS, + Gamma, + MulDown, + MulUp, + MulDivDown, + MulDivUp, + MulAddDown, + MulAddUp, + DivDown, + DivUp, + Dot3Down, + Dot3Up, + EllipPiIncrement, +} + +use Fun::*; + +pub const ALL: &[Fun] = &[ + Sin, Cos, Tan, Asin, Acos, Atan, Atan2, Sinh, Cosh, Tanh, Coth, Asinh, Acosh, Atanh, Acoth, + Exp, Pow2, Ln, Log2, Log10, Pow, Sqrt, SqrtOrZero, MulAdd, MulDiv, DivOrSaturate, Dot3, + Hypot, Norm3, Normalize3, ScaleBy, SqrtProduct, AsinFromGap, Rsqrt, + Expm1, Exprel, Exprel2, MomentE1, MomentE2, MomentE3, MomentE4, MomentE5, Log1p, Log1pRatio, Log1pResidual, Log1pResidualRatio, StumpffC0, StumpffC1, StumpffC2, StumpffC3, TanRatio, AtanRatio, AtanhRatio, TanhRatio, + EllipK, EllipKm, EllipE, JacobiSn, JacobiCn, JacobiDn, JacobiAm, JacobiInverse, CarlsonRf, CarlsonRfRoots, CarlsonRj, CarlsonRc, CarlsonRd, EllipPi, EllipPiComplete, + SinTurns, CosTurns, Atan2Turns, DivByTau, WrapRadians, WrapPrincipal, MulByTau, + CMulRe, CMulIm, CDivRe, CDivIm, CExpRe, CExpIm, CExprelRe, CExprelIm, CExprel2Re, CExprel2Im, FaddeevaRe, FaddeevaIm, CErfRe, CErfIm, E1Re, E1Im, GammaUpperRe, GammaUpperIm, CLnRe, CLnIm, CPowRe, CPowIm, WeightedK1Re, WeightedK1Im, WeightedK2Re, WeightedK2Im, KeplerUniversal, Erf, Erfc, FresnelC, FresnelS, Gamma, + MulDown, MulUp, MulDivDown, MulDivUp, MulAddDown, MulAddUp, DivDown, DivUp, Dot3Down, Dot3Up, EllipPiIncrement, +]; + +#[derive(Clone, Copy, PartialEq, Eq)] +pub enum Parity { + Odd, + Even, + Neither, +} + +/// Sampled points per function and wide type; pairs for two arguments. +const SAMPLES: usize = 40_000; +/// Pairs drawn per function of two arguments on 16-bit types. +const PAIRS_16: usize = 1 << 18; +/// Samples per type for the costly functions. +const HEAVY_SAMPLES: usize = 3_000; + +/// CI's quick mode: an eighth of the samples, and only 8-bit types swept +/// whole, so the run fits a small runner. Local runs, which write worst.txt, +/// stay full. +pub static QUICK: std::sync::atomic::AtomicBool = std::sync::atomic::AtomicBool::new(false); + +fn quick() -> bool { + QUICK.load(std::sync::atomic::Ordering::Relaxed) +} + +fn scaled(n: usize) -> usize { + if quick() { n / 8 } else { n } +} + +/// Types this narrow are swept whole for one argument. +fn exhaustive_bits() -> u32 { + if quick() { 8 } else { 16 } +} + +pub const fn sample_count() -> usize { + SAMPLES +} + +fn fun_roots(fun: Fun) -> bool { + fun == CarlsonRfRoots +} + +fn elliptic(fun: Fun, a: &[Float]) -> Float { + use crate::elliptic_ref as er; + let one = Float::with_val(er::P, 1); + let up = |v: &Float| Float::with_val(er::P, v); + match fun { + EllipK => er::k(&a[0]), + EllipKm => er::k(&(one - &a[0]).sqrt()), + EllipE => er::e(&a[0]), + JacobiSn | JacobiCn | JacobiDn | JacobiAm => { + let (sn, cn, dn, am) = er::jacobi(&up(&a[0]), &up(&a[1])); + match fun { JacobiSn => sn, JacobiCn => cn, JacobiDn => dn, _ => am } + } + JacobiInverse => er::inverse(&a[0], &a[1], &a[2], &a[3]), + CarlsonRf => er::rf(&a[0], &a[1], &a[2]), + CarlsonRfRoots => er::rf(&up(&a[0]).square(), &up(&a[1]).square(), &up(&a[2]).square()), + CarlsonRj => er::rj(&a[0], &a[1], &a[2], &a[3]), + CarlsonRc => er::rc(&up(&a[0]), &up(&a[1])), + CarlsonRd => er::rd(&a[0], &a[1], &a[2]), + EllipPi => er::pi(&up(&a[0]), &a[1], &a[2], &a[3], &a[4]), + EllipPiIncrement => er::pi_increment(&up(&a[0]), &up(&a[1]), &up(&a[2]), &up(&a[3])), + _ => er::pi_complete(&up(&a[0]), &a[1]), + } +} + +/// Points for the elliptic functions: moduli crowded toward both ends, +/// non-negative Carlson arguments, and `(sn, cn, dn)` rounded from an angle. +fn elliptic_points(fun: Fun, rng: &mut Rng) -> Vec> { + let any = candidates::(fun, rng, 4 * HEAVY_SAMPLES); + let one = ::one(); + let unit: Vec = any.iter().map(|&v| { let a = CordicNumber::abs(v); if a > one { T::from_rb(a.rb() % one.rb().max(1)) } else { a } }).collect(); + let pos: Vec = any.iter().map(|&v| CordicNumber::abs(v)).collect(); + let exps = exponent_candidates::(rng, 4 * HEAVY_SAMPLES); + let pick = |rng: &mut Rng, v: &[T]| v[rng.below(v.len())]; + let count = if is_weighted(fun) { HEAVY_SAMPLES / 8 } else { HEAVY_SAMPLES }; + (0..scaled(count)) + .map(|_| match fun { + EllipK | EllipKm | EllipE => vec![pick(rng, &unit)], + JacobiSn | JacobiCn | JacobiDn | JacobiAm => vec![pick(rng, &any), pick(rng, &unit)], + CarlsonRf | CarlsonRd => (0..3).map(|_| pick(rng, &pos)).collect(), + CarlsonRfRoots => (0..3).map(|_| pick(rng, &any)).collect(), + CarlsonRj => (0..4).map(|_| pick(rng, &pos)).collect(), + CarlsonRc => vec![pick(rng, &pos), pick(rng, &pos)], + EllipPiComplete => vec![pick(rng, &any), pick(rng, &unit)], + EllipPiIncrement => { + // Characteristics mostly within (−1, 1), and a third of the steps tiny. + let n = if rng.below(4) == 0 { pick(rng, &any) } else if rng.below(2) == 0 { pick(rng, &unit) } else { -pick(rng, &unit) }; + let n = if n >= one { n - one - one } else { n }; + let step = pick(rng, &any); + let step = match rng.below(3) { 0 => T::from_rb(step.rb() >> rng.below(T::bits() as usize)), 1 => T::from_rb(step.rb() % (one.rb() << 3).max(1)), _ => step }; + vec![n, pick(rng, &any), step, pick(rng, &unit)] + } + FaddeevaRe | FaddeevaIm | CErfRe | CErfIm | E1Re | E1Im => vec![pick(rng, &any), pick(rng, &any)], + GammaUpperRe | GammaUpperIm => vec![pick(rng, &pos), pick(rng, &any), pick(rng, &any)], + FresnelC | FresnelS => vec![pick(rng, &any)], + WeightedK1Re | WeightedK1Im | WeightedK2Re | WeightedK2Im => { + vec![pick(rng, &any), pick(rng, &any), pick(rng, &unit), pick(rng, &pos)] + } + KeplerUniversal => { + // A real orbit: α = (2 − w)/r₀ for w = r₀v₀²/μ ≥ 0, σ₀ = s·√(r₀w), |s| ≤ 1. + let r0 = pick(rng, &pos).to_float(); + let w = Float::with_val(PREC, pick(rng, &pos).to_float()) * f([0.01, 0.5, 1.0, 1.9, 2.0, 3.0, 100.0][rng.below(7)]); + let al = Float::with_val(PREC, Float::with_val(PREC, 2) - &w) / &r0; + let s = (rng.next() as f64 / u64::MAX as f64) * 2.0 - 1.0; + let sigma = Float::with_val(PREC, Float::with_val(PREC, &r0 * &w).sqrt() * s); + vec![T::from_float(&r0), T::from_float(&sigma), T::from_float(&al), pick(rng, &any)] + } + CPowRe | CPowIm => { + // A third of the points on an axis or diagonal, where parts vanish exactly. + let (x, y) = (pick(rng, &any), pick(rng, &any)); + let y = match rng.below(9) { 0 => ::zero(), 1 => x, 2 => -x, _ => y }; + let x = if rng.below(9) == 0 { ::zero() } else { x }; + vec![x, y, pick(rng, &exps)] + } + _ => { + let kc = pick(rng, &unit); + let phi = Float::with_val(PREC, rng.next() as f64 / u64::MAX as f64 * 6.2 - 3.1); + let m = Float::with_val(PREC, 1) - kc.to_float().square(); + let s = phi.clone().sin(); + let d = (Float::with_val(PREC, 1) - m * s.clone().square()).sqrt(); + let parts = vec![T::from_float(&s), T::from_float(&phi.cos()), T::from_float(&d), kc]; + if fun == JacobiInverse { parts } else { [vec![pick(rng, &any)], parts].concat() } + } + }) + .collect() +} + +/// References for the functions whose closed forms cancel near 0, worked at +/// three times the precision, which covers the cancellation for any input of +/// a 128-bit type. +fn special(fun: Fun, x: &Float) -> Float { + let p = PREC * 3; + let one = Float::with_val(p, 1); + let zero = x.is_zero(); + let at_zero = |v: f64| Float::with_val(p, v); + match fun { + Expm1 => x.clone().exp_m1(), + Exprel if zero => at_zero(1.0), + Exprel => x.clone().exp_m1() / x, + Exprel2 if zero => at_zero(0.5), + Exprel2 => (x.clone().exp_m1() - x) / x.clone().square(), + MomentE1 | MomentE2 | MomentE3 | MomentE4 | MomentE5 => { + let k = match fun { MomentE1 => 1u32, MomentE2 => 2, MomentE3 => 3, MomentE4 => 4, _ => 5 }; + // Σ (−x)ⁿ/(n!(n + k)) below 1, else (k − 1)!/x^k·(1 − e^(−x)·Σ_{j x.clone().ln_1p(), + Log1pRatio if zero => at_zero(1.0), + Log1pRatio => x.clone().ln_1p() / x, + Log1pResidual => x.clone().ln_1p() - x, + Log1pResidualRatio if zero => at_zero(-0.5), + Log1pResidualRatio => (x.clone().ln_1p() - x) / x.clone().square(), + StumpffC0 | StumpffC1 | StumpffC2 | StumpffC3 => { + let k = match fun { StumpffC0 => 0, StumpffC1 => 1, StumpffC2 => 2, _ => 3 }; + if zero { + return one / [1u32, 1, 2, 6][k]; + } + let w = x.clone().abs().sqrt(); + let (c, s) = if x.is_sign_positive() { + (w.clone().cos(), w.clone().sin()) + } else { + (w.clone().cosh(), w.clone().sinh()) + }; + let c1 = s / &w; + match k { + 0 => c, + 1 => c1, + 2 => (one - c) / x, + _ => (one - c1) / x, + } + } + TanRatio if zero => at_zero(1.0), + TanRatio => x.clone().tan() / x, + AtanRatio if zero => at_zero(1.0), + AtanRatio => x.clone().atan() / x, + AtanhRatio if zero => at_zero(1.0), + AtanhRatio => x.clone().atanh() / x, + TanhRatio if zero => at_zero(1.0), + TanhRatio => x.clone().tanh() / x, + _ => at_zero(f64::NAN), + } +} + +/// `χ` with `F(χ) = r₀χ + σ₀χ²c₂(αχ²) + (1 − αr₀)χ³c₃(αχ²) = τ`, by a doubled +/// bracket and Newton's method at three times the precision, the Stumpff +/// functions from MPFR's sines and cosines or their series below 1. +fn kepler(a: &[Float]) -> Float { + let p = PREC * 3; + let up = |v: &Float| Float::with_val(p, v); + let (r0, s0, al, tau) = (up(&a[0]), up(&a[1]), up(&a[2]), up(&a[3])); + if tau.is_zero() { + return f(0.0); + } + let beta = Float::with_val(p, 1) - Float::with_val(p, &al * &r0); + let eval = |chi: &Float| { + let x2 = Float::with_val(p, chi.square_ref()); + let z = Float::with_val(p, &al * &x2); + if z < -1e8 { + // e^√|z| swamps the rest, its coefficient of χ's sign on a real orbit. + let inf = Float::with_val(p, if chi.is_sign_negative() { f64::NEG_INFINITY } else { f64::INFINITY }); + return (inf, Float::with_val(p, f64::INFINITY)); + } + let (c0, c1, c2, c3) = if z.clone().abs() < 1 { + let mut c = [Float::with_val(p, 0), Float::with_val(p, 0), Float::with_val(p, 0), Float::with_val(p, 0)]; + for (k, ck) in c.iter_mut().enumerate() { + let mut term = Float::with_val(p, 1) / Float::with_val(p, Float::factorial(k as u32)); + let mut n = k as u32; + for _ in 0..400 { + *ck += &term; + term = -term * &z / ((n + 1) * (n + 2)); + n += 2; + } + } + let [a0, a1, a2, a3] = c; + (a0, a1, a2, a3) + } else { + let w = Float::with_val(p, z.clone().abs().sqrt()); + let (c, s) = if z.is_sign_positive() { (w.clone().cos(), w.clone().sin()) } else { (w.clone().cosh(), w.clone().sinh()) }; + let c1 = Float::with_val(p, &s / &w); + let c2 = Float::with_val(p, Float::with_val(p, 1) - &c) / &z; + let c3 = Float::with_val(p, Float::with_val(p, 1) - &c1) / &z; + (c, c1, c2, c3) + }; + let fx = Float::with_val(p, &r0 * chi) + Float::with_val(p, &s0 * &x2) * &c2 + + Float::with_val(p, &beta * &x2) * chi * &c3 - τ + let rx = Float::with_val(p, &r0 * &c0) + Float::with_val(p, &s0 * chi) * &c1 + Float::with_val(p, &x2 * &c2); + (fx, rx) + }; + let neg = tau.is_sign_negative(); + let past = |m: &Float| { + let chi = if neg { Float::with_val(p, -m) } else { m.clone() }; + let (fx, _) = eval(&chi); + fx.is_zero() || fx.is_sign_negative() == neg + }; + let start = Float::with_val(p, &tau / &r0).abs(); + let (mut near, mut far) = (start.clone(), start.clone()); + if past(&start) { + for _ in 0..1400 { + near = Float::with_val(p, &far / 2u32); + if !past(&near) { break; } + far = near.clone(); + } + } else { + for _ in 0..1400 { + far = Float::with_val(p, &near * 2u32); + if past(&far) { break; } + if far > Float::with_val(p, Float::i_exp(1, 300)) { + return f(if neg { f64::NEG_INFINITY } else { f64::INFINITY }); + } + near = far.clone(); + } + } + let (mut lo, mut hi) = if neg { (Float::with_val(p, -&far), Float::with_val(p, -&near)) } else { (near, far) }; + let mut chi = Float::with_val(p, &lo + &hi) / 2u32; + let mut stride = Float::with_val(64, Float::with_val(p, &hi - &lo).abs()); + for _ in 0..4000 { + let (fx, rx) = eval(&chi); + if fx.is_zero() { + break; + } + if fx.is_sign_negative() { lo = chi.clone() } else { hi = chi.clone() } + let newton = Float::with_val(p, &chi - Float::with_val(p, &fx / &rx)); + let slow = Float::with_val(64, Float::with_val(p, &fx / &rx).abs()) * 2u32 > stride; + let next = if slow || !rx.is_finite() || rx <= 0 || newton <= lo || newton >= hi { Float::with_val(p, &lo + &hi) / 2u32 } else { newton }; + stride = Float::with_val(64, Float::with_val(p, &next - &chi).abs()); + let step = Float::with_val(64, &next - &chi).abs(); + chi = next; + let scale = Float::with_val(64, chi.abs_ref()).max(&Float::with_val(64, Float::i_exp(1, -400))); + if step.is_zero() || step < scale * Float::with_val(64, Float::i_exp(1, -(p as i32) + 8)) { + break; + } + } + chi +} + +/// `num / den`, with 0 for 0/0 and an infinity of the numerator's sign for a +/// zero denominator, as the saturating forms define it. +fn ratio(num: Float, den: &Float) -> Float { + if num.is_zero() { + return f(0.0); + } + if den.is_zero() { + return f(if num.is_sign_negative() { f64::NEG_INFINITY } else { f64::INFINITY }); + } + num / den +} + +fn f(x: f64) -> Float { + Float::with_val(PREC, x) +} + +fn pi() -> Float { + Float::with_val(PREC, Constant::Pi) +} + +impl Fun { + pub fn name(self) -> &'static str { + match self { + Sin => "sin", + Cos => "cos", + Tan => "tan", + Asin => "asin", + Acos => "acos", + Atan => "atan", + Atan2 => "atan2", + Sinh => "sinh", + Cosh => "cosh", + Tanh => "tanh", + Coth => "coth", + Asinh => "asinh", + Acosh => "acosh", + Atanh => "atanh", + Acoth => "acoth", + Exp => "exp", + Pow2 => "pow2", + Ln => "ln", + Log2 => "log2", + Log10 => "log10", + Pow => "pow", + Sqrt => "sqrt", + SqrtOrZero => "sqrt_or_zero", + MulAdd => "mul_add", + MulDiv => "mul_div", + DivOrSaturate => "div_or_saturate", + Dot3 => "dot3", + Hypot => "hypot", + Norm3 => "norm3", + Normalize3 => "normalize3", + ScaleBy => "scale_by", + SqrtProduct => "sqrt_product", + AsinFromGap => "asin_from_gap", + Rsqrt => "rsqrt", + Expm1 => "expm1", + Exprel => "exprel", + Exprel2 => "exprel2", + MomentE1 => "exp_moments_e1", + MomentE2 => "exp_moments_e2", + MomentE3 => "exp_moments_e3", + MomentE4 => "exp_moments_e4", + MomentE5 => "exp_moments_e5", + Log1p => "log1p", + Log1pRatio => "log1p_ratio", + Log1pResidual => "log1p_residual", + Log1pResidualRatio => "log1p_residual_ratio", + StumpffC0 => "stumpff_c0", + StumpffC1 => "stumpff_c1", + StumpffC2 => "stumpff_c2", + StumpffC3 => "stumpff_c3", + TanRatio => "tan_ratio", + AtanRatio => "atan_ratio", + AtanhRatio => "atanh_ratio", + TanhRatio => "tanh_ratio", + EllipK => "ellip_k", + EllipKm => "ellip_k_from_m", + EllipE => "ellip_e", + JacobiSn => "jacobi_sn", + JacobiCn => "jacobi_cn", + JacobiDn => "jacobi_dn", + JacobiAm => "jacobi_am", + JacobiInverse => "jacobi_inverse", + CarlsonRf => "carlson_rf", + CarlsonRfRoots => "carlson_rf_roots", + CarlsonRj => "carlson_rj", + CarlsonRc => "carlson_rc", + CarlsonRd => "carlson_rd", + EllipPi => "ellip_pi", + EllipPiComplete => "ellip_pi_complete", + SinTurns => "sin_turns", + CosTurns => "cos_turns", + Atan2Turns => "atan2_turns", + DivByTau => "div_by_tau", + WrapRadians => "wrap_radians", + WrapPrincipal => "wrap_principal", + MulByTau => "mul_by_tau", + CMulRe => "c_mul_re", + CMulIm => "c_mul_im", + CDivRe => "c_div_re", + CDivIm => "c_div_im", + CExpRe => "c_exp_re", + CExpIm => "c_exp_im", + CExprelRe => "c_exprel_re", + CExprelIm => "c_exprel_im", + CExprel2Re => "c_exprel2_re", + CExprel2Im => "c_exprel2_im", + FaddeevaRe => "faddeeva_re", + FaddeevaIm => "faddeeva_im", + CErfRe => "c_erf_re", + CErfIm => "c_erf_im", + E1Re => "e1_re", + E1Im => "e1_im", + GammaUpperRe => "gamma_upper_re", + GammaUpperIm => "gamma_upper_im", + CLnRe => "c_ln_re", + CLnIm => "c_ln_im", + CPowRe => "c_pow_re", + CPowIm => "c_pow_im", + WeightedK1Re => "exprel_weighted_k1_re", + WeightedK1Im => "exprel_weighted_k1_im", + WeightedK2Re => "exprel_weighted_k2_re", + WeightedK2Im => "exprel_weighted_k2_im", + KeplerUniversal => "kepler_universal", + Erf => "erf", + Erfc => "erfc", + FresnelC => "fresnel_c", + FresnelS => "fresnel_s", + Gamma => "gamma", + MulDown => "mul_down", + MulUp => "mul_up", + MulDivDown => "mul_div_down", + MulDivUp => "mul_div_up", + MulAddDown => "mul_add_down", + MulAddUp => "mul_add_up", + DivDown => "div_down", + DivUp => "div_up", + Dot3Down => "dot3_down", + Dot3Up => "dot3_up", + EllipPiIncrement => "ellip_pi_increment", + } + } + + pub fn arity(self) -> usize { + match self { + Atan2 | Pow | DivOrSaturate | Hypot | ScaleBy | SqrtProduct => 2, + MulAdd | MulDiv | Norm3 | Normalize3 | CarlsonRf | CarlsonRfRoots | CarlsonRd => 3, + JacobiSn | JacobiCn | JacobiDn | JacobiAm | CarlsonRc | EllipPiComplete | Atan2Turns => 2, + JacobiInverse | CarlsonRj | CMulRe | CMulIm | CDivRe | CDivIm => 4, + CExpRe | CExpIm | CExprelRe | CExprelIm | CExprel2Re | CExprel2Im | FaddeevaRe + | FaddeevaIm | CErfRe | CErfIm | E1Re | E1Im => 2, + GammaUpperRe | GammaUpperIm | CPowRe | CPowIm => 3, + CLnRe | CLnIm => 2, + WeightedK1Re | WeightedK1Im | WeightedK2Re | WeightedK2Im | KeplerUniversal => 4, + EllipPi => 5, + Dot3 | Dot3Down | Dot3Up => 6, + MulDown | MulUp | DivDown | DivUp => 2, + MulDivDown | MulDivUp | MulAddDown | MulAddUp => 3, + EllipPiIncrement => 4, + _ => 1, + } + } + + pub fn eval(self, a: &[T]) -> Option { + use fixed_analytics as fa; + let x = a[0]; + match self { + Sin => Some(fa::sin(x)), + Cos => Some(fa::cos(x)), + Tan => Some(fa::tan(x)), + Asin => fa::asin(x).ok(), + Acos => fa::acos(x).ok(), + Atan => Some(fa::atan(x)), + Atan2 => Some(fa::atan2(x, a[1])), + Sinh => Some(fa::sinh(x)), + Cosh => Some(fa::cosh(x)), + Tanh => Some(fa::tanh(x)), + Coth => fa::coth(x).ok(), + Asinh => Some(fa::asinh(x)), + Acosh => fa::acosh(x).ok(), + Atanh => fa::atanh(x).ok(), + Acoth => fa::acoth(x).ok(), + Exp => Some(fa::exp(x)), + Pow2 => Some(fa::pow2(x)), + Ln => fa::ln(x).ok(), + Log2 => fa::log2(x).ok(), + Log10 => fa::log10(x).ok(), + Pow => fa::pow(x, a[1]).ok(), + Sqrt => fa::sqrt(x).ok(), + SqrtOrZero => Some(fa::sqrt_or_zero(x)), + MulAdd => Some(fa::mul_add(x, a[1], a[2])), + MulDiv => Some(fa::mul_div(x, a[1], a[2])), + DivOrSaturate => Some(fa::div_or_saturate(x, a[1])), + Dot3 => Some(fa::dot(&a[0..3], &a[3..6])), + Hypot => Some(fa::hypot(x, a[1])), + Norm3 => Some(fa::norm(&a[0..3])), + Normalize3 => fa::normalize([x, a[1], a[2]]).ok().map(|v| v[0]), + ScaleBy => Some(fa::scale_by(x, exponent_of(a[1]))), + SqrtProduct => fa::sqrt_product(x, a[1]).ok(), + AsinFromGap => fa::asin_from_gap(x).ok(), + Rsqrt => fa::rsqrt(x).ok(), + Expm1 => Some(fa::expm1(x)), + Exprel => Some(fa::exprel(x)), + Exprel2 => Some(fa::exprel2(x)), + MomentE1 => Some(fa::exp_moments(x)[0]), + MomentE2 => Some(fa::exp_moments(x)[1]), + MomentE3 => Some(fa::exp_moments(x)[2]), + MomentE4 => Some(fa::exp_moments(x)[3]), + MomentE5 => Some(fa::exp_moments(x)[4]), + Log1p => fa::log1p(x).ok(), + Log1pRatio => fa::log1p_ratio(x).ok(), + Log1pResidual => fa::log1p_residual(x).ok(), + Log1pResidualRatio => fa::log1p_residual_ratio(x).ok(), + StumpffC0 => Some(fa::stumpff(x)[0]), + StumpffC1 => Some(fa::stumpff(x)[1]), + StumpffC2 => Some(fa::stumpff(x)[2]), + StumpffC3 => Some(fa::stumpff(x)[3]), + TanRatio => Some(fa::tan_ratio(x).0), + AtanRatio => Some(fa::atan_ratio(x)), + AtanhRatio => fa::atanh_ratio(x).ok(), + TanhRatio => Some(fa::tanh_ratio(x)), + EllipK => fa::ellip_k(fa::Modulus::from_complement(x)?).ok(), + EllipKm => fa::ellip_k(fa::Modulus::from_parameter(x)?).ok(), + EllipE => Some(fa::ellip_e(fa::Modulus::from_complement(x)?)), + JacobiSn | JacobiCn | JacobiDn => { + let (sn, cn, dn) = fa::jacobi_sn_cn_dn(x, fa::Modulus::from_complement(a[1])?); + Some(match self { JacobiSn => sn, JacobiCn => cn, _ => dn }) + } + JacobiAm => Some(fa::jacobi_am(x, fa::Modulus::from_complement(a[1])?)), + JacobiInverse => fa::jacobi_inverse(x, a[1], a[2], fa::Modulus::from_complement(a[3])?).ok(), + CarlsonRf => fa::carlson_rf(x, a[1], a[2]).ok(), + CarlsonRfRoots => fa::carlson_rf_roots(x, a[1], a[2]).ok(), + CarlsonRj => fa::carlson_rj(x, a[1], a[2], a[3]).ok(), + CarlsonRc => fa::carlson_rc(x, a[1]).ok(), + CarlsonRd => fa::carlson_rd(x, a[1], a[2]).ok(), + EllipPi => fa::ellip_pi(x, a[1], a[2], a[3], fa::Modulus::from_complement(a[4])?).ok(), + EllipPiComplete => fa::ellip_pi_complete(x, fa::Modulus::from_complement(a[1])?).ok(), + SinTurns => Some(fa::sin_cos_turns(x).0), + CosTurns => Some(fa::sin_cos_turns(x).1), + Atan2Turns => Some(fa::atan2_turns(x, a[1])), + DivByTau => Some(fa::div_by_tau(x)), + WrapRadians => Some(fa::wrap_radians(x)), + WrapPrincipal => Some(fa::wrap_principal(x)), + CMulRe | CMulIm | CDivRe | CDivIm => { + let (u, v) = (fa::Complex::new(x, a[1]), fa::Complex::new(a[2], a[3])); + let r = if matches!(self, CMulRe | CMulIm) { u * v } else { u / v }; + Some(if matches!(self, CMulRe | CDivRe) { r.re } else { r.im }) + } + CExpRe | CExpIm | CExprelRe | CExprelIm | CExprel2Re | CExprel2Im | FaddeevaRe + | FaddeevaIm | CErfRe | CErfIm | E1Re | E1Im => { + let z = fa::Complex::new(x, a[1]); + let r = match self { + CExpRe | CExpIm => z.exp(), + CExprelRe | CExprelIm => z.exprel(), + CExprel2Re | CExprel2Im => z.exprel2(), + FaddeevaRe | FaddeevaIm => z.faddeeva(), + CErfRe | CErfIm => z.erf(), + _ => z.e1().ok()?, + }; + Some(if (self as u32) % 2 == (CExpRe as u32) % 2 { r.re } else { r.im }) + } + GammaUpperRe | GammaUpperIm => { + let r = fa::Complex::new(a[1], a[2]).gamma_upper(x).ok()?; + Some(if self == GammaUpperRe { r.re } else { r.im }) + } + CLnRe | CLnIm | CPowRe | CPowIm => { + let z = fa::Complex::new(x, a[1]); + let r = if matches!(self, CLnRe | CLnIm) { z.ln() } else { z.pow(a[2]) }.ok()?; + Some(if matches!(self, CLnRe | CPowRe) { r.re } else { r.im }) + } + WeightedK1Re | WeightedK1Im | WeightedK2Re | WeightedK2Im => { + let (k1, k2) = fa::Complex::new(x, a[1]).exprel_weighted(a[2], a[3]).ok()?; + Some(match self { + WeightedK1Re => k1.re, + WeightedK1Im => k1.im, + WeightedK2Re => k2.re, + _ => k2.im, + }) + } + KeplerUniversal => fa::kepler_universal(x, a[1], a[2], a[3]).ok(), + MulByTau => Some(fa::mul_by_tau(x)), + Erf => Some(fa::erf(x)), + Erfc => Some(fa::erfc(x)), + FresnelC => Some(fa::fresnel(x).0), + FresnelS => Some(fa::fresnel(x).1), + Gamma => fa::gamma(x).ok(), + MulDown | MulUp | MulDivDown | MulDivUp | MulAddDown | MulAddUp | DivDown | DivUp + | Dot3Down | Dot3Up => { + let m = if self.direction() == Some(true) { fa::Rounding::Up } else { fa::Rounding::Down }; + Some(match self { + MulDown | MulUp => fa::mul_rounded(x, a[1], m), + MulDivDown | MulDivUp => fa::mul_div_rounded(x, a[1], a[2], m), + MulAddDown | MulAddUp => fa::mul_add_rounded(x, a[1], a[2], m), + DivDown | DivUp => fa::div_rounded(x, a[1], m), + _ => fa::dot_rounded(&a[0..3], &a[3..6], m), + }) + } + EllipPiIncrement => fa::ellip_pi_increment(x, a[1], a[2], fa::Modulus::from_complement(a[3])?).ok(), + } + } + + /// For the directed forms, whether they round up rather than down. + pub fn direction(self) -> Option { + match self { + MulDown | MulDivDown | MulAddDown | DivDown | Dot3Down => Some(false), + MulUp | MulDivUp | MulAddUp | DivUp | Dot3Up => Some(true), + _ => None, + } + } + + /// Whether the fallible form should succeed at `a`. + pub fn defined(self, a: &[T]) -> bool { + let x = a[0]; + let zero = T::zero(); + let one = T::one(); + match self { + Asin | Acos => x >= -one && x <= one, + Coth => x != zero, + Acosh => x >= one, + Atanh => x > -one && x < one, + Acoth => x < -one || x > one, + Ln | Log2 | Log10 | Rsqrt => x > zero, + Log1p | Log1pRatio | Log1pResidual | Log1pResidualRatio => x > -one, + AtanhRatio => x > -one && x < one, + EllipK => x > zero && x <= one, + EllipKm => x >= zero && x < one, + EllipE => x >= zero && x <= one, + JacobiSn | JacobiCn | JacobiDn | JacobiAm => a[1] >= zero && a[1] <= one, + JacobiInverse => a[3] >= zero && a[3] <= one && (a[1] != zero || a[2] != zero), + CarlsonRf | CarlsonRfRoots => a.iter().filter(|&&v| v == zero).count() <= 1 && (fun_roots(self) || a.iter().all(|&v| v >= zero)), + CarlsonRj => a[..3].iter().filter(|&&v| v == zero).count() <= 1 && a.iter().all(|&v| v >= zero) && a[3] > zero, + CarlsonRc => x >= zero && a[1] > zero, + CarlsonRd => x >= zero && a[1] >= zero && a[2] > zero && (x != zero || a[1] != zero), + EllipPi => { + let sn = a[1].to_float(); + let pole = Float::with_val(PREC, 1) - Float::with_val(PREC, a[0].to_float() * sn.square()); + let reachable = a[2] >= zero || a[0] < one; + a[4] >= zero && a[4] <= one && pole.is_sign_positive() && !pole.is_zero() && reachable && (a[2] != zero || a[3] != zero) + } + EllipPiComplete => x < one && a[1] > zero && a[1] <= one, + EllipPiIncrement => x < one && a[3] >= zero && a[3] <= one, + E1Re | E1Im => x != zero || a[1] != zero, + GammaUpperRe | GammaUpperIm => x > zero && x.to_float() <= 128, + CLnRe | CLnIm => x != zero || a[1] != zero, + CPowRe | CPowIm => x != zero || a[1] != zero || a[2] >= zero, + WeightedK1Re | WeightedK1Im | WeightedK2Re | WeightedK2Im => a[2] >= zero && a[2] < one && a[3] >= zero, + KeplerUniversal => { + let (r, s, al) = (a[0].to_float(), a[1].to_float(), a[2].to_float()); + let h2 = Float::with_val(PREC * 3, &r * (Float::with_val(PREC * 3, 2) - Float::with_val(PREC * 3, &al * &r))) - Float::with_val(PREC * 3, s.square_ref()); + x > zero && h2 > 0 + } + Gamma => x > zero, + Sqrt => x >= zero, + AsinFromGap => x >= zero && x <= one + one, + SqrtProduct => !((x < zero && a[1] > zero) || (x > zero && a[1] < zero)), + Normalize3 => a.iter().any(|&v| v != zero), + ScaleBy => { + let e = a[1]; + (e >> T::frac_n()) << T::frac_n() == e && exponent_of(e).abs() <= 2 * T::bits() as i32 + } + Pow => { + let e = a[1]; + if x == zero { + e >= zero + } else if x < zero { + e == zero || (e >> T::frac_n()) << T::frac_n() == e + } else { + true + } + } + _ => true, + } + } + + /// The exact value at `PREC` bits; infinite past the range of `Float`. + pub fn reference(self, a: &[Float]) -> Float { + let x = a[0].clone(); + let one = f(1.0); + match self { + Sin => x.sin(), + Cos => x.cos(), + Tan => x.tan(), + Asin => x.asin(), + Acos => x.acos(), + Atan => x.atan(), + Atan2 => x.atan2(&a[1]), + Sinh => x.sinh(), + Cosh => x.cosh(), + Tanh => x.tanh(), + Coth => one / x.tanh(), + Asinh => x.asinh(), + Acosh => x.acosh(), + Atanh => x.atanh(), + Acoth => (one / x).atanh(), + Exp => x.exp(), + Pow2 => x.exp2(), + Ln => x.ln(), + Log2 => x.log2(), + Log10 => x.log10(), + Pow => { + if a[1].is_zero() { + one + } else { + x.pow(&a[1]) + } + } + Sqrt => x.sqrt(), + SqrtOrZero => { + if x.is_sign_positive() && !x.is_zero() { + x.sqrt() + } else { + f(0.0) + } + } + MulAdd | MulAddDown | MulAddUp => Float::with_val(PREC * 3, &x * &a[1]) + &a[2], + MulDiv | MulDivDown | MulDivUp => ratio(Float::with_val(PREC * 3, &x * &a[1]), &a[2]), + DivOrSaturate | DivDown | DivUp => ratio(x, &a[1]), + MulDown | MulUp => Float::with_val(PREC * 3, &x * &a[1]), + Dot3 | Dot3Down | Dot3Up => { + let mut acc = Float::with_val(PREC * 3, 0); + for i in 0..3 { + acc += Float::with_val(PREC * 3, &a[i] * &a[i + 3]); + } + acc + } + Hypot => (Float::with_val(PREC * 3, x.clone().square()) + a[1].clone().square()).sqrt(), + Norm3 | Normalize3 => { + let mut acc = Float::with_val(PREC * 3, 0); + for v in &a[0..3] { + acc += v.clone().square(); + } + if self == Norm3 { acc.sqrt() } else { x / acc.sqrt() } + } + ScaleBy => { + let e = a[1].to_i32_saturating().unwrap_or(0); + Float::with_val(PREC * 3, x) * Float::with_val(PREC, Float::i_exp(1, e)) + } + SqrtProduct => Float::with_val(PREC * 3, &x * &a[1]).abs().sqrt(), + AsinFromGap => (one - x).asin(), + Rsqrt => one / x.sqrt(), + EllipK | EllipKm | EllipE | JacobiSn | JacobiCn | JacobiDn | JacobiAm | JacobiInverse + | CarlsonRf | CarlsonRfRoots | CarlsonRj | CarlsonRc | CarlsonRd | EllipPi + | EllipPiComplete | EllipPiIncrement => elliptic(self, a), + SinTurns | CosTurns | DivByTau | WrapRadians | WrapPrincipal => { + let p = PREC * 3; + let tau = Float::with_val(p, Constant::Pi) * 2u32; + let t = Float::with_val(p, &x); + match self { + SinTurns => (t * tau).sin(), + CosTurns => (t * tau).cos(), + DivByTau => t / tau, + _ => { + let r = Float::with_val(p, &t - (t.clone() / &tau).floor() * &tau); + let half = Float::with_val(p, &tau / 2u32); + if self == WrapPrincipal && r > half { r - tau } else { r } + } + } + } + CMulRe | CMulIm | CDivRe | CDivIm => { + let p = PREC * 3; + let u = rug::Complex::with_val(p, (&a[0], &a[1])); + let v = rug::Complex::with_val(p, (&a[2], &a[3])); + let r = if matches!(self, CMulRe | CMulIm) { + u * v + } else if v.is_zero() { + let part = |c: &Float| ratio(c.clone(), &Float::with_val(PREC, 0)); + return if self == CDivRe { part(u.real()) } else { part(u.imag()) }; + } else { + u / v + }; + if matches!(self, CMulRe | CDivRe) { r.real().clone() } else { r.imag().clone() } + } + CExpRe | CExpIm | CExprelRe | CExprelIm | CExprel2Re | CExprel2Im => { + let p = PREC * 3; + let z = rug::Complex::with_val(p, (&a[0], &a[1])); + let r = if z.is_zero() { + let at = match self { CExpRe | CExpIm => 1.0, CExprelRe | CExprelIm => 1.0, _ => 0.5 }; + rug::Complex::with_val(p, (at, 0)) + } else { + let e = crate::complex_ref::exp_capped(&z); + let em1 = rug::Complex::with_val(p, &e - 1u32); + match self { + CExpRe | CExpIm => e, + CExprelRe | CExprelIm => em1 / &z, + _ => (em1 - &z) / rug::Complex::with_val(p, &z * &z), + } + }; + if (self as u32) % 2 == (CExpRe as u32) % 2 { r.real().clone() } else { r.imag().clone() } + } + FaddeevaRe | FaddeevaIm | CErfRe | CErfIm | E1Re | E1Im => { + use crate::complex_ref as cr; + let r = match self { + FaddeevaRe | FaddeevaIm => cr::faddeeva(&a[0], &a[1]), + CErfRe | CErfIm => cr::erf(&a[0], &a[1]), + _ => cr::e1(&a[0], &a[1]), + }; + if (self as u32) % 2 == (CExpRe as u32) % 2 { r.real().clone() } else { r.imag().clone() } + } + GammaUpperRe | GammaUpperIm => { + let r = crate::complex_ref::gamma_upper(&a[0], &a[1], &a[2]); + if self == GammaUpperRe { r.real().clone() } else { r.imag().clone() } + } + CLnRe | CLnIm => { + let r = rug::Complex::with_val(PREC * 3, (&a[0], &a[1])).ln(); + if self == CLnRe { r.real().clone() } else { r.imag().clone() } + } + CPowRe | CPowIm => { + let r = crate::complex_ref::pow(&a[0], &a[1], &a[2]); + if self == CPowRe { r.real().clone() } else { r.imag().clone() } + } + WeightedK1Re | WeightedK1Im | WeightedK2Re | WeightedK2Im => { + let (k1, k2) = weighted_reference(a); + let r = match self { + WeightedK1Re | WeightedK1Im => k1, + _ => k2, + }; + if matches!(self, WeightedK1Re | WeightedK2Re) { r.real().clone() } else { r.imag().clone() } + } + KeplerUniversal => kepler(a), + MulByTau => { + let p = PREC * 3; + Float::with_val(p, &x) * (Float::with_val(p, Constant::Pi) * 2u32) + } + Erf => Float::with_val(PREC * 3, &x).erf(), + Erfc => Float::with_val(PREC * 3, &x).erfc(), + FresnelC | FresnelS => { + let p = PREC * 3; + let c = Float::with_val(p, Constant::Pi).sqrt() / 2u32 * &x; + let e = crate::complex_ref::erf(&c, &Float::with_val(p, -&c)); + let half = rug::Complex::with_val(p, (0.5, 0.5)); + let r = half * e; + if self == FresnelC { r.real().clone() } else { r.imag().clone() } + } + Gamma => Float::with_val(PREC * 3, &x).gamma(), + Atan2Turns => { + let p = PREC * 3; + Float::with_val(p, x.atan2(&a[1])) / (Float::with_val(p, Constant::Pi) * 2u32) + } + _ => special(self, &Float::with_val(PREC * 3, &x)), + } + } + + /// The period an error is taken modulo, for the wraps. + pub fn period(self) -> Option { + matches!(self, WrapRadians | WrapPrincipal).then(|| Float::with_val(PREC, Constant::Pi) * 2u32) + } + + /// Too costly to sweep exhaustively; sampled on every type instead. + pub fn heavy(self) -> bool { + matches!( + self, + FaddeevaRe | FaddeevaIm | CErfRe | CErfIm | E1Re | E1Im | GammaUpperRe | GammaUpperIm + | FresnelC | FresnelS | CPowRe | CPowIm | WeightedK1Re | WeightedK1Im | WeightedK2Re + | WeightedK2Im | + EllipK | EllipKm | EllipE | JacobiSn | JacobiCn | JacobiDn | JacobiAm | JacobiInverse + | CarlsonRf | CarlsonRfRoots | CarlsonRj | CarlsonRc | CarlsonRd | EllipPi + | EllipPiComplete | EllipPiIncrement + ) + } + + /// Whether the result is exact before its one rounding, so its error is at + /// most half an ulp. + pub fn rounds_once(self) -> bool { + matches!(self, Sqrt | SqrtOrZero | MulAdd | MulDiv | DivOrSaturate | Dot3) + } + + pub fn parity(self) -> Parity { + match self { + Sin | Tan | Asin | Atan | Sinh | Tanh | Coth | Asinh | Atanh | Acoth | SinTurns + | DivByTau | MulByTau | Erf | FresnelC | FresnelS => Parity::Odd, + Cos | Cosh | TanRatio | AtanRatio | AtanhRatio | TanhRatio | CosTurns => Parity::Even, + _ => Parity::Neither, + } + } + + /// Open intervals on which the function is monotone, and whether it rises. + pub fn monotone(self) -> Vec<(Float, Float, bool)> { + let inf = f(f64::INFINITY); + let ninf = f(f64::NEG_INFINITY); + let half_pi = pi() / 2u32; + let one = f(1.0); + let zero = f(0.0); + match self { + Sin | Tan => vec![(-half_pi.clone(), half_pi, true)], + Cos => vec![(-pi(), zero.clone(), true), (zero, pi(), false)], + Asin => vec![(f(-1.0) - f(1e-300), one + f(1e-300), true)], + Acos => vec![(f(-1.0) - f(1e-300), one + f(1e-300), false)], + Atan | Sinh | Tanh | Asinh | Exp | Pow2 => vec![(ninf, inf, true)], + Cosh => vec![(ninf, zero.clone() + f(1e-300), false), (zero - f(1e-300), inf, true)], + Coth => vec![(ninf, zero.clone(), false), (zero, inf, false)], + Acosh => vec![(one - f(1e-300), inf, true)], + Atanh => vec![(f(-1.0), one, true)], + Acoth => vec![(ninf, f(-1.0), false), (one, inf, false)], + Ln | Log2 | Log10 => vec![(zero, inf, true)], + Rsqrt => vec![(zero, inf, false)], + SinTurns => vec![(f(-0.25), f(0.25), true)], + DivByTau | MulByTau | Erf => vec![(ninf, inf, true)], + Erfc => vec![(ninf, inf, false)], + Expm1 | Exprel | Exprel2 | Log1p => vec![(ninf, inf, true)], + MomentE1 | MomentE2 | MomentE3 | MomentE4 | MomentE5 | Log1pRatio => vec![(ninf, inf, false)], + StumpffC0 | StumpffC1 | StumpffC2 | StumpffC3 => vec![(ninf, zero.clone() + f(1e-300), false)], + Log1pResidual => vec![(f(-1.0), zero.clone(), true), (zero, inf, false)], + AsinFromGap => vec![(zero.clone() - f(1e-300), f(2.0) + f(1e-300), false)], + Sqrt => vec![(zero - f(1e-300), inf, true)], + _ => vec![], + } + } + + /// Points where the function is hardest, for the sampled types. + fn hot(self) -> Vec { + let max = T::max_value().to_float(); + let int_bits = T::bits() - T::frac_n(); + let ln2 = Float::with_val(PREC, Constant::Log2); + let min_exp = -(i64::from(T::frac_n()) + 1); + let mut h = vec![f(1.0), f(-1.0), f(0.5), f(-0.5), f(2.0), f(-2.0)]; + match self { + Sin | Cos | Tan => { + for j in 0..int_bits + 1 { + let k = Float::with_val(PREC, 1u32) << j; + h.push(pi() * k.clone() / 2u32); + h.push(-(pi() * k.clone() / 2u32)); + h.push(pi() * k.clone() * 3u32 / 2u32); + h.push(pi() * k); + } + } + Exp => { + h.push(max.clone().ln()); + h.push(ln2 * min_exp); + } + Pow2 => { + h.push(max.clone().log2()); + h.push(f(min_exp as f64)); + } + Sinh | Cosh => { + h.push(max.clone().asinh()); + h.push(-(max.clone().asinh())); + } + _ => {} + } + h + } +} + +/// splitmix64. +pub struct Rng(u64); + +impl Rng { + pub fn new(seed: u64) -> Self { + Self(seed) + } + + pub fn next(&mut self) -> u64 { + self.0 = self.0.wrapping_add(0x9E37_79B9_7F4A_7C15); + let mut z = self.0; + z = (z ^ (z >> 30)).wrapping_mul(0xBF58_476D_1CE4_E5B9); + z = (z ^ (z >> 27)).wrapping_mul(0x94D0_49BB_1331_11EB); + z ^ (z >> 31) + } + + pub fn below(&mut self, n: usize) -> usize { + (self.next() % n as u64) as usize + } + + fn wide(&mut self) -> u128 { + (u128::from(self.next()) << 64) | u128::from(self.next()) + } + + /// Uniform over the raw range of `T`. + fn uniform(&mut self) -> i128 { + (self.wide() as i128) >> (128 - T::bits()) + } + + /// A magnitude whose bit length is uniform, with a random sign. + fn log_uniform(&mut self) -> i128 { + let len = self.below(T::bits() as usize) as u32; + let m = if len == 0 { + 0 + } else { + ((self.wide() >> (128 - len)) | (1u128 << (len - 1))) as i128 + }; + if self.next() & 1 == 0 { m } else { -m } + } +} + +fn is_weighted(fun: Fun) -> bool { + matches!(fun, WeightedK1Re | WeightedK1Im | WeightedK2Re | WeightedK2Im) +} + +/// The four parts of `exprel_weighted` share their points and references. +fn weighted_reference(a: &[Float]) -> (rug::Complex, rug::Complex) { + use std::collections::HashMap; + use std::sync::Mutex; + static CACHE: std::sync::OnceLock>> = std::sync::OnceLock::new(); + let key = a.iter().map(|v| v.to_string_radix(16, None)).collect::>().join(","); + let cache = CACHE.get_or_init(|| Mutex::new(HashMap::new())); + if let Some(hit) = cache.lock().unwrap().get(&key) { + return hit.clone(); + } + let value = crate::complex_ref::exprel_weighted(&a[0], &a[1], &a[2], &a[3]); + cache.lock().unwrap().insert(key, value.clone()); + value +} + +fn seed(fun: Fun, ty: &str) -> u64 { + let mut h: u64 = 0xCBF2_9CE4_8422_2325; + let name = if is_weighted(fun) { "exprel_weighted" } else { fun.name() }; + for b in name.bytes().chain(ty.bytes()) { + h = (h ^ u64::from(b)).wrapping_mul(0x0100_0000_01B3); + } + h +} + +fn clamp_raw(r: i128) -> i128 { + r.clamp(T::min_raw(), T::max_raw()) +} + +/// Candidate arguments: the ends of the range, each hot point and its +/// neighbours at every power-of-two distance, then uniform, log-uniform and +/// small values, and values crowding the hot points. +fn candidates(fun: Fun, rng: &mut Rng, n: usize) -> Vec { + let mut centers = vec![0, T::min_raw(), T::max_raw()]; + let pi_t = T::from_float(&pi()).rb(); + centers.extend([pi_t, -pi_t, pi_t / 2, -pi_t / 2]); + centers.extend(fun.hot::().iter().map(|h| T::from_float(h).rb())); + let mut raws = Vec::with_capacity(n + centers.len() * 2 * 130); + for &c in ¢ers { + for d in -3..=3 { + raws.push(c.saturating_add(d)); + } + for k in 0..T::bits() - 1 { + raws.push(c.saturating_add(1 << k)); + raws.push(c.saturating_sub(1 << k)); + } + } + let small = (8i128 << T::frac_n()).min(T::max_raw()); + for i in 0..n { + let r = match i % 4 { + 0 => rng.uniform::(), + 1 => rng.log_uniform::(), + 2 => (rng.wide() % (2 * small as u128 + 1)) as i128 - small, + _ => { + let c = centers[rng.below(centers.len())]; + c.saturating_add(rng.log_uniform::() >> rng.below(T::bits() as usize)) + } + }; + raws.push(r); + } + raws.into_iter() + .map(|r| T::from_rb(clamp_raw::(r))) + .collect() +} + +/// The integer part of `e`, the exponent `scale_by` reads from its sample. +fn exponent_of(e: T) -> i32 { + e.to_float().to_i32_saturating().unwrap_or(0) +} + +fn exponent_candidates(rng: &mut Rng, n: usize) -> Vec { + let mut v: Vec = (-16..=16).map(|k: i32| ::from_num(k)).collect(); + v.extend(candidates::(Pow, rng, n)); + v +} + +fn all() -> Vec { + (T::min_raw()..=T::max_raw()).map(T::from_rb).collect() +} + +fn points(fun: Fun, rng: &mut Rng) -> Vec> { + if fun.heavy() { + return elliptic_points(fun, rng); + } + if fun.arity() == 1 { + let xs = if T::bits() <= exhaustive_bits() { + all::() + } else { + candidates::(fun, rng, scaled(SAMPLES)) + }; + return xs.into_iter().map(|x| vec![x]).collect(); + } + if fun.arity() > 2 { + let xs = if T::bits() <= 16 { + all::() + } else { + candidates::(fun, rng, scaled(SAMPLES)) + }; + let n = scaled(if T::bits() <= 16 { PAIRS_16 } else { SAMPLES }); + return (0..n) + .map(|_| (0..fun.arity()).map(|_| xs[rng.below(xs.len())]).collect()) + .collect(); + } + if T::bits() <= 8 { + let xs = all::(); + return xs + .iter() + .flat_map(|&x| xs.iter().map(move |&y| vec![x, y])) + .collect(); + } + let n = scaled(if T::bits() <= 16 { PAIRS_16 } else { SAMPLES }); + let xs = if T::bits() <= 16 { + all::() + } else { + candidates::(fun, rng, n) + }; + let ys = if fun == Pow || fun == ScaleBy { + exponent_candidates::(rng, n) + } else if T::bits() <= 16 { + all::() + } else { + candidates::(fun, rng, n) + }; + (0..n) + .map(|_| vec![xs[rng.below(xs.len())], ys[rng.below(ys.len())]]) + .collect() +} + +#[derive(Clone, Debug, Default)] +pub struct FunStats { + pub fun: String, + pub ty: String, + pub samples: u64, + pub max_ulps: f64, + pub worst: String, + pub max_rel_ulps: f64, + pub worst_rel: String, + pub parity_bad: u64, + pub parity_n: u64, + pub mono_bad: u64, + pub mono_n: u64, + pub errors: u64, + pub example: String, +} + +fn show(a: &[T]) -> String { + a.iter() + .map(|x| format!("{:e}", x.to_float().to_f64())) + .collect::>() + .join(",") +} + +/// Every check on one function and type. +pub fn run(fun: Fun) -> FunStats { + let name = T::name(); + let mut rng = Rng::new(seed(fun, &name)); + let pts = points::(fun, &mut rng); + let mut s = FunStats { + fun: fun.name().to_string(), + ty: name, + ..FunStats::default() + }; + let lo = T::min_value().to_float(); + let hi = T::max_value().to_float(); + + for p in &pts { + if !fun.defined(p) { + continue; + } + s.samples += 1; + let Some(c) = fun.eval(p) else { + s.errors += 1; + if s.example.is_empty() { + s.example = format!("error at {}", show(p)); + } + continue; + }; + let args: Vec = p.iter().map(|x| x.to_float()).collect(); + let r = fun.reference(&args); + if r.is_nan() { + continue; + } + let expected = if r < lo { + lo.clone() + } else if r > hi { + hi.clone() + } else { + r + }; + let mut diff = Float::with_val(PREC, c.to_float() - &expected); + if let Some(period) = fun.period() { + let half = Float::with_val(PREC, &period / 2u32); + if diff > half { + diff -= . + } else if diff < -half.clone() { + diff += . + } + } + if let Some(up) = fun.direction() { + // Exactly: the wrong way, or an ulp or more off, is a failure. + let exact = Float::with_val(PREC * 4, c.to_float() - &expected); + let wrong = !exact.is_zero() && exact.is_sign_negative() == up; + if wrong || (Float::with_val(PREC * 4, exact.abs()) << T::frac_n()) >= 1 { + s.errors += 1; + if s.example.is_empty() { + s.example = format!("not rounded {} at {}", if up { "up" } else { "down" }, show(p)); + } + } + } + let err = diff.abs() << T::frac_n(); + let ulps = err.to_f64(); + let scale = expected.abs().to_f64().max(1.0); + if ulps > s.max_ulps { + s.max_ulps = ulps; + s.worst = show(p); + } + if ulps / scale > s.max_rel_ulps { + s.max_rel_ulps = ulps / scale; + s.worst_rel = show(p); + } + } + + if fun.arity() == 1 { + parity::(fun, &pts, &mut s); + monotone::(fun, &pts, &mut s); + } + s +} + +fn parity(fun: Fun, pts: &[Vec], s: &mut FunStats) { + let kind = fun.parity(); + if kind == Parity::Neither { + return; + } + for p in pts { + let x = p[0]; + let nx = x.neg_sat(); + if x == T::min_value() || !fun.defined(&[x]) || !fun.defined(&[nx]) { + continue; + } + let (Some(y), Some(z)) = (fun.eval(&[x]), fun.eval(&[nx])) else { + continue; + }; + s.parity_n += 1; + let want = if kind == Parity::Odd { y.neg_sat() } else { y }; + // −MAX is MIN + ulp, so an odd function saturating at MAX meets MIN. + let saturated = kind == Parity::Odd && y == T::max_value() && z == T::min_value(); + if z != want && !saturated { + s.parity_bad += 1; + if s.example.is_empty() { + s.example = format!( + "f({}) = {:e}, f(-x) = {:e}", + show(&[x]), + y.to_float().to_f64(), + z.to_float().to_f64() + ); + } + } + } +} + +fn monotone(fun: Fun, pts: &[Vec], s: &mut FunStats) { + for (lo, hi, rising) in fun.monotone() { + let mut raws: Vec = Vec::new(); + for p in pts { + let r = p[0].rb(); + raws.push(r); + if T::bits() > 16 { + raws.push(clamp_raw::(r.saturating_add(1))); + raws.push(clamp_raw::(r.saturating_add(2))); + } + } + raws.sort_unstable(); + raws.dedup(); + let mut prev: Option<(T, T)> = None; + for r in raws { + let x = T::from_rb(r); + let xf = x.to_float(); + if xf <= lo || xf >= hi || !fun.defined(&[x]) { + continue; + } + let Some(y) = fun.eval(&[x]) else { continue }; + if let Some((px, py)) = prev { + s.mono_n += 1; + let bad = if rising { y < py } else { y > py }; + if bad { + s.mono_bad += 1; + if s.example.is_empty() { + s.example = format!( + "f({}) = {:e} then f({}) = {:e}", + show(&[px]), + py.to_float().to_f64(), + show(&[x]), + y.to_float().to_f64() + ); + } + } + } + prev = Some((x, y)); + } + } +} diff --git a/tools/accuracy-bench/src/worst.rs b/tools/accuracy-bench/src/worst.rs new file mode 100644 index 0000000..1a98497 --- /dev/null +++ b/tools/accuracy-bench/src/worst.rs @@ -0,0 +1,217 @@ +//! The worst-case table: running the sweep, writing it as text, comparing it +//! with a baseline, and its section of the README. + +use crate::sut::types; +use crate::sweep::{ALL, FunStats}; +use rayon::prelude::*; +use std::collections::BTreeMap; +use std::fmt::Write; + +pub const WORST_PATH: &str = "worst.txt"; +const MARKER_START: &str = ""; +const MARKER_END: &str = ""; +const README_TYPES: [&str; 3] = ["I16F16", "I32F32", "I64F64"]; + +/// Runs every function on every type, in the order of `ALL` then `types()`. +pub fn sweep() -> Vec { + let types = types(); + let jobs: Vec<(usize, usize)> = (0..ALL.len()) + .flat_map(|f| (0..types.len()).map(move |t| (f, t))) + .collect(); + let mut out: Vec<((usize, usize), FunStats)> = jobs + .par_iter() + .map(|&(f, t)| ((f, t), (types[t].run)(ALL[f]))) + .collect(); + out.sort_by_key(|(k, _)| *k); + out.into_iter().map(|(_, s)| s).collect() +} + +const HEADER: &str = "# function\ttype\tsamples\tmax_ulps\tmax_rel_ulps\tparity_bad\tparity_n\tmono_bad\tmono_n\terrors\tworst\tworst_rel\texample"; + +pub fn to_text(stats: &[FunStats]) -> String { + let mut out = String::new(); + writeln!(out, "{HEADER}").unwrap(); + for s in stats { + writeln!( + out, + "{}\t{}\t{}\t{:e}\t{:e}\t{}\t{}\t{}\t{}\t{}\t{}\t{}\t{}", + s.fun, + s.ty, + s.samples, + s.max_ulps, + s.max_rel_ulps, + s.parity_bad, + s.parity_n, + s.mono_bad, + s.mono_n, + s.errors, + s.worst, + s.worst_rel, + s.example + ) + .unwrap(); + } + out +} + +pub fn parse(text: &str) -> BTreeMap<(String, String), FunStats> { + let mut map = BTreeMap::new(); + for line in text.lines().filter(|l| !l.starts_with('#') && !l.is_empty()) { + let c: Vec<&str> = line.split('\t').collect(); + if c.len() < 10 { + continue; + } + let s = FunStats { + fun: c[0].to_string(), + ty: c[1].to_string(), + samples: c[2].parse().unwrap_or(0), + max_ulps: c[3].parse().unwrap_or(f64::INFINITY), + max_rel_ulps: c[4].parse().unwrap_or(f64::INFINITY), + parity_bad: c[5].parse().unwrap_or(u64::MAX), + parity_n: c[6].parse().unwrap_or(0), + mono_bad: c[7].parse().unwrap_or(u64::MAX), + mono_n: c[8].parse().unwrap_or(0), + errors: c[9].parse().unwrap_or(u64::MAX), + ..FunStats::default() + }; + map.insert((s.fun.clone(), s.ty.clone()), s); + } + map +} + +fn worse(cur: f64, base: f64) -> bool { + cur > base * (1.0 + 1e-9) + 1e-12 +} + +/// Prints every row past the crate's stated bound, or that breaks a symmetry +/// or monotonicity, or errs; true if none. +pub fn check_bounds(stats: &[FunStats]) -> bool { + let limit = f64::from(fixed_analytics::ERROR_ULPS); + let mut ok = true; + for s in stats { + let fun = crate::sweep::ALL.iter().find(|f| f.name() == s.fun); + let exact = fun.is_some_and(|f| f.rounds_once()); + // A directed form's error reaches toward a whole ulp, which the sweep checks exactly. + let directed = fun.is_some_and(|f| f.direction().is_some()); + let bound = if exact { 0.5 } else { limit }; + let over = if exact || directed { s.max_ulps > bound } else { s.max_ulps >= bound }; + if over || s.parity_bad > 0 || s.mono_bad > 0 || s.errors > 0 { + ok = false; + println!( + "BOUND {} {}: {:.3e} ulps (bound {bound}), symmetry {}, monotonicity {}, errors {}: {}", + s.fun, s.ty, s.max_ulps, s.parity_bad, s.mono_bad, s.errors, s.example + ); + } + } + println!("Bounds: {}", if ok { "every function within its bound" } else { "FAILED" }); + ok +} + +/// Prints every row where `current` is worse than `baseline`; true if none. +pub fn compare(current: &[FunStats], baseline: &str) -> bool { + let base = parse(baseline); + let mut ok = true; + let mut improved = 0; + for s in current { + let Some(b) = base.get(&(s.fun.clone(), s.ty.clone())) else { + continue; + }; + let mut why = Vec::new(); + if worse(s.max_ulps, b.max_ulps) { + why.push(format!("max {:.3e} -> {:.3e} ulps", b.max_ulps, s.max_ulps)); + } + if worse(s.max_rel_ulps, b.max_rel_ulps) { + why.push(format!( + "relative {:.3e} -> {:.3e} ulps", + b.max_rel_ulps, s.max_rel_ulps + )); + } + if s.parity_bad > b.parity_bad { + why.push(format!("symmetry {} -> {}", b.parity_bad, s.parity_bad)); + } + if s.mono_bad > b.mono_bad { + why.push(format!("monotonicity {} -> {}", b.mono_bad, s.mono_bad)); + } + if s.errors > b.errors { + why.push(format!("errors {} -> {}", b.errors, s.errors)); + } + if why.is_empty() { + if s.max_ulps < b.max_ulps || s.parity_bad < b.parity_bad || s.mono_bad < b.mono_bad { + improved += 1; + } + } else { + ok = false; + println!("REGRESS {} {}: {}", s.fun, s.ty, why.join(", ")); + } + } + println!("Worst case: {improved} function-type pairs improve"); + ok +} + +fn fmt_ulps(u: f64) -> String { + if u < 100.0 { + format!("{u:.2}") + } else { + format!("{u:.2e}") + } +} + +pub fn readme_section(stats: &[FunStats]) -> String { + let n_types = types().len(); + let mut out = String::new(); + writeln!(out, "### Worst-case error\n").unwrap(); + writeln!( + out, + "The largest error seen, in ulps of the type, against MPFR on exact inputs. The sweep covers {n_types} layouts from 8 to 128 bits, every value of each 8- and 16-bit one and {} seeded samples of each wider one; tools/accuracy-bench/worst.txt holds every layout. The directed forms, `_down` and `_up`, come within a rounding of a whole ulp, and the sweep checks exactly that each is below one and on its side.\n", + crate::sweep::sample_count() + ) + .unwrap(); + writeln!(out, "| Function | {} |", README_TYPES.join(" | ")).unwrap(); + writeln!(out, "|----------|{}", "------|".repeat(README_TYPES.len())).unwrap(); + let mut by_fun: BTreeMap> = BTreeMap::new(); + for s in stats { + if let Some(col) = README_TYPES.iter().position(|t| *t == s.ty) { + let row = ALL.iter().position(|f| f.name() == s.fun).unwrap_or(usize::MAX); + let cells = by_fun.entry(row).or_insert_with(|| vec![String::new(); README_TYPES.len()]); + cells[col] = fmt_ulps(s.max_ulps); + } + } + for (row, cells) in by_fun { + writeln!(out, "| {} | {} |", ALL[row].name(), cells.join(" | ")).unwrap(); + } + out +} + +fn section_bounds(content: &str) -> Result<(usize, usize), String> { + let start = content + .find(MARKER_START) + .ok_or("README has no marker")?; + let end = content + .find(MARKER_END) + .ok_or("README has no marker")?; + if end <= start { + return Err("WORST_END must follow WORST_START".to_string()); + } + Ok((start + MARKER_START.len(), end)) +} + +pub fn update_readme(path: &str, stats: &[FunStats]) -> Result { + let content = std::fs::read_to_string(path).map_err(|e| e.to_string())?; + let (a, b) = section_bounds(&content)?; + let new = format!("{}\n{}{}", &content[..a], readme_section(stats), &content[b..]); + if new == content { + return Ok(false); + } + std::fs::write(path, new).map_err(|e| e.to_string())?; + Ok(true) +} + +pub fn verify_readme(path: &str, stats: &[FunStats]) -> Result<(), String> { + let content = std::fs::read_to_string(path).map_err(|e| e.to_string())?; + let (a, b) = section_bounds(&content)?; + if content[a..b].trim() == readme_section(stats).trim() { + Ok(()) + } else { + Err("README worst-case section is out of date; run `cargo run --release` in tools/accuracy-bench".to_string()) + } +} diff --git a/tools/accuracy-bench/worst.txt b/tools/accuracy-bench/worst.txt new file mode 100644 index 0000000..9832afc --- /dev/null +++ b/tools/accuracy-bench/worst.txt @@ -0,0 +1,12377 @@ +# function type samples max_ulps max_rel_ulps parity_bad parity_n mono_bad mono_n errors worst worst_rel example +sin I8F0 256 4.987131538963941e-1 4.987131538963941e-1 0 255 0 2 0 -1.22e2 -1.22e2 +sin I7F1 256 4.980255064566478e-1 4.980255064566478e-1 0 255 0 6 0 -1.8e1 -1.8e1 +sin I6F2 256 4.9889581501676966e-1 4.9889581501676966e-1 0 255 0 12 0 -8.75e0 -8.75e0 +sin I5F3 256 4.9600018580208916e-1 4.9600018580208916e-1 0 255 0 24 0 -7.5e0 -7.5e0 +sin I4F4 256 4.976262190118162e-1 4.976262190118162e-1 0 255 0 50 0 -6.0625e0 -6.0625e0 +sin I3F5 256 4.993901499432876e-1 4.993901499432876e-1 0 255 0 100 0 -1.96875e0 -1.96875e0 +sin I16F0 65536 4.9999288793871693e-1 4.9999288793871693e-1 0 65535 0 2 0 -1.7273e4 -1.7273e4 +sin I15F1 65536 4.9998509377195255e-1 4.9998509377195255e-1 0 65535 0 6 0 -5.4195e3 -5.4195e3 +sin I14F2 65536 4.999745202643424e-1 4.999745202643424e-1 0 65535 0 12 0 -4.25125e3 -4.25125e3 +sin I13F3 65536 4.9999382999567e-1 4.9999382999567e-1 0 65535 0 24 0 -3.944625e3 -3.944625e3 +sin I12F4 65536 4.999807943747819e-1 4.999807943747819e-1 0 65535 0 50 0 -1.0973125e3 -1.0973125e3 +sin I11F5 65536 4.999873345238048e-1 4.999873345238048e-1 0 65535 0 100 0 -1.4525e2 -1.4525e2 +sin I10F6 65536 4.999447687264412e-1 4.999447687264412e-1 0 65535 0 200 0 -3.3903125e2 -3.3903125e2 +sin I9F7 65536 4.999928765065806e-1 4.999928765065806e-1 0 65535 0 402 0 -2.61015625e1 -2.61015625e1 +sin I8F8 65536 4.999743105923841e-1 4.999743105923841e-1 0 65535 0 804 0 -1.891796875e1 -1.891796875e1 +sin I7F9 65536 4.9998295266452697e-1 4.9998295266452697e-1 0 65535 0 1608 0 -4.7216796875e1 -4.7216796875e1 +sin I6F10 65536 4.999776120678055e-1 4.999776120678055e-1 0 65535 0 3216 0 -2.36318359375e1 -2.36318359375e1 +sin I5F11 65536 4.9999409528552485e-1 4.9999409528552485e-1 0 65535 0 6432 0 -9.20849609375e0 -9.20849609375e0 +sin I4F12 65536 4.999999450472921e-1 4.999999450472921e-1 0 65535 0 12866 0 -7.79345703125e0 -7.79345703125e0 +sin I3F13 65536 4.999846636082491e-1 4.999846636082491e-1 0 65535 0 25734 0 -2.093505859375e-1 -2.093505859375e-1 +sin I32F0 50005 4.9997303545007793e-1 4.9997303545007793e-1 0 49733 0 2 0 4.4692639e7 4.4692639e7 +sin I31F1 49729 4.999746234927378e-1 4.999746234927378e-1 0 49456 0 6 0 2.45207776e8 2.45207776e8 +sin I30F2 49453 4.999338077115528e-1 4.999338077115528e-1 0 49193 0 12 0 -1.9192693225e8 -1.9192693225e8 +sin I29F3 49177 4.999476622312598e-1 4.999476622312598e-1 0 48887 0 24 0 -6.588409125e6 -6.588409125e6 +sin I28F4 48901 4.999823363844383e-1 4.999823363844383e-1 0 48628 0 50 0 -2.4008125e3 -2.4008125e3 +sin I27F5 48625 4.999756609758986e-1 4.999756609758986e-1 0 48333 0 100 0 1.61130646875e6 1.61130646875e6 +sin I26F6 48349 4.9987589427421053e-1 4.9987589427421053e-1 0 48090 0 200 0 5.5641875e3 5.5641875e3 +sin I25F7 48073 4.999766189458372e-1 4.999766189458372e-1 0 47779 0 402 0 4.680314140625e5 4.680314140625e5 +sin I24F8 47797 4.999691475885791e-1 4.999691475885791e-1 0 47480 0 804 0 1.124296875e2 1.124296875e2 +sin I23F9 47521 4.999918908231776e-1 4.999918908231776e-1 0 47185 0 1603 0 -2.458493333984375e6 -2.458493333984375e6 +sin I22F10 47245 4.9994280600411084e-1 4.9994280600411084e-1 0 46912 0 3069 0 -3.88900705078125e5 -3.88900705078125e5 +sin I21F11 46969 4.9996263785701467e-1 4.9996263785701467e-1 0 46632 0 5202 0 4.88232421875e0 4.88232421875e0 +sin I20F12 46693 4.999846298033023e-1 4.999846298033023e-1 0 46334 0 7427 0 3.58955078125e1 3.58955078125e1 +sin I19F13 46417 4.9998979328730786e-1 4.9998979328730786e-1 0 46081 0 9365 0 -9.585295495605469e4 -9.585295495605469e4 +sin I18F14 46141 4.9999662532800077e-1 4.9999662532800077e-1 0 45801 0 11411 0 -6.5523433654785156e4 -6.5523433654785156e4 +sin I17F15 45865 4.9999324211133295e-1 4.9999324211133295e-1 0 45468 0 12653 0 -5.800395236206055e4 -5.800395236206055e4 +sin I16F16 45589 4.9996348434667054e-1 4.9996348434667054e-1 0 45199 0 13958 0 -3.0401611328125e-1 -3.0401611328125e-1 +sin I15F17 45313 4.9996024447582593e-1 4.9996024447582593e-1 0 44904 0 15030 0 7.547645965576172e3 7.547645965576172e3 +sin I14F18 45037 4.9999812612296496e-1 4.9999812612296496e-1 0 44607 0 15968 0 -1.4722251892089844e0 -1.4722251892089844e0 +sin I13F19 44761 4.999774779000173e-1 4.999774779000173e-1 0 44332 0 17333 0 8.69903564453125e-2 8.69903564453125e-2 +sin I12F20 44485 4.99991551448018e-1 4.99991551448018e-1 0 44011 0 18489 0 7.949206495285034e2 7.949206495285034e2 +sin I11F21 44209 4.999950594614313e-1 4.999950594614313e-1 0 43743 0 19473 0 -4.281147480010986e0 -4.281147480010986e0 +sin I10F22 43933 4.999755874816856e-1 4.999755874816856e-1 0 43442 0 20634 0 4.71287727355957e0 4.71287727355957e0 +sin I9F23 43657 4.99964238823313e-1 4.99964238823313e-1 0 43093 0 21988 0 2.123876428604126e1 2.123876428604126e1 +sin I8F24 43381 4.9995475031945497e-1 4.9995475031945497e-1 0 42810 0 23414 0 7.634103536605835e0 7.634103536605835e0 +sin I7F25 43105 4.9999530442890944e-1 4.9999530442890944e-1 0 42508 0 24937 0 -1.9999899864196777e0 -1.9999899864196777e0 +sin I6F26 42829 4.9999187335727124e-1 4.9999187335727124e-1 0 42181 0 26755 0 1.570674255490303e0 1.570674255490303e0 +sin I5F27 42553 4.9995614022505686e-1 4.9995614022505686e-1 0 41854 0 29344 0 1.2274702832102776e1 1.2274702832102776e1 +sin I4F28 42277 4.9999247896855614e-1 4.9999247896855614e-1 0 41519 0 33139 0 2.2357814013957977e-3 2.2357814013957977e-3 +sin I3F29 42001 4.9994450435187854e-1 4.9994450435187854e-1 0 41121 0 46687 0 -1.5092310681939125e0 -1.5092310681939125e0 +sin I64F0 76309 4.999748891971395e-1 4.999748891971395e-1 0 76031 0 2 0 2.7165083962905487e18 2.7165083962905487e18 +sin I63F1 75777 4.9999422748037387e-1 4.9999422748037387e-1 0 75497 0 6 0 6.616120875e8 6.616120875e8 +sin I62F2 75245 4.9993971979963264e-1 4.9993971979963264e-1 0 74948 0 12 0 -6.15211862075e9 -6.15211862075e9 +sin I61F3 74713 4.9999699561393623e-1 4.9999699561393623e-1 0 74413 0 24 0 6.3504505e7 6.3504505e7 +sin I60F4 74181 4.9996594975182046e-1 4.9996594975182046e-1 0 73884 0 50 0 -3.364830914889284e17 -3.364830914889284e17 +sin I56F8 72053 4.999887361367688e-1 4.999887361367688e-1 0 71757 0 804 0 -4.268613706734375e9 -4.268613706734375e9 +sin I52F12 69925 4.999862134682241e-1 4.999862134682241e-1 0 69611 0 6376 0 2.3056051835998535e9 2.3056051835998535e9 +sin I48F16 67797 4.9997430329156306e-1 4.9997430329156306e-1 0 67484 0 10430 0 2.1269350494384766e5 2.1269350494384766e5 +sin I44F20 65669 4.999968994042372e-1 4.999968994042372e-1 0 65360 0 12587 0 1.0792752442017615e11 1.0792752442017615e11 +sin I40F24 63541 4.9998719776458606e-1 4.9998719776458606e-1 0 63210 0 14593 0 8.954816081913211e10 8.954816081913211e10 +sin I36F28 61413 4.999920655111608e-1 4.999920655111608e-1 0 61063 0 16781 0 2.725777483071484e10 2.725777483071484e10 +sin I33F31 59817 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0 32803 0 4.71238898410998e0 4.71238898410998e0 +sin I6F58 45453 4.9999999990139365e-1 4.9999999990139365e-1 0 44672 0 35689 0 4.712388982247335e0 4.712388982247335e0 +sin I5F59 44921 4.9993698095072003e-1 4.9993698095072003e-1 0 44104 0 37849 0 1.541169855644659e1 1.541169855644659e1 +sin I4F60 44389 4.9999999980278736e-1 4.9999999980278736e-1 0 43463 0 41337 0 4.712388981316012e0 4.712388981316012e0 +sin I3F61 43857 4.999972998743619e-1 4.999972998743619e-1 0 42874 0 53926 0 6.373394312597807e-1 6.373394312597807e-1 +sin I128F0 178069 4.9997923360267327e-1 4.9997923360267327e-1 0 177633 0 2 0 -2.9514790517935276e20 -2.9514790517935276e20 +sin I127F1 177025 4.999803559256365e-1 4.999803559256365e-1 0 176584 0 6 0 8.263905321065603e16 8.263905321065603e16 +sin I126F2 175981 4.9999888753245997e-1 4.9999888753245997e-1 0 175547 0 12 0 -8.470370245344897e36 -8.470370245344897e36 +sin I124F4 173893 4.999978911691846e-1 4.999978911691846e-1 0 173458 0 50 0 -2.658455991569832e36 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4.9999999999999983e-1 0 0 0 0 0 -5.960464477539063e-8,2e0 -5.960464477539063e-8,2e0 +atan2 I36F28 40000 5e-1 5e-1 0 0 0 0 0 1.9999999962747097e0,1.073741822e9 1.9999999962747097e0,1.073741822e9 +atan2 I33F31 40000 5e-1 5e-1 0 0 0 0 0 4.656612873077393e-10,2e0 4.656612873077393e-10,2e0 +atan2 I32F32 40000 5e-1 5e-1 0 0 0 0 0 6.984919309616089e-10,2e0 6.984919309616089e-10,2e0 +atan2 I31F33 40000 5e-1 5e-1 0 0 0 0 0 1.1641532182693481e-10,2e0 1.1641532182693481e-10,2e0 +atan2 I28F36 40000 5e-1 5e-1 0 0 0 0 0 7.275957614183426e-11,2e0 7.275957614183426e-11,2e0 +atan2 I24F40 40000 5e-1 5e-1 0 0 0 0 0 -9.00399754755199e-11,2e0 -9.00399754755199e-11,2e0 +atan2 I20F44 40000 5e-1 5e-1 0 0 0 0 0 -5.115907697472721e-13,2e0 -5.115907697472721e-13,2e0 +atan2 I16F48 40000 5e-1 5e-1 0 0 0 0 0 4.511946372076636e-13,2e0 4.511946372076636e-13,2e0 +atan2 I12F52 40000 5e-1 5e-1 0 0 0 0 0 -2.220446049250313e-16,2e0 -2.220446049250313e-16,2e0 +atan2 I8F56 40000 5e-1 5e-1 0 0 0 0 0 2.4272250875867485e-14,2e0 2.4272250875867485e-14,2e0 +atan2 I6F58 40000 5e-1 5e-1 0 0 0 0 0 -3.469446951953614e-18,2e0 -3.469446951953614e-18,2e0 +atan2 I5F59 40000 5e-1 5e-1 0 0 0 0 0 -1.9051305671924368e-12,2e0 -1.9051305671924368e-12,2e0 +atan2 I4F60 40000 5e-1 5e-1 0 0 0 0 0 -6.071532165918825e-18,2e0 -6.071532165918825e-18,2e0 +atan2 I3F61 40000 5e-1 5e-1 0 0 0 0 0 -9.454242944073599e-17,4e0 -9.454242944073599e-17,4e0 +atan2 I128F0 40000 4.998202893373962e-1 4.9959131722958705e-1 0 0 0 0 0 3.9487812797602514e37,2.8074042554648936e36 9.303885819091845e37,1.7014118346046923e38 +atan2 I127F1 40000 4.999502396489142e-1 4.9982489124599966e-1 0 0 0 0 0 8.364675004468243e37,-6.753523577401181e37 6.023756361076769e37,6.467192803909859e37 +atan2 I126F2 40000 4.9994668470515213e-1 4.9984659364925876e-1 0 0 0 0 0 4.407944409909784e34,-2.658455991569832e36 -6.225e1,4.9525e2 +atan2 I124F4 40000 4.9999466737233816e-1 4.9999466737233816e-1 0 0 0 0 0 1.6875e0,2.5e0 1.6875e0,2.5e0 +atan2 I120F8 40000 4.999759016528231e-1 4.999759016528231e-1 0 0 0 0 0 9.0234375e-1,2e0 9.0234375e-1,2e0 +atan2 I112F16 40000 4.999999999902987e-1 4.999999999902987e-1 0 0 0 0 0 1.52587890625e-5,2e0 1.52587890625e-5,2e0 +atan2 I104F24 40000 4.9999999999999983e-1 4.9999999999999983e-1 0 0 0 0 0 1.1805916207174113e21,3.961408125713217e28 1.1805916207174113e21,3.961408125713217e28 +atan2 I96F32 40000 4.999999999999763e-1 4.999999999999763e-1 0 0 0 0 0 5.098991096019745e-8,2e0 5.098991096019745e-8,2e0 +atan2 I88F40 40000 4.9999999999999817e-1 4.9999999999999817e-1 0 0 0 0 0 -3.4042386687360704e-9,2e0 -3.4042386687360704e-9,2e0 +atan2 I80F48 40000 5e-1 5e-1 0 0 0 0 0 -3.552713678800501e-15,2e0 -3.552713678800501e-15,2e0 +atan2 I72F56 40000 5e-1 5e-1 0 0 0 0 0 -2e0,2.8823037615171174e17 -2e0,2.8823037615171174e17 +atan2 I68F60 40000 5e-1 5e-1 0 0 0 0 0 -8.673617379884035e-19,2e0 -8.673617379884035e-19,2e0 +atan2 I67F61 40000 5e-1 5e-1 0 0 0 0 0 -4.336808689942018e-19,2e0 -4.336808689942018e-19,2e0 +atan2 I66F62 40000 5e-1 5e-1 0 0 0 0 0 1.0842021724855044e-18,2e0 1.0842021724855044e-18,2e0 +atan2 I65F63 40000 5e-1 5e-1 0 0 0 0 0 1e0,1.8446744073709552e19 1e0,1.8446744073709552e19 +atan2 I64F64 40000 5e-1 5e-1 0 0 0 0 0 -4.87890977618477e-19,2e0 -4.87890977618477e-19,2e0 +atan2 I63F65 40000 5e-1 5e-1 0 0 0 0 0 4.0657581468206416e-19,2e0 4.0657581468206416e-19,2e0 +atan2 I62F66 40000 5e-1 5e-1 0 0 0 0 0 -1.0794587879808804e-16,2e0 -1.0794587879808804e-16,2e0 +atan2 I56F72 40000 5e-1 5e-1 0 0 0 0 0 -1.9761088077029695e-16,2e0 -1.9761088077029695e-16,2e0 +atan2 I48F80 40000 5.000000000000031e-1 5.000000000000031e-1 0 0 0 0 0 -8.271806125530277e-25,1.9999999999999878e0 -8.271806125530277e-25,1.9999999999999878e0 +atan2 I40F88 40000 5.000000000063256e-1 5.000000000063256e-1 0 0 0 0 0 3.088690090033148e-15,2e0 3.088690090033148e-15,2e0 +atan2 I32F96 40000 5.000000002572472e-1 5.000000002572472e-1 0 0 0 0 0 -5.87923517620564e-24,1.9999999999999978e0 -5.87923517620564e-24,1.9999999999999978e0 +atan2 I24F104 40000 5e-1 5e-1 0 0 0 0 0 -4.930380657631324e-32,2e0 -4.930380657631324e-32,2e0 +atan2 I16F112 40000 5e-1 5e-1 0 0 0 0 0 -8.940186061144993e-29,2e0 -8.940186061144993e-29,2e0 +atan2 I10F118 40000 5e-1 5e-1 0 0 0 0 0 -3.009265538105056e-36,2e0 -3.009265538105056e-36,2e0 +atan2 I8F120 40000 5e-1 5e-1 0 0 0 0 0 -6.355605521994279e-28,2e0 -6.355605521994279e-28,2e0 +atan2 I6F122 40000 5e-1 5e-1 0 0 0 0 0 -9.4039548065783e-37,2e0 -9.4039548065783e-37,2e0 +atan2 I5F123 40000 5e-1 5e-1 0 0 0 0 0 3.1070666680934703e-34,1.6e1 3.1070666680934703e-34,1.6e1 +atan2 I4F124 40000 5e-1 5e-1 0 0 0 0 0 2.4323799304955102e-33,2e0 2.4323799304955102e-33,2e0 +atan2 I3F125 40000 5e-1 5e-1 0 0 0 0 0 9.541493553329977e-27,4e0 9.541493553329977e-27,4e0 +sinh I8F0 256 3.7313959215298126e-1 1.4908187176067847e-1 0 255 0 255 0 -2e0 -1e0 +sinh I7F1 256 4.2016560574449513e-1 2.9816374352135694e-1 0 255 0 255 0 -4e0 -1e0 +sinh I6F2 256 4.9686858127731987e-1 2.892669277433199e-1 0 255 0 255 0 -3.25e0 -7.5e-1 +sinh I5F3 256 4.739705520964517e-1 4.214661445133602e-1 0 255 0 255 0 -1.625e0 -7.5e-1 +sinh I4F4 256 4.889743585320395e-1 4.821746578350044e-1 0 255 0 255 0 -1.5625e0 -5.625e-1 +sinh I3F5 256 4.9583701310784994e-1 4.598834477815151e-1 0 255 0 255 0 -1.59375e0 -6.5625e-1 +sinh I16F0 65536 4.788257895501861e-1 1.4908187176067847e-1 0 65535 0 65535 0 -8e0 -1e0 +sinh I15F1 65536 4.6574940678675447e-1 2.9816374352135694e-1 0 65535 0 65535 0 -1e1 -1e0 +sinh I14F2 65536 4.9686858127731987e-1 2.892669277433199e-1 0 65535 0 65535 0 -3.25e0 -7.5e-1 +sinh I13F3 65536 4.977592746042975e-1 4.214661445133602e-1 0 65535 0 65535 0 -8.25e0 -7.5e-1 +sinh I12F4 65536 4.998200391815078e-1 4.821746578350044e-1 0 65535 0 65535 0 -5.75e0 -5.625e-1 +sinh I11F5 65536 4.994593677642524e-1 4.598834477815151e-1 0 65535 0 65535 0 -7.21875e0 -6.5625e-1 +sinh I10F6 65536 4.982833119024726e-1 4.982833119024726e-1 0 65535 0 65535 0 -3.59375e-1 -3.59375e-1 +sinh I9F7 65536 4.9849866329116754e-1 4.875167388162345e-1 0 65535 0 65535 0 -1.9296875e0 -7.8125e-1 +sinh I8F8 65536 4.9996901774069447e-1 4.9747053252599305e-1 0 65535 0 65535 0 -2.47265625e0 -2.265625e-1 +sinh I7F9 65536 4.9993302845390436e-1 4.991789831762988e-1 0 65535 0 65535 0 -3.056640625e0 -2.59765625e-1 +sinh I6F10 65536 4.999886167884685e-1 4.999886167884685e-1 0 65535 0 65535 0 -8.232421875e-1 -8.232421875e-1 +sinh I5F11 65536 4.998631523415222e-1 4.997161072326665e-1 0 65535 0 65535 0 -2.36474609375e0 -5.849609375e-1 +sinh I4F12 65536 4.999576980893051e-1 4.998959456136278e-1 0 65535 0 65535 0 -2.7451171875e0 -3.5791015625e-1 +sinh I3F13 65536 4.9997150307433236e-1 4.998881294684262e-1 0 65535 0 65535 0 -1.6793212890625e0 -7.1533203125e-2 +sinh I32F0 41035 4.815936276210568e-1 1.4908187176067847e-1 0 40520 0 58138 0 1.9e1 -1e0 +sinh I31F1 41035 4.973532898116108e-1 2.9816374352135694e-1 0 40507 0 57967 0 -1.05e1 -1e0 +sinh I30F2 41035 4.9953850397740324e-1 2.892669277433199e-1 0 40474 0 58539 0 -1.85e1 -7.5e-1 +sinh I29F3 41035 4.977592746042975e-1 4.214661445133602e-1 0 40491 0 58889 0 -8.25e0 7.5e-1 +sinh I28F4 41035 4.998200391815078e-1 4.821746578350044e-1 0 40471 0 58843 0 5.75e0 5.625e-1 +sinh I27F5 41035 4.994593677642524e-1 4.598834477815151e-1 0 40512 0 58967 0 -7.21875e0 6.5625e-1 +sinh I26F6 41035 4.982833119024726e-1 4.982833119024726e-1 0 40458 0 59204 0 3.59375e-1 3.59375e-1 +sinh I25F7 41035 4.9928704001788077e-1 4.875167388162345e-1 0 40496 0 59502 0 -7.984375e0 7.8125e-1 +sinh I24F8 41035 4.9996901774069447e-1 4.9747053252599305e-1 0 40471 0 60552 0 -2.47265625e0 2.265625e-1 +sinh I23F9 41035 4.999411819705965e-1 4.991789831762988e-1 0 40489 0 63334 0 -7.341796875e0 -2.59765625e-1 +sinh I22F10 41035 4.999886167884685e-1 4.999886167884685e-1 0 40529 0 69070 0 8.232421875e-1 8.232421875e-1 +sinh I21F11 41035 4.999613054099979e-1 4.997161072326665e-1 0 40523 0 75773 0 4.18505859375e0 -5.849609375e-1 +sinh I20F12 41035 4.9991731902809866e-1 4.99326003427682e-1 0 40455 0 81790 0 -3.478271484375e0 -9.0087890625e-2 +sinh I19F13 41035 4.998960247617765e-1 4.997504200647426e-1 0 40504 0 86229 0 2.5421142578125e0 7.73681640625e-1 +sinh I18F14 41035 4.999705120048214e-1 4.999705120048214e-1 0 40472 0 89033 0 -3.40576171875e-1 -3.40576171875e-1 +sinh I17F15 41035 4.999986629224434e-1 4.998662475588752e-1 0 40519 0 90862 0 5.775054931640625e0 -8.067626953125e-1 +sinh I16F16 41035 4.9996869334097865e-1 4.9996869334097865e-1 0 40516 0 91421 0 -5.15899658203125e-2 -5.15899658203125e-2 +sinh I15F17 41035 4.9999147008990713e-1 4.9982842477480155e-1 0 40509 0 91821 0 -3.9953384399414063e0 5.347976684570313e-1 +sinh I14F18 41035 4.9997955688600115e-1 4.9997955688600115e-1 0 40519 0 92157 0 4.8436737060546875e-1 4.8436737060546875e-1 +sinh I13F19 41035 4.999818718420879e-1 4.999818718420879e-1 0 40502 0 92163 0 4.7725677490234375e-2 4.7725677490234375e-2 +sinh I12F20 41035 4.999998875442057e-1 4.99985736567628e-1 0 40478 0 92520 0 -3.042074203491211e0 1.7515182495117188e-1 +sinh I11F21 41035 4.999967495364934e-1 4.9997340296272e-1 0 40492 0 92542 0 3.1218252182006836e0 -2.897167205810547e-2 +sinh I10F22 41035 4.9997128201582297e-1 4.9997128201582297e-1 0 40474 0 92319 0 5.058574676513672e-2 5.058574676513672e-2 +sinh I9F23 41035 4.999878283230903e-1 4.999878283230903e-1 0 40452 0 92529 0 -7.857644557952881e-2 -7.857644557952881e-2 +sinh I8F24 41035 4.9999087679521953e-1 4.9999087679521953e-1 0 40530 0 92596 0 -4.99908447265625e-1 -4.99908447265625e-1 +sinh I7F25 41035 4.999700848361183e-1 4.998920426376167e-1 0 40553 0 92693 0 -3.291543126106262e0 1.4626681804656982e-2 +sinh I6F26 41035 4.999336821807727e-1 4.9976570354764666e-1 0 40538 0 92240 0 2.5806129723787308e0 1.8151269853115082e-1 +sinh I5F27 41035 4.9988569942488276e-1 4.9988569942488276e-1 0 40495 0 92744 0 -2.4388276040554047e-2 -2.4388276040554047e-2 +sinh I4F28 41035 4.999830211940046e-1 4.9995822246715127e-1 0 40532 0 92167 0 1.79728327319026e0 -8.18135216832161e-3 +sinh I3F29 41035 4.999749941256499e-1 4.998258513774211e-1 0 40489 0 92731 0 -1.8678440023213625e0 4.7589447908103466e-1 +sinh I64F0 41995 4.815936276210568e-1 1.4908187176067847e-1 0 41377 0 67907 0 1.9e1 -1e0 +sinh I63F1 41995 4.973532898116108e-1 2.9816374352135694e-1 0 41459 0 68849 0 1.05e1 -1e0 +sinh I62F2 41995 4.9953850397740324e-1 2.892669277433199e-1 0 41419 0 69742 0 -1.85e1 -7.5e-1 +sinh I61F3 41995 4.977592746042975e-1 4.214661445133602e-1 0 41423 0 70273 0 -8.25e0 7.5e-1 +sinh I60F4 41995 4.998200391815078e-1 4.821746578350044e-1 0 41401 0 70456 0 -5.75e0 5.625e-1 +sinh I56F8 41995 4.9996901774069447e-1 4.9747053252599305e-1 0 41421 0 72749 0 2.47265625e0 -2.265625e-1 +sinh I52F12 41995 4.99986268958669e-1 4.99326003427682e-1 0 41422 0 93741 0 6.810302734375e0 -9.0087890625e-2 +sinh I48F16 41995 4.9996869334097865e-1 4.9996869334097865e-1 0 41391 0 102201 0 -5.15899658203125e-2 -5.15899658203125e-2 +sinh I44F20 41995 4.998997126212114e-1 4.9956067929084763e-1 0 41415 0 102592 0 -9.371776580810547e-1 6.292839050292969e-1 +sinh I40F24 41995 4.999021067798479e-1 4.999021067798479e-1 0 41404 0 102806 0 -1.674301028251648e-1 -1.674301028251648e-1 +sinh I36F28 41995 4.9986704245851254e-1 4.998005489832974e-1 0 41399 0 102525 0 -3.198125172406435e0 -4.369455873966217e-1 +sinh I33F31 41995 4.999210593744916e-1 4.999210593744916e-1 0 41390 0 102558 0 6.414891164749861e-1 6.414891164749861e-1 +sinh I32F32 41995 4.998102203923425e-1 4.9960809163229475e-1 0 41399 0 102354 0 -3.200757318874821e0 6.618139334022999e-3 +sinh I31F33 41995 4.9994556943429347e-1 4.998995900723517e-1 0 41410 0 102905 0 -1.3418556104879826e1 -7.042705547064543e-4 +sinh I28F36 41995 4.9998589605421556e-1 4.9998589605421556e-1 0 41410 0 102503 0 -3.5210839996580034e-4 -3.5210839996580034e-4 +sinh I24F40 41995 4.999863858054863e-1 4.999412173150789e-1 0 41423 0 102792 0 1.570796328036522e0 -8.06264343865223e-1 +sinh I20F44 41995 4.999756601265024e-1 4.999756601265024e-1 0 41428 0 102741 0 3.1820155597227995e-1 3.1820155597227995e-1 +sinh I16F48 41995 4.999962960376079e-1 4.999370536017962e-1 0 41426 0 102722 0 -4.4523688740774965e0 1.1061862376848808e-4 +sinh I12F52 41995 4.999860885617106e-1 4.998389923843655e-1 0 41416 0 102449 0 8.317765500207996e0 4.884807987010993e-1 +sinh I8F56 41995 4.99796778634048e-1 4.995337493043222e-1 0 41441 0 102604 0 3.9713653970490714e0 -5.6327899276478605e-3 +sinh I6F58 41995 4.9999320697505345e-1 4.9999320697505345e-1 0 41395 0 102359 0 -3.733512103084452e-6 -3.733512103084452e-6 +sinh I5F59 41995 4.9993459782776106e-1 4.9993459782776106e-1 0 41448 0 102511 0 6.105035971107486e-1 6.105035971107486e-1 +sinh I4F60 41995 4.9997664124538604e-1 4.999352652836482e-1 0 41382 0 102580 0 1.9785839803993632e0 -7.643249966296704e-1 +sinh I3F61 41995 4.9997886969569466e-1 4.9988593625092603e-1 0 41445 0 102735 0 -1.367504405096587e0 -5.066836671449706e-1 +sinh I128F0 43915 4.8788833240758406e-1 1.4908187176067847e-1 0 43337 0 74060 0 -7.6e1 -1e0 +sinh I127F1 43915 4.973532898116108e-1 2.9816374352135694e-1 0 43290 0 76073 0 1.05e1 -1e0 +sinh I126F2 43915 4.9953850397740324e-1 2.892669277433199e-1 0 43275 0 78320 0 1.85e1 -7.5e-1 +sinh I124F4 43915 4.998200391815078e-1 4.821746578350044e-1 0 43296 0 80402 0 5.75e0 5.625e-1 +sinh I120F8 43915 4.9996901774069447e-1 4.9747053252599305e-1 0 43312 0 83440 0 -2.47265625e0 -2.265625e-1 +sinh I112F16 43915 4.998170384978388e-1 4.998170384978388e-1 0 43281 0 110688 0 5.624542236328125e-1 5.624542236328125e-1 +sinh I104F24 43915 4.999825313385148e-1 4.9967227990950414e-1 0 43295 0 111367 0 2.29372638463974e0 -5.59354841709137e-1 +sinh I96F32 43915 4.9996927243860156e-1 4.995078431059302e-1 0 43279 0 111509 0 1.9800195728894323e0 -3.66088637849316e-1 +sinh I88F40 43915 4.9996391135435014e-1 4.9984144977904305e-1 0 43276 0 111280 0 -1.940453020404675e0 -7.34768487218389e-2 +sinh I80F48 43915 4.999862209248259e-1 4.999780410673888e-1 0 43278 0 111678 0 -3.146277929077719e0 5.426894497981145e-1 +sinh I72F56 43915 4.999982982535434e-1 4.9997841782450075e-1 0 43303 0 111579 0 -4.981917972300561e0 -1.8831522387287672e-1 +sinh I68F60 43915 4.999941174911676e-1 4.9997390423896115e-1 0 43289 0 111473 0 3.057601739981294e0 5.822403633562065e-1 +sinh I67F61 43915 4.9999585646581707e-1 4.9962962083827916e-1 0 43274 0 111432 0 -4.644086108967002e1 -3.5215727681033815e-1 +sinh I66F62 43915 4.9999127142394767e-1 4.999213762452672e-1 0 43316 0 111461 0 -9.999999999926587e-1 -8.72866784723078e-1 +sinh I65F63 43915 4.999042029329741e-1 4.995121494840743e-1 0 43285 0 111404 0 -4.994295604724256e0 -1.7692187827280542e-1 +sinh I64F64 43915 4.9997733417438894e-1 4.998468778072302e-1 0 43304 0 111440 0 3.141592653473378e0 5.001220703125e-1 +sinh I63F65 43915 4.999993778215755e-1 4.999993778215755e-1 0 43311 0 111239 0 -1.8592753922697956e-2 -1.8592753922697956e-2 +sinh I62F66 43915 4.9997229045560454e-1 4.9975934192662674e-1 0 43291 0 111601 0 1.490646797672522e0 7.961454607260843e-1 +sinh I56F72 43915 4.999779152454582e-1 4.999779152454582e-1 0 43297 0 111429 0 -2.695278096384482e-1 -2.695278096384482e-1 +sinh I48F80 43915 4.999973444261847e-1 4.996991035337925e-1 0 43282 0 111322 0 1.1674117416368404e0 3.1961866305299904e-5 +sinh I40F88 43915 4.999953690836758e-1 4.9972784726515407e-1 0 43244 0 111395 0 -2.7725853316774888e1 5e-1 +sinh I32F96 43915 4.9998091742816186e-1 4.99957387107964e-1 0 43304 0 111522 0 2.7222540966542237e0 -8.811913409886519e-8 +sinh I24F104 43915 4.999683329142315e-1 4.9991949864087687e-1 0 43290 0 111261 0 1.5707963267948966e0 4.693080349882529e-3 +sinh I16F112 43915 4.999891956272262e-1 4.9995922434760426e-1 0 43252 0 111142 0 -7.206617960986798e0 -7.456428544176342e-8 +sinh I10F118 43915 4.999288070593385e-1 4.998790744418235e-1 0 43290 0 111261 0 -6.244776983790173e0 6.263359520520951e-11 +sinh I8F120 43915 4.9997888716370914e-1 4.9961006768242927e-1 0 43278 0 111649 0 3.141592653589793e0 -4.084337001844779e-4 +sinh I6F122 43915 4.999687052082472e-1 4.999687052082472e-1 0 43251 0 111405 0 4.004095033450909e-5 4.004095033450909e-5 +sinh I5F123 43915 4.999978168881393e-1 4.9985418622598515e-1 0 43280 0 111379 0 -1.471982285025897e0 -5e-1 +sinh I4F124 43915 4.9999640946111185e-1 4.9999640946111185e-1 0 33275 0 81532 0 -1.797044694919115e-11 -1.797044694919115e-11 +sinh I3F125 43915 4.9988109028503486e-1 4.9974841978707973e-1 0 43258 0 81522 0 1.1387791170790635e0 5.203658889526317e-2 +cosh I8F0 256 4.5691936518475623e-1 2.961085473277708e-1 0 255 0 255 0 -1e0 -1e0 +cosh I7F1 256 4.7560861783273706e-1 2.263622320598522e-1 0 255 0 255 0 -2e0 -5e-1 +cosh I6F2 256 4.894961391744769e-1 4.340944198503696e-1 0 255 0 255 0 -5e-1 -5e-1 +cosh I5F3 256 4.919442275946145e-1 4.02260289876037e-1 0 255 0 255 0 -2.625e0 -3.75e-1 +cosh I4F4 256 4.973904019268292e-1 4.82241695383648e-1 0 255 0 255 0 -2.5e-1 -2.5e-1 +cosh I3F5 256 4.918817458448762e-1 4.282993365033043e-1 0 255 0 255 0 -1.34375e0 -1.875e-1 +cosh I16F0 65536 4.7916125217808864e-1 2.961085473277708e-1 0 65535 0 65535 0 -8e0 -1e0 +cosh I15F1 65536 4.7560861783273706e-1 2.263622320598522e-1 0 65535 0 65535 0 -2e0 -5e-1 +cosh I14F2 65536 4.894961391744769e-1 4.340944198503696e-1 0 65535 0 65535 0 -5e-1 -5e-1 +cosh I13F3 65536 4.956692061458841e-1 4.02260289876037e-1 0 65535 0 65535 0 -8.25e0 -3.75e-1 +cosh I12F4 65536 4.973904019268292e-1 4.82241695383648e-1 0 65535 0 65535 0 -2.5e-1 -2.5e-1 +cosh I11F5 65536 4.95557195407704e-1 4.282993365033043e-1 0 65535 0 65535 0 -2.375e0 -1.875e-1 +cosh I10F6 65536 4.997335144378396e-1 4.9547269588317444e-1 0 65535 0 65535 0 -3.328125e0 -1.25e-1 +cosh I9F7 65536 4.9894575422925314e-1 4.7120615249657605e-1 0 65535 0 65535 0 -4.484375e0 -8.59375e-2 +cosh I8F8 65536 4.999833448819451e-1 4.988625602415457e-1 0 65535 0 65535 0 -1.7109375e0 -6.25e-2 +cosh I7F9 65536 4.996254511989109e-1 4.917213286228726e-1 0 65535 0 65535 0 -1.18359375e0 -1.46484375e-1 +cosh I6F10 65536 4.999593085712626e-1 4.9971528710868124e-1 0 65535 0 65535 0 -3.125e-2 -3.125e-2 +cosh I5F11 65536 4.9986058403470124e-1 4.9429374541438653e-1 0 65535 0 65535 0 -3.01513671875e0 -7.32421875e-2 +cosh I4F12 65536 4.9999080749360414e-1 4.999287996847717e-1 0 65535 0 65535 0 -4.59716796875e-1 -1.5625e-2 +cosh I3F13 65536 4.999961830103438e-1 4.983275646504195e-1 0 65535 0 65535 0 -1.2684326171875e0 -5.2978515625e-2 +cosh I32F0 41035 4.8159363322385323e-1 2.961085473277708e-1 0 40494 0 57881 0 1.9e1 -1e0 +cosh I31F1 41035 4.972982169129113e-1 2.263622320598522e-1 0 40506 0 58268 0 -1.05e1 -5e-1 +cosh I30F2 41035 4.9953846702760457e-1 4.340944198503696e-1 0 40506 0 58434 0 1.85e1 -5e-1 +cosh I29F3 41035 4.956692061458841e-1 4.02260289876037e-1 0 40484 0 58983 0 8.25e0 -3.75e-1 +cosh I28F4 41035 4.9808483702670336e-1 4.82241695383648e-1 0 40504 0 58852 0 1.9375e1 2.5e-1 +cosh I27F5 41035 4.9817204298213497e-1 4.282993365033043e-1 0 40461 0 58768 0 1.3625e1 1.875e-1 +cosh I26F6 41035 4.997335144378396e-1 4.9547269588317444e-1 0 40536 0 58913 0 3.328125e0 1.25e-1 +cosh I25F7 41035 4.9894575422925314e-1 4.7120615249657605e-1 0 40472 0 59542 0 -4.484375e0 8.59375e-2 +cosh I24F8 41035 4.999833448819451e-1 4.988625602415457e-1 0 40505 0 60708 0 -1.7109375e0 6.25e-2 +cosh I23F9 41035 4.999232520105492e-1 4.917213286228726e-1 0 40448 0 63440 0 6.0390625e0 -1.46484375e-1 +cosh I22F10 41035 4.999593085712626e-1 4.9971528710868124e-1 0 40544 0 68843 0 3.125e-2 3.125e-2 +cosh I21F11 41035 4.9998802910353e-1 4.9429374541438653e-1 0 40455 0 75571 0 -6.0634765625e0 -7.32421875e-2 +cosh I20F12 41035 4.9999080749360414e-1 4.999287996847717e-1 0 40471 0 81853 0 -4.59716796875e-1 1.5625e-2 +cosh I19F13 41035 4.99991813993134e-1 4.983275646504195e-1 0 40514 0 86590 0 3.93505859375e0 -5.2978515625e-2 +cosh I18F14 41035 4.9999745686331554e-1 4.999821985399029e-1 0 40489 0 88686 0 7.8125e-3 7.8125e-3 +cosh I17F15 41035 4.999374494040958e-1 4.99886833442706e-1 0 40503 0 90696 0 1.1725982666015625e1 -5.523681640625e-3 +cosh I16F16 41035 4.99999364216799e-1 4.99995549548637e-1 0 40505 0 91305 0 3.90625e-3 3.90625e-3 +cosh I15F17 41035 4.9996023412436447e-1 4.9989159970331365e-1 0 40513 0 91811 0 -5.320762634277344e0 -2.7618408203125e-3 +cosh I14F18 41035 4.999998410542604e-1 4.999988873817629e-1 0 40464 0 92190 0 1.953125e-3 1.953125e-3 +cosh I13F19 41035 4.9998632157203937e-1 4.9966740703298895e-1 0 40493 0 92417 0 -7.375787734985352e0 -2.391815185546875e-3 +cosh I12F20 41035 4.999999602635689e-1 4.9999972184510344e-1 0 40476 0 92423 0 9.765625e-4 9.765625e-4 +cosh I11F21 41035 4.9996680385091674e-1 4.998930891779456e-1 0 40512 0 92280 0 3.0826640129089355e0 -6.90460205078125e-4 +cosh I10F22 41035 4.9999999006589246e-1 4.9999993046125474e-1 0 40480 0 92556 0 4.8828125e-4 4.8828125e-4 +cosh I9F23 41035 4.999839720575445e-1 4.999212871167922e-1 0 40496 0 92457 0 -5.169707775115967e0 9.135007858276367e-4 +cosh I8F24 41035 4.999999975164731e-1 4.999999826153123e-1 0 40480 0 92725 0 2.44140625e-4 2.44140625e-4 +cosh I7F25 41035 4.9997422637516015e-1 4.999341728783361e-1 0 40538 0 92201 0 2.9167268574237823e0 -1.7264485359191895e-4 +cosh I6F26 41035 4.9999999937911827e-1 4.99999995653828e-1 0 40450 0 92418 0 1.220703125e-4 1.220703125e-4 +cosh I5F27 41035 4.999588845482963e-1 4.991584351503833e-1 0 40508 0 92048 0 9.999990463256836e-1 -6.7696645855903625e-3 +cosh I4F28 41035 4.9999999984477955e-1 4.99999998913457e-1 0 40509 0 92392 0 6.103515625e-5 6.103515625e-5 +cosh I3F29 41035 4.9998111041484417e-1 4.997428657959879e-1 0 40446 0 92561 0 2.512412406504154e-1 -7.474608719348907e-5 +cosh I64F0 41995 4.8159363322385323e-1 2.961085473277708e-1 0 41410 0 67950 0 1.9e1 -1e0 +cosh I63F1 41995 4.972982169129113e-1 2.263622320598522e-1 0 41422 0 68609 0 1.05e1 -5e-1 +cosh I62F2 41995 4.9953846702760457e-1 4.340944198503696e-1 0 41411 0 69547 0 -1.85e1 -5e-1 +cosh I61F3 41995 4.957169928502976e-1 4.02260289876037e-1 0 41447 0 70251 0 3.35e1 -3.75e-1 +cosh I60F4 41995 4.9808483702670336e-1 4.82241695383648e-1 0 41443 0 70275 0 1.9375e1 2.5e-1 +cosh I56F8 41995 4.999833448819451e-1 4.988625602415457e-1 0 41455 0 73076 0 -1.7109375e0 6.25e-2 +cosh I52F12 41995 4.9998982739117376e-1 4.999287996847717e-1 0 41407 0 93596 0 1.5625e-2 1.5625e-2 +cosh I48F16 41995 4.99999364216799e-1 4.99995549548637e-1 0 41399 0 101899 0 3.90625e-3 3.90625e-3 +cosh I44F20 41995 4.999999602635689e-1 4.9999972184510344e-1 0 41419 0 102277 0 9.765625e-4 9.765625e-4 +cosh I40F24 41995 4.999999975164731e-1 4.999999826153123e-1 0 41443 0 102659 0 2.44140625e-4 2.44140625e-4 +cosh I36F28 41995 4.9999999984477955e-1 4.99999998913457e-1 0 41470 0 102686 0 6.103515625e-5 6.103515625e-5 +cosh I33F31 41995 4.9992505935271897e-1 4.997987128658706e-1 0 41407 0 102546 0 7.599877671804279e0 1.5988434199243784e-2 +cosh I32F32 41995 5.000000000097012e-1 4.999999999514936e-1 0 41445 0 102415 0 1.52587890625e-5 1.52587890625e-5 +cosh I31F33 41995 4.9994596698256566e-1 4.996199206657312e-1 0 41444 0 102751 0 -2.197499219328165e0 -1.50231271982193e-3 +cosh I28F36 41995 5.000000000006063e-1 4.999999999969683e-1 0 41396 0 102424 0 3.814697265625e-6 3.814697265625e-6 +cosh I24F40 41995 4.999999999999621e-1 4.999999999997347e-1 0 41428 0 102560 0 9.5367431640625e-7 9.5367431640625e-7 +cosh I20F44 41995 4.999999999999976e-1 4.999999999999834e-1 0 41408 0 102912 0 2.384185791015625e-7 2.384185791015625e-7 +cosh I16F48 41995 4.9999999999999983e-1 4.9999999999999895e-1 0 41389 0 102639 0 5.960464477539063e-8 5.960464477539063e-8 +cosh I12F52 41995 5e-1 4.999999999999999e-1 0 41408 0 102527 0 1.4901161193847656e-8 1.4901161193847656e-8 +cosh I8F56 41995 5e-1 5e-1 0 41383 0 102516 0 3.725290298461914e-9 3.725290298461914e-9 +cosh I6F58 41995 5e-1 5e-1 0 41392 0 102177 0 1.862645149230957e-9 1.862645149230957e-9 +cosh I5F59 41995 4.9998444943827675e-1 4.9993914087677344e-1 0 41435 0 102623 0 1.9130487557069746e0 -9.370834922435064e-4 +cosh I4F60 41995 5e-1 5e-1 0 41428 0 102654 0 9.313225746154785e-10 9.313225746154785e-10 +cosh I3F61 41995 4.9997897880938974e-1 4.9991646378009347e-1 0 41367 0 102623 0 -1.3116662129906536e0 -8.355988717796692e-9 +cosh I128F0 43915 4.8788833240758406e-1 2.961085473277708e-1 0 43243 0 74563 0 -7.6e1 -1e0 +cosh I127F1 43915 4.972982169129113e-1 2.263622320598522e-1 0 43313 0 76086 0 -1.05e1 -5e-1 +cosh I126F2 43915 4.9953846702760457e-1 4.340944198503696e-1 0 43268 0 78278 0 1.85e1 -5e-1 +cosh I124F4 43915 4.9808483702670336e-1 4.82241695383648e-1 0 43299 0 80343 0 1.9375e1 2.5e-1 +cosh I120F8 43915 4.999833448819451e-1 4.988625602415457e-1 0 43292 0 82940 0 -1.7109375e0 6.25e-2 +cosh I112F16 43915 4.99999364216799e-1 4.99995549548637e-1 0 43291 0 110955 0 3.90625e-3 3.90625e-3 +cosh I104F24 43915 4.999999975164731e-1 4.999999826153123e-1 0 43298 0 111799 0 2.44140625e-4 2.44140625e-4 +cosh I96F32 43915 4.999999999902987e-1 4.9999999993209104e-1 0 43301 0 111559 0 1.52587890625e-5 1.52587890625e-5 +cosh I88F40 43915 4.999999999999621e-1 4.999999999997347e-1 0 43254 0 111293 0 9.5367431640625e-7 9.5367431640625e-7 +cosh I80F48 43915 4.9999999999999983e-1 4.9999999999999895e-1 0 43291 0 111377 0 5.960464477539063e-8 5.960464477539063e-8 +cosh I72F56 43915 5e-1 5e-1 0 43297 0 111481 0 3.725290298461914e-9 3.725290298461914e-9 +cosh I68F60 43915 5e-1 5e-1 0 43270 0 111519 0 9.313225746154785e-10 9.313225746154785e-10 +cosh I67F61 43915 4.9998516856434516e-1 4.9992899773045296e-1 0 43296 0 111611 0 3.435865116292277e0 -1.6409640803869384e-6 +cosh I66F62 43915 5e-1 5e-1 0 43279 0 111450 0 4.656612873077393e-10 4.656612873077393e-10 +cosh I65F63 43915 4.9998284409050325e-1 4.999778577828195e-1 0 43277 0 111681 0 4.349724365900083e0 3.8540383095229946e-3 +cosh I64F64 43915 5e-1 5e-1 0 43349 0 111266 0 2.3283064365386963e-10 2.3283064365386963e-10 +cosh I63F65 43915 4.999889957205336e-1 4.9981791116834073e-1 0 43273 0 111392 0 4.6440784075084257e-1 9.516196526339034e-9 +cosh I62F66 43915 5e-1 5e-1 0 43275 0 111688 0 1.1641532182693481e-10 1.1641532182693481e-10 +cosh I56F72 43915 5e-1 5e-1 0 43278 0 111505 0 1.4551915228366852e-11 1.4551915228366852e-11 +cosh I48F80 43915 5e-1 5e-1 0 43280 0 111402 0 9.094947017729282e-13 9.094947017729282e-13 +cosh I40F88 43915 5e-1 5e-1 0 43305 0 111550 0 5.684341886080802e-14 5.684341886080802e-14 +cosh I32F96 43915 5e-1 5e-1 0 43280 0 111629 0 3.552713678800501e-15 3.552713678800501e-15 +cosh I24F104 43915 5e-1 5e-1 0 43278 0 111484 0 2.220446049250313e-16 2.220446049250313e-16 +cosh I16F112 43915 5e-1 5e-1 0 43255 0 111796 0 1.3877787807814457e-17 1.3877787807814457e-17 +cosh I10F118 43915 5e-1 5e-1 0 43315 0 111444 0 1.734723475976807e-18 1.734723475976807e-18 +cosh I8F120 43915 5e-1 5e-1 0 43310 0 111315 0 8.673617379884035e-19 8.673617379884035e-19 +cosh I6F122 43915 5e-1 5e-1 0 43278 0 111675 0 4.336808689942018e-19 4.336808689942018e-19 +cosh I5F123 43915 4.999795733751506e-1 4.999243385627716e-1 0 43325 0 111524 0 5e-1 -7.031795203059504e-12 +cosh I4F124 43915 5e-1 5e-1 0 33254 0 81598 0 2.168404344971009e-19 2.168404344971009e-19 +cosh I3F125 43915 4.999888054688586e-1 4.9995005762231165e-1 0 43242 0 81731 0 1e0 -4.6757071311166946e-11 +tanh I8F0 256 2.3840584404423512e-1 2.3840584404423512e-1 0 255 0 255 0 -1e0 -1e0 +tanh I7F1 256 4.7681168808847024e-1 4.7681168808847024e-1 0 255 0 255 0 -1e0 -1e0 +tanh I6F2 256 4.5940419045085074e-1 4.5940419045085074e-1 0 255 0 255 0 -7.5e-1 -7.5e-1 +tanh I5F3 256 4.744085616142481e-1 4.744085616142481e-1 0 255 0 255 0 -1.125e0 -1.125e0 +tanh I4F4 256 4.909954652219292e-1 4.909954652219292e-1 0 255 0 255 0 -2.0625e0 -2.0625e0 +tanh I3F5 256 4.9028862733892775e-1 4.9028862733892775e-1 0 255 0 255 0 -9.375e-1 -9.375e-1 +tanh I16F0 65536 2.3840584404423512e-1 2.3840584404423512e-1 0 65535 0 65535 0 -1e0 -1e0 +tanh I15F1 65536 4.7681168808847024e-1 4.7681168808847024e-1 0 65535 0 65535 0 -1e0 -1e0 +tanh I14F2 65536 4.5940419045085074e-1 4.5940419045085074e-1 0 65535 0 65535 0 -7.5e-1 -7.5e-1 +tanh I13F3 65536 4.744085616142481e-1 4.744085616142481e-1 0 65535 0 65535 0 -1.125e0 -1.125e0 +tanh I12F4 65536 4.909954652219292e-1 4.909954652219292e-1 0 65535 0 65535 0 -2.0625e0 -2.0625e0 +tanh I11F5 65536 4.9028862733892775e-1 4.9028862733892775e-1 0 65535 0 65535 0 -9.375e-1 -9.375e-1 +tanh I10F6 65536 4.9854033930153774e-1 4.9854033930153774e-1 0 65535 0 65535 0 -1.65625e0 -1.65625e0 +tanh I9F7 65536 4.990036633101739e-1 4.990036633101739e-1 0 65535 0 65535 0 -3.1171875e0 -3.1171875e0 +tanh I8F8 65536 4.995965987108789e-1 4.995965987108789e-1 0 65535 0 65535 0 -3.46484375e0 -3.46484375e0 +tanh I7F9 65536 4.997556228347729e-1 4.997556228347729e-1 0 65535 0 65535 0 -1.103515625e0 -1.103515625e0 +tanh I6F10 65536 4.999179216814521e-1 4.999179216814521e-1 0 65535 0 65535 0 -2.4716796875e0 -2.4716796875e0 +tanh I5F11 65536 4.999860482897728e-1 4.999860482897728e-1 0 65535 0 65535 0 -1.55615234375e0 -1.55615234375e0 +tanh I4F12 65536 4.999989703224892e-1 4.999989703224892e-1 0 65535 0 65535 0 -1.570556640625e0 -1.570556640625e0 +tanh I3F13 65536 4.99992476901521e-1 4.99992476901521e-1 0 65535 0 65535 0 -4.549560546875e-1 -4.549560546875e-1 +tanh I32F0 40897 2.3840584404423512e-1 2.3840584404423512e-1 0 40303 0 57594 0 -1e0 -1e0 +tanh I31F1 40897 4.7681168808847024e-1 4.7681168808847024e-1 0 40282 0 57750 0 -1e0 -1e0 +tanh I30F2 40897 4.5940419045085074e-1 4.5940419045085074e-1 0 40277 0 58232 0 -7.5e-1 -7.5e-1 +tanh I29F3 40897 4.744085616142481e-1 4.744085616142481e-1 0 40280 0 58878 0 1.125e0 1.125e0 +tanh I28F4 40897 4.909954652219292e-1 4.909954652219292e-1 0 40275 0 58776 0 2.0625e0 2.0625e0 +tanh I27F5 40897 4.9028862733892775e-1 4.9028862733892775e-1 0 40246 0 58472 0 9.375e-1 9.375e-1 +tanh I26F6 40897 4.9854033930153774e-1 4.9854033930153774e-1 0 40256 0 58568 0 1.65625e0 1.65625e0 +tanh I25F7 40897 4.990036633101739e-1 4.990036633101739e-1 0 40273 0 58568 0 3.1171875e0 3.1171875e0 +tanh I24F8 40897 4.995965987108789e-1 4.995965987108789e-1 0 40311 0 59609 0 -3.46484375e0 -3.46484375e0 +tanh I23F9 40897 4.997556228347729e-1 4.997556228347729e-1 0 40287 0 62907 0 1.103515625e0 1.103515625e0 +tanh I22F10 40897 4.999179216814521e-1 4.999179216814521e-1 0 40287 0 68051 0 -2.4716796875e0 -2.4716796875e0 +tanh I21F11 40897 4.999860482897728e-1 4.999860482897728e-1 0 40272 0 74986 0 -1.55615234375e0 -1.55615234375e0 +tanh I20F12 40897 4.999989703224892e-1 4.999989703224892e-1 0 40268 0 81280 0 1.570556640625e0 1.570556640625e0 +tanh I19F13 40897 4.99992476901521e-1 4.99992476901521e-1 0 40243 0 85449 0 -4.549560546875e-1 -4.549560546875e-1 +tanh I18F14 40897 4.999867363952529e-1 4.999867363952529e-1 0 40263 0 88360 0 -2.44525146484375e0 -2.44525146484375e0 +tanh I17F15 40897 4.9987971404733955e-1 4.9987971404733955e-1 0 40299 0 89937 0 -2.101959228515625e0 -2.101959228515625e0 +tanh I16F16 40897 4.999448018321617e-1 4.999448018321617e-1 0 40291 0 90763 0 2.83966064453125e-2 2.83966064453125e-2 +tanh I15F17 40897 4.9997824010277114e-1 4.9997824010277114e-1 0 40227 0 91626 0 3.953369140625e0 3.953369140625e0 +tanh I14F18 40897 4.9990460166537426e-1 4.9990460166537426e-1 0 40301 0 91700 0 -1.81243896484375e-1 -1.81243896484375e-1 +tanh I13F19 40897 4.99962690142321e-1 4.99962690142321e-1 0 40311 0 91848 0 -5.332122802734375e0 -5.332122802734375e0 +tanh I12F20 40897 4.9998357911679997e-1 4.9998357911679997e-1 0 40279 0 91584 0 -2.137022018432617e0 -2.137022018432617e0 +tanh I11F21 40897 4.9999227506394034e-1 4.9999227506394034e-1 0 40327 0 91804 0 7.421888828277588e0 7.421888828277588e0 +tanh I10F22 40897 4.999521901940085e-1 4.999521901940085e-1 0 40259 0 91613 0 -6.215417146682739e0 -6.215417146682739e0 +tanh I9F23 40897 4.999279668709366e-1 4.999279668709366e-1 0 40287 0 92099 0 5.187482118606567e0 5.187482118606567e0 +tanh I8F24 40897 4.9999853924822196e-1 4.9999853924822196e-1 0 40304 0 91848 0 -6.470209956169128e0 -6.470209956169128e0 +tanh I7F25 40897 4.999216413205263e-1 4.999216413205263e-1 0 40279 0 91776 0 2.465987503528595e-1 2.465987503528595e-1 +tanh I6F26 40897 4.9993939495410367e-1 4.9993939495410367e-1 0 40269 0 91797 0 -6.675667583942413e0 -6.675667583942413e0 +tanh I5F27 40897 4.9998445544202885e-1 4.9998445544202885e-1 0 40293 0 91624 0 -1.999999575316906e0 -1.999999575316906e0 +tanh I4F28 40897 4.999991795553994e-1 4.999991795553994e-1 0 40253 0 92018 0 -3.394562751054764e-3 -3.394562751054764e-3 +tanh I3F29 40897 4.999942221962028e-1 4.999942221962028e-1 0 40294 0 91934 0 2.9947158694267273e0 2.9947158694267273e0 +tanh I64F0 41729 2.3840584404423512e-1 2.3840584404423512e-1 0 41048 0 67642 0 -1e0 -1e0 +tanh I63F1 41729 4.7681168808847024e-1 4.7681168808847024e-1 0 41095 0 67621 0 -1e0 -1e0 +tanh I62F2 41729 4.5940419045085074e-1 4.5940419045085074e-1 0 41049 0 68947 0 -7.5e-1 -7.5e-1 +tanh I61F3 41729 4.744085616142481e-1 4.744085616142481e-1 0 41067 0 69717 0 1.125e0 1.125e0 +tanh I60F4 41729 4.909954652219292e-1 4.909954652219292e-1 0 41032 0 69453 0 2.0625e0 2.0625e0 +tanh I56F8 41729 4.995965987108789e-1 4.995965987108789e-1 0 41052 0 71361 0 -3.46484375e0 -3.46484375e0 +tanh I52F12 41729 4.999989703224892e-1 4.999989703224892e-1 0 41076 0 92816 0 1.570556640625e0 1.570556640625e0 +tanh I48F16 41729 4.999448018321617e-1 4.999448018321617e-1 0 41021 0 100616 0 2.83966064453125e-2 2.83966064453125e-2 +tanh I44F20 41729 4.9999111931090523e-1 4.9999111931090523e-1 0 41122 0 101567 0 6.27059268951416e0 6.27059268951416e0 +tanh I40F24 41729 4.998765847510266e-1 4.998765847510266e-1 0 41082 0 101819 0 1.9199623465538025e0 1.9199623465538025e0 +tanh I36F28 41729 4.9996296560003994e-1 4.9996296560003994e-1 0 41079 0 101640 0 -3.829348783940077e0 -3.829348783940077e0 +tanh I33F31 41729 4.999068306720504e-1 4.999068306720504e-1 0 41113 0 101799 0 1.6020463267341256e0 1.6020463267341256e0 +tanh I32F32 41729 4.9999226114570033e-1 4.9999226114570033e-1 0 41054 0 101383 0 5.572413152549416e0 5.572413152549416e0 +tanh I31F33 41729 4.9996817253806347e-1 4.9996817253806347e-1 0 41082 0 101812 0 7.615726250223815e0 7.615726250223815e0 +tanh I28F36 41729 4.9999533992429435e-1 4.9999533992429435e-1 0 41087 0 101484 0 -9.059155790164368e-1 -9.059155790164368e-1 +tanh I24F40 41729 4.9997222250220674e-1 4.9997222250220674e-1 0 41105 0 101388 0 -3.6184744063448306e0 -3.6184744063448306e0 +tanh I20F44 41729 4.999874512180074e-1 4.999874512180074e-1 0 41077 0 101887 0 6.944815422518786e-1 6.944815422518786e-1 +tanh I16F48 41729 4.99923143331462e-1 4.99923143331462e-1 0 41063 0 101615 0 -3.1415926535931398e0 -3.1415926535931398e0 +tanh I12F52 41729 4.999808146330764e-1 4.999808146330764e-1 0 41078 0 101799 0 6.931676296728639e-6 6.931676296728639e-6 +tanh I8F56 41729 4.999983531133055e-1 4.999983531133055e-1 0 41092 0 101961 0 4.294958459333783e0 4.294958459333783e0 +tanh I6F58 41729 4.999710119179436e-1 4.999710119179436e-1 0 41105 0 101426 0 9.847625208387933e0 9.847625208387933e0 +tanh I5F59 41729 4.999641314216109e-1 4.999641314216109e-1 0 41073 0 101281 0 -8.25487837768923e0 -8.25487837768923e0 +tanh I4F60 41729 4.999954920629952e-1 4.999954920629952e-1 0 41092 0 101913 0 5.70365331733283e-1 5.70365331733283e-1 +tanh I3F61 41729 4.9999456749555404e-1 4.9999456749555404e-1 0 41113 0 101915 0 -1.570398029421752e0 -1.570398029421752e0 +tanh I128F0 43393 2.3840584404423512e-1 2.3840584404423512e-1 0 42720 0 72978 0 -1e0 -1e0 +tanh I127F1 43393 4.7681168808847024e-1 4.7681168808847024e-1 0 42697 0 74361 0 -1e0 -1e0 +tanh I126F2 43393 4.5940419045085074e-1 4.5940419045085074e-1 0 42683 0 76940 0 -7.5e-1 -7.5e-1 +tanh I124F4 43393 4.909954652219292e-1 4.909954652219292e-1 0 42656 0 78714 0 2.0625e0 2.0625e0 +tanh I120F8 43393 4.995965987108789e-1 4.995965987108789e-1 0 42716 0 81470 0 -3.46484375e0 -3.46484375e0 +tanh I112F16 43393 4.9992969978715973e-1 4.9992969978715973e-1 0 42634 0 109479 0 -2.0689697265625e0 -2.0689697265625e0 +tanh I104F24 43393 4.999540858046217e-1 4.999540858046217e-1 0 42698 0 109872 0 -5.989593684673309e0 -5.989593684673309e0 +tanh I96F32 43393 4.999903317085449e-1 4.999903317085449e-1 0 42692 0 110017 0 -2.1555381510406733e0 -2.1555381510406733e0 +tanh I88F40 43393 4.999912639252093e-1 4.999912639252093e-1 0 42695 0 109745 0 7.472347842853196e0 7.472347842853196e0 +tanh I80F48 43393 4.9998917428339656e-1 4.9998917428339656e-1 0 42695 0 109816 0 1.9999999999998934e0 1.9999999999998934e0 +tanh I72F56 43393 4.999996262346593e-1 4.999996262346593e-1 0 42693 0 110182 0 1.932931171917767e0 1.932931171917767e0 +tanh I68F60 43393 4.999732388669244e-1 4.999732388669244e-1 0 42689 0 109917 0 1.0155161076041286e0 1.0155161076041286e0 +tanh I67F61 43393 4.999339238019618e-1 4.999339238019618e-1 0 42674 0 109852 0 -1.3484490427989657e0 -1.3484490427989657e0 +tanh I66F62 43393 4.9998814296210986e-1 4.9998814296210986e-1 0 42657 0 109470 0 -4.9243003038108873e-1 -4.9243003038108873e-1 +tanh I65F63 43393 4.9998529774578054e-1 4.9998529774578054e-1 0 42652 0 109786 0 -6.310363869029949e-1 -6.310363869029949e-1 +tanh I64F64 43393 4.999663846865015e-1 4.999663846865015e-1 0 42686 0 110021 0 -6.639358758315289e0 -6.639358758315289e0 +tanh I63F65 43393 4.9999176046033345e-1 4.9999176046033345e-1 0 42692 0 109692 0 -2e0 -2e0 +tanh I62F66 43393 4.999204716697616e-1 4.999204716697616e-1 0 42665 0 110204 0 2.2060686018363318e-1 2.2060686018363318e-1 +tanh I56F72 43393 4.9989277308502567e-1 4.9989277308502567e-1 0 42693 0 109886 0 -5.985843146467216e0 -5.985843146467216e0 +tanh I48F80 43393 4.9998887906141326e-1 4.9998887906141326e-1 0 42706 0 109776 0 -1.456660563921564e-1 -1.456660563921564e-1 +tanh I40F88 43393 4.999811994411405e-1 4.999811994411405e-1 0 42662 0 109522 0 -2.688412447869595e-8 -2.688412447869595e-8 +tanh I32F96 43393 4.999926004671408e-1 4.999926004671408e-1 0 42691 0 109793 0 6.056137871919631e0 6.056137871919631e0 +tanh I24F104 43393 4.9999443196373505e-1 4.9999443196373505e-1 0 42716 0 109547 0 3.7965010168501e0 3.7965010168501e0 +tanh I16F112 43393 4.999804616645284e-1 4.999804616645284e-1 0 42704 0 109759 0 -2.9719465883503795e0 -2.9719465883503795e0 +tanh I10F118 43393 4.999750397398889e-1 4.999750397398889e-1 0 42671 0 109988 0 -6.620144220035864e-1 -6.620144220035864e-1 +tanh I8F120 43393 4.9992983474568004e-1 4.9992983474568004e-1 0 42717 0 109422 0 -5.4532080810562915e0 -5.4532080810562915e0 +tanh I6F122 43393 4.99971014028299e-1 4.99971014028299e-1 0 42692 0 109793 0 -1.2193696167590286e1 -1.2193696167590286e1 +tanh I5F123 43393 4.9996577786550056e-1 4.9996577786550056e-1 0 42685 0 109815 0 1.5803562863265957e-11 1.5803562863265957e-11 +tanh I4F124 43393 4.9996251450065426e-1 4.9996251450065426e-1 0 32643 0 79944 0 -4.748550800876172e0 -4.748550800876172e0 +tanh I3F125 43393 4.9999021319719955e-1 4.9999021319719955e-1 0 42684 0 79840 0 -1.5707963267948966e0 -1.5707963267948966e0 +coth I8F0 255 3.130352854993313e-1 2.3840584404423512e-1 0 254 0 253 0 -1e0 -1e0 +coth I7F1 255 3.739294290013374e-1 2.8478246786729466e-1 0 254 0 253 0 -1e0 -1e0 +coth I6F2 255 4.191655719300476e-1 3.7940698542053425e-1 0 254 0 253 0 -1.5e0 -1.5e0 +coth I5F3 255 4.982025481247966e-1 4.689956920217011e-1 0 254 0 253 0 -1.75e0 -1.75e0 +coth I4F4 255 4.991069234320825e-1 4.5918268179829985e-1 0 254 0 253 0 -1.3125e0 -2.0625e0 +coth I3F5 255 4.9240559310428234e-1 4.849434411430919e-1 0 254 0 253 0 -2.4375e0 -2.4375e0 +coth I16F0 65535 3.130352854993313e-1 2.3840584404423512e-1 0 65534 0 65533 0 -1e0 -1e0 +coth I15F1 65535 3.739294290013374e-1 2.8478246786729466e-1 0 65534 0 65533 0 -1e0 -1e0 +coth I14F2 65535 4.191655719300476e-1 3.7940698542053425e-1 0 65534 0 65533 0 -1.5e0 -1.5e0 +coth I13F3 65535 4.982025481247966e-1 4.689956920217011e-1 0 65534 0 65533 0 -1.75e0 -1.75e0 +coth I12F4 65535 4.991069234320825e-1 4.5918268179829985e-1 0 65534 0 65533 0 -1.3125e0 -2.0625e0 +coth I11F5 65535 4.9240559310428234e-1 4.849434411430919e-1 0 65534 0 65533 0 -2.4375e0 -2.4375e0 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I28F4 41035 5e-1 5e-1 0 0 0 58749 0 -5e0 -5e0 +pow2 I27F5 41035 5e-1 5e-1 0 0 0 58964 0 -6e0 -6e0 +pow2 I26F6 41035 5e-1 5e-1 0 0 0 58818 0 -7e0 -7e0 +pow2 I25F7 41035 5e-1 5e-1 0 0 0 59453 0 -8e0 -8e0 +pow2 I24F8 41035 5e-1 5e-1 0 0 0 60434 0 -9e0 -9e0 +pow2 I23F9 41035 5e-1 5e-1 0 0 0 63622 0 -1e1 -1e1 +pow2 I22F10 41035 5e-1 5e-1 0 0 0 68779 0 -1.1e1 -1.1e1 +pow2 I21F11 41035 5e-1 5e-1 0 0 0 75340 0 -1.2e1 -1.2e1 +pow2 I20F12 41035 5e-1 5e-1 0 0 0 82175 0 -1.3e1 -1.3e1 +pow2 I19F13 41035 5e-1 5e-1 0 0 0 86675 0 -1.4e1 -1.4e1 +pow2 I18F14 41035 5e-1 5e-1 0 0 0 89088 0 -1.5e1 -1.5e1 +pow2 I17F15 41035 5e-1 5e-1 0 0 0 90256 0 -1.6e1 -1.6e1 +pow2 I16F16 41035 5e-1 5e-1 0 0 0 91979 0 -1.7e1 -1.7e1 +pow2 I15F17 41035 5e-1 5e-1 0 0 0 91860 0 -1.8e1 -1.8e1 +pow2 I14F18 41035 5e-1 5e-1 0 0 0 92401 0 -1.9e1 -1.9e1 +pow2 I13F19 41035 5e-1 5e-1 0 0 0 92393 0 -2e1 -2e1 +pow2 I12F20 41035 5e-1 5e-1 0 0 0 92310 0 -2.1e1 -2.1e1 +pow2 I11F21 41035 5e-1 5e-1 0 0 0 92180 0 -2.2e1 -2.2e1 +pow2 I10F22 41035 5e-1 5e-1 0 0 0 92643 0 -2.3e1 -2.3e1 +pow2 I9F23 41035 5e-1 5e-1 0 0 0 92564 0 -2.4e1 -2.4e1 +pow2 I8F24 41035 5e-1 5e-1 0 0 0 92253 0 -2.5e1 -2.5e1 +pow2 I7F25 41035 5e-1 5e-1 0 0 0 92607 0 -2.6e1 -2.6e1 +pow2 I6F26 41035 5e-1 5e-1 0 0 0 92613 0 -2.7e1 -2.7e1 +pow2 I5F27 41035 4.9999638489859916e-1 4.9999272857935867e-1 0 0 0 91845 0 1.055002212524414e-5 1.055002212524414e-5 +pow2 I4F28 41035 4.9998378736791405e-1 4.999721260492914e-1 0 0 0 92259 0 1.3890490904450417e0 -1.983046282082796e0 +pow2 I3F29 41035 4.9997828171156206e-1 4.9997828171156206e-1 0 0 0 91468 0 -8.400822430849075e-1 -8.400822430849075e-1 +pow2 I64F0 41995 5e-1 5e-1 0 0 0 67705 0 -1e0 -1e0 +pow2 I63F1 41995 5e-1 5e-1 0 0 0 68522 0 -2e0 -2e0 +pow2 I62F2 41995 5e-1 5e-1 0 0 0 69397 0 -3e0 -3e0 +pow2 I61F3 41995 5e-1 5e-1 0 0 0 70480 0 -4e0 -4e0 +pow2 I60F4 41995 5e-1 5e-1 0 0 0 70434 0 -5e0 -5e0 +pow2 I56F8 41995 5e-1 5e-1 0 0 0 72777 0 -9e0 -9e0 +pow2 I52F12 41995 5e-1 5e-1 0 0 0 93579 0 -1.3e1 -1.3e1 +pow2 I48F16 41995 5e-1 5e-1 0 0 0 101642 0 -1.7e1 -1.7e1 +pow2 I44F20 41995 5e-1 5e-1 0 0 0 102425 0 -2.1e1 -2.1e1 +pow2 I40F24 41995 5e-1 5e-1 0 0 0 102790 0 -2.5e1 -2.5e1 +pow2 I36F28 41995 5e-1 5e-1 0 0 0 102271 0 -2.9e1 -2.9e1 +pow2 I33F31 41995 5e-1 5e-1 0 0 0 102516 0 -3.2e1 -3.2e1 +pow2 I32F32 41995 5e-1 5e-1 0 0 0 102505 0 -3.3e1 -3.3e1 +pow2 I31F33 41995 5e-1 5e-1 0 0 0 102721 0 -3.4e1 -3.4e1 +pow2 I28F36 41995 5e-1 5e-1 0 0 0 102689 0 -3.7e1 -3.7e1 +pow2 I24F40 41995 5e-1 5e-1 0 0 0 102565 0 -4.1e1 -4.1e1 +pow2 I20F44 41995 5e-1 5e-1 0 0 0 102593 0 -4.5e1 -4.5e1 +pow2 I16F48 41995 5e-1 5e-1 0 0 0 102552 0 -4.9e1 -4.9e1 +pow2 I12F52 41995 5e-1 5e-1 0 0 0 103007 0 -5.3e1 -5.3e1 +pow2 I8F56 41995 5e-1 5e-1 0 0 0 102613 0 -5.7e1 -5.7e1 +pow2 I6F58 41995 4.999862454933232e-1 4.999862454930802e-1 0 0 0 101545 0 7.011474734142098e-13 7.011474734142098e-13 +pow2 I5F59 41995 4.9998059641487136e-1 4.9996977246917823e-1 0 0 0 101835 0 2.967971466608026e0 -5.000000000000039e-1 +pow2 I4F60 41995 4.999597637048219e-1 4.999495638419245e-1 0 0 0 101825 0 1.5129014608063112e0 -5.051874286376866e0 +pow2 I3F61 41995 4.999871909772485e-1 4.999871909772485e-1 0 0 0 101266 0 -9.205440229531625e-14 -9.205440229531625e-14 +pow2 I128F0 43915 5e-1 5e-1 0 0 0 73797 0 -1e0 -1e0 +pow2 I127F1 43915 5e-1 5e-1 0 0 0 75291 0 -2e0 -2e0 +pow2 I126F2 43915 5e-1 5e-1 0 0 0 77914 0 -3e0 -3e0 +pow2 I124F4 43915 5e-1 5e-1 0 0 0 80276 0 -5e0 -5e0 +pow2 I120F8 43915 5e-1 5e-1 0 0 0 83004 0 -9e0 -9e0 +pow2 I112F16 43915 5e-1 5e-1 0 0 0 110907 0 -1.7e1 -1.7e1 +pow2 I104F24 43915 5e-1 5e-1 0 0 0 111377 0 -2.5e1 -2.5e1 +pow2 I96F32 43915 5e-1 5e-1 0 0 0 111186 0 -3.3e1 -3.3e1 +pow2 I88F40 43915 5e-1 5e-1 0 0 0 111756 0 -4.1e1 -4.1e1 +pow2 I80F48 43915 5e-1 5e-1 0 0 0 111217 0 -4.9e1 -4.9e1 +pow2 I72F56 43915 5e-1 5e-1 0 0 0 111224 0 -5.7e1 -5.7e1 +pow2 I68F60 43915 5e-1 5e-1 0 0 0 111617 0 -6.1e1 -6.1e1 +pow2 I67F61 43915 5e-1 5e-1 0 0 0 111433 0 -6.2e1 -6.2e1 +pow2 I66F62 43915 5e-1 5e-1 0 0 0 111635 0 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4.9960759661672655e-1 0 0 0 54676 0 2.283711189494397e21 1.2312898328087212e0 +log2 I68F60 21380 4.999767373591694e-1 4.9984587431566646e-1 0 0 0 54574 0 3.830609180797814e19 1.9999999982697996e0 +log2 I67F61 21433 4.999774700831157e-1 4.999774700831157e-1 0 0 0 54785 0 1.2387314753633474e0 1.2387314753633474e0 +log2 I66F62 21456 4.9996766422073013e-1 4.9987737125034e-1 0 0 0 54962 0 3.2718316503556706e19 1.5707963267968976e0 +log2 I65F63 21471 4.999895870826254e-1 4.999164432578653e-1 0 0 0 55049 0 2.974394290060645e18 1.0900932177046596e0 +log2 I64F64 21524 4.9999253297405144e-1 4.9999253297405144e-1 0 0 0 55180 0 1.2992881609819715e0 1.2992881609819715e0 +log2 I63F65 21404 4.9997151668289835e-1 4.997057980779533e-1 0 0 0 54772 0 1.15026583569147e-3 2e0 +log2 I62F66 21594 4.9993389831585766e-1 4.9977741257057323e-1 0 0 0 55047 0 7.39834047477007e17 1.5707352916386466e0 +log2 I56F72 21419 4.999378763784793e-1 4.9979312943548804e-1 0 0 0 55086 0 5.563175898090281e15 1e0 +log2 I48F80 21358 4.999734101601717e-1 4.993988148356882e-1 0 0 0 54762 0 6.661832276402874e13 5.000000063978018e-1 +log2 I40F88 21437 4.999755969846273e-1 4.995090636313036e-1 0 0 0 54872 0 1.156044836247925e11 2e0 +log2 I32F96 21397 4.999586320429907e-1 4.999586319967878e-1 0 0 0 54850 0 2.000000000128112e0 2.000000000128112e0 +log2 I24F104 21347 4.9999509565946504e-1 4.999788178716964e-1 0 0 0 54697 0 1.833897832023328e6 5.000000000000002e-1 +log2 I16F112 21395 4.9989148791001276e-1 4.997174457083156e-1 0 0 0 54807 0 1.561611763566659e4 9.999999998835847e-1 +log2 I10F118 21486 4.9999245674132675e-1 4.999378992767685e-1 0 0 0 54804 0 3.141592653589793e0 1.187065965062111e0 +log2 I8F120 21460 4.999825624002918e-1 4.9991600352858734e-1 0 0 0 54989 0 3.405740558078022e1 1e0 +log2 I6F122 21364 4.9996230603254e-1 4.9972129790385794e-1 0 0 0 54619 0 2.2561552587719003e1 1.2842895990352543e0 +log2 I5F123 21363 4.9994787330491025e-1 4.9994787330491025e-1 0 0 0 54654 0 5e-1 5e-1 +log2 I4F124 16521 4.999713588826298e-1 4.99897333496758e-1 0 0 0 40207 0 2.2194007029787685e-1 5e-1 +log2 I3F125 16321 4.9990240284376297e-1 4.9990240284376297e-1 0 0 0 39630 0 1.5707963267948966e0 1.5707963267948966e0 +log10 I8F0 127 4.94850021680094e-1 4.7712125471966244e-1 0 0 0 126 0 3.2e1 3e0 +log10 I7F1 127 4.963760540124008e-1 4.807253789884877e-1 0 0 0 126 0 5.6e1 5.5e0 +log10 I6F2 127 4.997549464331998e-1 4.997549464331998e-1 0 0 0 126 0 7.5e-1 7.5e-1 +log10 I5F3 127 4.9669314685715543e-1 4.9669314685715543e-1 0 0 0 126 0 6.5e0 6.5e0 +log10 I4F4 127 4.8764600447668294e-1 4.8764600447668294e-1 0 0 0 126 0 2.9375e0 2.9375e0 +log10 I3F5 127 4.9753493368837715e-1 4.9753493368837715e-1 0 0 0 126 0 1.84375e0 1.84375e0 +log10 I16F0 32767 4.999969319483068e-1 4.7712125471966244e-1 0 0 0 32766 0 3.1623e4 3e0 +log10 I15F1 32767 4.9998660104553616e-1 4.807253789884877e-1 0 0 0 32766 0 5.6235e3 5.5e0 +log10 I14F2 32767 4.9998658559983583e-1 4.997549464331998e-1 0 0 0 32766 0 7.499e3 7.5e-1 +log10 I13F3 32767 4.9999169649924513e-1 4.9669314685715543e-1 0 0 0 32766 0 3.65175e3 6.5e0 +log10 I12F4 32767 4.999985399036318e-1 4.964843577068131e-1 0 0 0 32766 0 1.2409375e3 8.0625e0 +log10 I11F5 32767 4.999943097833602e-1 4.9753493368837715e-1 0 0 0 32766 0 8.976875e2 1.84375e0 +log10 I10F6 32767 4.9998339748427967e-1 4.9923807944892484e-1 0 0 0 32766 0 2.788125e2 5.328125e0 +log10 I9F7 32767 4.999917274276471e-1 4.9935972125366324e-1 0 0 0 32766 0 1.860078125e2 3.078125e0 +log10 I8F8 32767 4.99958324363831e-1 4.9989972520647424e-1 0 0 0 32766 0 8.022265625e1 9.34765625e0 +log10 I7F9 32767 4.999915415141137e-1 4.9983684038433246e-1 0 0 0 32766 0 1.3365234375e1 5.560546875e0 +log10 I6F10 32767 4.9999940673817267e-1 4.999744610745826e-1 0 0 0 32766 0 2.81015625e1 6.1181640625e0 +log10 I5F11 32767 4.9999938529695226e-1 4.9998769248374986e-1 0 0 0 32766 0 1.278466796875e1 1.03955078125e0 +log10 I4F12 32767 4.9999125243758e-1 4.9999125243758e-1 0 0 0 32766 0 7.671630859375e0 7.671630859375e0 +log10 I3F13 32767 4.9999539971558826e-1 4.9999539971558826e-1 0 0 0 32766 0 3.2301025390625e0 3.2301025390625e0 +log10 I32F0 19005 4.9990080808427717e-1 4.7712125471966244e-1 0 0 0 28847 0 3.163e3 3e0 +log10 I31F1 19740 4.999361670188058e-1 4.807253789884877e-1 0 0 0 29093 0 5.623e3 5.5e0 +log10 I30F2 19964 4.9998263404965454e-1 4.997549464331998e-1 0 0 0 28866 0 2.37135e5 7.5e-1 +log10 I29F3 20100 4.9983423330829546e-1 4.9669314685715543e-1 0 0 0 29024 0 1.54e2 6.5e0 +log10 I28F4 19772 4.999854334267107e-1 4.949394278984119e-1 0 0 0 28898 0 1.074605575625e8 9.3125e0 +log10 I27F5 20136 4.999891510610382e-1 4.9753493368837715e-1 0 0 0 29534 0 2.12875e2 1.84375e0 +log10 I26F6 19843 4.999615774362125e-1 4.9923807944892484e-1 0 0 0 29224 0 1.2188125e3 5.328125e0 +log10 I25F7 20067 4.9994895146039503e-1 4.9935972125366324e-1 0 0 0 29365 0 1.34557109375e4 3.078125e0 +log10 I24F8 20111 4.999515959897575e-1 4.998553867499855e-1 0 0 0 29746 0 8.09474764453125e6 6.58203125e0 +log10 I23F9 20215 4.999915415141137e-1 4.9983684038433246e-1 0 0 0 31499 0 1.3365234375e1 5.560546875e0 +log10 I22F10 20017 4.999844825870094e-1 4.99946154660236e-1 0 0 0 34071 0 8.306796640625e5 2.0146484375e0 +log10 I21F11 19973 4.99996805151728e-1 4.999783304053115e-1 0 0 0 37308 0 1.07400390625e3 2.490234375e-1 +log10 I20F12 20294 4.9996842132391595e-1 4.9996842132391595e-1 0 0 0 40998 0 9.58984375e-1 9.58984375e-1 +log10 I19F13 20137 4.999947234889895e-1 4.999947234889895e-1 0 0 0 42779 0 2.8447265625e0 2.8447265625e0 +log10 I18F14 20136 4.999939529568647e-1 4.9983973079162086e-1 0 0 0 44401 0 1.324462890625e-2 1.846923828125e-1 +log10 I17F15 20143 4.999061360026008e-1 4.999061360026008e-1 0 0 0 44991 0 3.54193115234375e0 3.54193115234375e0 +log10 I16F16 20184 4.999719442103142e-1 4.9996975134398725e-1 0 0 0 45424 0 1.6300328231811523e4 6.2756805419921875e0 +log10 I15F17 20057 4.9996573274455064e-1 4.9996573274455064e-1 0 0 0 45432 0 1.70098876953125e0 1.70098876953125e0 +log10 I14F18 20164 4.999984143321246e-1 4.99963769136698e-1 0 0 0 46011 0 7.403876266479492e3 5.629566192626953e0 +log10 I13F19 20195 4.999914037155659e-1 4.999914037155659e-1 0 0 0 46419 0 2.1676063537597656e0 2.1676063537597656e0 +log10 I12F20 20132 4.999572070045254e-1 4.999572070045254e-1 0 0 0 45814 0 6.30235481262207e0 6.30235481262207e0 +log10 I11F21 20219 4.9998333017548596e-1 4.9993572029818084e-1 0 0 0 46121 0 3.2611748933792114e2 2.7242484092712402e0 +log10 I10F22 20185 4.9999726580416937e-1 4.99913551414531e-1 0 0 0 45840 0 1.327991485595703e-4 7.162179470062256e0 +log10 I9F23 20116 4.999367999071796e-1 4.999367999071796e-1 0 0 0 46015 0 1.8360564708709717e0 1.8360564708709717e0 +log10 I8F24 20017 4.999798174540321e-1 4.996846855310562e-1 0 0 0 46010 0 1.3728737831115723e-3 5.924713015556335e-1 +log10 I7F25 19815 4.999749518854896e-1 4.9993595343495906e-1 0 0 0 45111 0 1.1533889651298523e1 9.995232224464417e-1 +log10 I6F26 20064 4.999982658324662e-1 4.999564528983628e-1 0 0 0 45672 0 9.834766387939453e-7 3.497573971748352e0 +log10 I5F27 20044 4.999725004251862e-1 4.999559181733356e-1 0 0 0 45747 0 2.1949410438537598e-5 5.110195130109787e-1 +log10 I4F28 20205 4.9998514627046126e-1 4.9998514627046126e-1 0 0 0 45875 0 6.065712012350559e0 6.065712012350559e0 +log10 I3F29 20122 4.999666787220988e-1 4.9995846456883597e-1 0 0 0 45902 0 2.0051660016179085e-2 1.5891960430890322e0 +log10 I64F0 19283 4.9999555162793513e-1 4.7712125471966244e-1 0 0 0 33124 0 3.1622452698983e13 3e0 +log10 I63F1 20316 4.999998511212197e-1 4.807253789884877e-1 0 0 0 33690 0 1.7782797148414282e18 5.5e0 +log10 I62F2 20483 4.998845445144752e-1 4.997549464331998e-1 0 0 0 34260 0 2.371531316e9 7.5e-1 +log10 I61F3 20560 4.9990133967785405e-1 4.9669314685715543e-1 0 0 0 34551 0 8.659889142585619e17 6.5e0 +log10 I60F4 20542 4.999280150721581e-1 4.964843577068131e-1 0 0 0 34643 0 1.9109727715212646e17 8.0625e0 +log10 I56F8 20576 4.999812469951394e-1 4.9989972520647424e-1 0 0 0 35844 0 2.3395950738140068e16 9.34765625e0 +log10 I52F12 20661 4.999821481568942e-1 4.9991518285755043e-1 0 0 0 46375 0 4.8737831997511644e14 5.739990234375e0 +log10 I48F16 20584 4.999573246827441e-1 4.999573246827441e-1 0 0 0 50295 0 1.863250732421875e-1 1.863250732421875e-1 +log10 I44F20 20717 4.9999929687074346e-1 4.9999929687074346e-1 0 0 0 51145 0 4.9441680908203125e0 4.9441680908203125e0 +log10 I40F24 20649 4.9993746145334683e-1 4.9992228424840945e-1 0 0 0 50901 0 5.0655219995074243e11 3.5556814074516296e0 +log10 I36F28 20585 4.999527013470254e-1 4.999527013470254e-1 0 0 0 50865 0 1.6458539329469204e0 1.6458539329469204e0 +log10 I33F31 20636 4.999913658762062e-1 4.999913658762062e-1 0 0 0 51112 0 9.04186611995101e-1 9.04186611995101e-1 +log10 I32F32 20464 4.9997864463176e-1 4.9997864463176e-1 0 0 0 50615 0 7.541903460398316e0 7.541903460398316e0 +log10 I31F33 20590 4.9998404722581613e-1 4.9998404722581613e-1 0 0 0 51147 0 1.6035851484630257e0 1.6035851484630257e0 +log10 I28F36 20592 4.9999642227740654e-1 4.9999642227740654e-1 0 0 0 50831 0 4.008965185668785e0 4.008965185668785e0 +log10 I24F40 20611 4.9999460271320956e-1 4.997274858099076e-1 0 0 0 50861 0 7.593540688830832e6 4.203056249193651e0 +log10 I20F44 20477 4.9999463752814555e-1 4.999220122249967e-1 0 0 0 50640 0 6.236774822809821e-6 1.3738197243260402e0 +log10 I16F48 20655 4.999943802678029e-1 4.999943802678029e-1 0 0 0 50873 0 4.999784007592609e-1 4.999784007592609e-1 +log10 I12F52 20659 4.9998677198090213e-1 4.9998677198090213e-1 0 0 0 51010 0 3.011633203305572e0 3.011633203305572e0 +log10 I8F56 20599 4.9998329411696535e-1 4.9998329411696535e-1 0 0 0 50966 0 4.999999999950395e-1 4.999999999950395e-1 +log10 I6F58 20555 4.999845137809715e-1 4.999845137809715e-1 0 0 0 50790 0 1.82041703234067e0 1.82041703234067e0 +log10 I5F59 20399 4.999787694950875e-1 4.9994333493932147e-1 0 0 0 50655 0 1.2090166391870175e1 5.8948901178595525e0 +log10 I4F60 20706 4.9997105154703086e-1 4.9997105154703086e-1 0 0 0 51123 0 2.2239419961415687e0 2.2239419961415687e0 +log10 I3F61 20615 4.9999662603035855e-1 4.9999662603035855e-1 0 0 0 50955 0 3.0841430301866573e0 3.0841430301866573e0 +log10 I128F0 20249 4.999943449133103e-1 4.7712125471966244e-1 0 0 0 36438 0 3.162236483412466e37 3e0 +log10 I127F1 21147 4.99909123022846e-1 4.807253789884877e-1 0 0 0 37447 0 5.624001637841855e37 5.5e0 +log10 I126F2 21385 4.9998658559983583e-1 4.997549464331998e-1 0 0 0 38205 0 7.499e3 7.5e-1 +log10 I124F4 21430 4.999974283525152e-1 4.949394278984119e-1 0 0 0 39222 0 3.39820958658511e36 9.3125e0 +log10 I120F8 21526 4.999924244148632e-1 4.998553867499855e-1 0 0 0 40716 0 1.9195662053056143e35 6.58203125e0 +log10 I112F16 21421 4.9997218969276735e-1 4.998841973815972e-1 0 0 0 54289 0 1.7440312394111303e33 5.5296630859375e0 +log10 I104F24 21503 4.999834266286663e-1 4.998816702911783e-1 0 0 0 55015 0 1.2027741128206253e2 2.793554365634918e0 +log10 I96F32 21444 4.9999328650878244e-1 4.9991151766848163e-1 0 0 0 54823 0 2.2441025136570134e28 3.946776813128963e0 +log10 I88F40 21531 4.99979717470206e-1 4.99979717470206e-1 0 0 0 55117 0 4.932397669746933e0 4.932397669746933e0 +log10 I80F48 21556 4.9994885148907586e-1 4.9994885148907586e-1 0 0 0 55162 0 7.384860361971278e0 7.384860361971278e0 +log10 I72F56 21639 4.9999281189676303e-1 4.9987398833172497e-1 0 0 0 55282 0 5.6821611882383743e20 7.940806489052493e-1 +log10 I68F60 21352 4.999932398685626e-1 4.999390056054362e-1 0 0 0 54612 0 1.9655088117147174e6 6.506058610986345e0 +log10 I67F61 21501 4.9994415693597144e-1 4.9994415693597144e-1 0 0 0 54915 0 2.6696535896213947e-1 2.6696535896213947e-1 +log10 I66F62 21429 4.9998648329463635e-1 4.999834248547445e-1 0 0 0 54750 0 2.3288082438557543e4 3.69569800458686e0 +log10 I65F63 21641 4.9999316721512477e-1 4.999366937807141e-1 0 0 0 55251 0 3.913983673004372e2 5.3786106783757335e0 +log10 I64F64 21302 4.999932671883264e-1 4.998591848886926e-1 0 0 0 54341 0 6.336307592539077e8 6.969467630673624e0 +log10 I63F65 21334 4.999836744078189e-1 4.9984562171807656e-1 0 0 0 54825 0 3.778672231720037e18 7.562665043904379e0 +log10 I62F66 21498 4.999332801517996e-1 4.998516496399995e-1 0 0 0 54943 0 5.0386604077110406e17 6.93128750460088e0 +log10 I56F72 21484 4.9999714962302944e-1 4.9993765053007055e-1 0 0 0 55033 0 1.3226644952583015e-8 3.141592653589793e0 +log10 I48F80 21483 4.9997371288106796e-1 4.9997154944403677e-1 0 0 0 55004 0 4.3843478904378914e13 5e-1 +log10 I40F88 21461 4.9999681568343735e-1 4.998979699686728e-1 0 0 0 54519 0 8.284077070125439e1 2.0955022644558725e0 +log10 I32F96 21491 4.99950525278862e-1 4.998965660321585e-1 0 0 0 55220 0 1.2239398176271922e-10 2e0 +log10 I24F104 21625 4.999761162267988e-1 4.999761162267988e-1 0 0 0 55447 0 1.916143095719421e0 1.916143095719421e0 +log10 I16F112 21456 4.999833568300236e-1 4.999253861780381e-1 0 0 0 55045 0 3.2222437070722153e4 2.0000000000000018e0 +log10 I10F118 21306 4.9998832060143356e-1 4.9998560556506483e-1 0 0 0 54405 0 2.458498324112024e-28 2.717672458683524e-1 +log10 I8F120 21593 4.999290004042814e-1 4.999290004042814e-1 0 0 0 55447 0 2.7092939068070385e0 2.7092939068070385e0 +log10 I6F122 21316 4.999638305423811e-1 4.999591221448898e-1 0 0 0 54518 0 3.3297006429723426e-6 7.823236580423288e0 +log10 I5F123 21498 4.999845766466993e-1 4.999845766466993e-1 0 0 0 55084 0 3.141592653589793e0 3.141592653589793e0 +log10 I4F124 16455 4.999409915500722e-1 4.999409915500722e-1 0 0 0 40109 0 1.0659250631515918e-1 1.0659250631515918e-1 +log10 I3F125 16664 4.999792418346618e-1 4.999792418346618e-1 0 0 0 40401 0 3.681542022686901e-1 3.681542022686901e-1 +pow I8F0 65408 5e-1 5e-1 0 0 0 0 0 -2e0,-1e0 -2e0,-1e0 +pow I7F1 49024 5e-1 5e-1 0 0 0 0 0 -7.5e0,2e0 -4e0,-1e0 +pow I6F2 40832 5e-1 5e-1 0 0 0 0 0 -8e0,-1e0 -8e0,-1e0 +pow I5F3 36736 5e-1 5e-1 0 0 0 0 0 -1.6e1,-1e0 -1.6e1,-1e0 +pow I4F4 34688 5e-1 5e-1 0 0 0 0 0 -2e0,-5e0 -2e0,-5e0 +pow I3F5 33664 5e-1 5e-1 0 0 0 0 0 -4e0,-3e0 -4e0,-3e0 +pow I16F0 262140 5e-1 5e-1 0 0 0 0 0 -2e0,-1e0 -2e0,-1e0 +pow I15F1 197479 5e-1 4.714045207910317e-1 0 0 0 0 0 1.015e2,2e0 1.8e1,-5e-1 +pow I14F2 166489 5e-1 4.999313118653995e-1 0 0 0 0 0 1.45e1,3e0 4.09375e3,-2.5e-1 +pow I13F3 151881 5e-1 4.99956598369219e-1 0 0 0 0 0 5.925e1,2e0 1.625875e3,-3.75e-1 +pow I12F4 145153 4.99969479627623e-1 4.99969479627623e-1 0 0 0 0 0 1.023875e3,-5e-1 1.023875e3,-5e-1 +pow I11F5 141766 5e-1 4.999716231503821e-1 0 0 0 0 0 -1.875e0,2e0 4.1775e2,-3.125e-2 +pow I10F6 140265 5e-1 4.9999029754419416e-1 0 0 0 0 0 5.0625e0,1.75e0 3.50375e2,-6.40625e-1 +pow I9F7 139760 5e-1 4.9999219165345676e-1 0 0 0 0 0 -1.9375e0,2e0 2.344609375e2,-4.765625e-1 +pow I8F8 139900 4.999845673360945e-1 4.999845673360945e-1 0 0 0 0 0 2.579296875e1,-4.1015625e-1 2.579296875e1,-4.1015625e-1 +pow I7F9 139155 5e-1 5e-1 0 0 0 0 0 3.78125e0,2e0 3.125e-2,2e0 +pow I6F10 138955 4.999813954863553e-1 4.999813954863553e-1 0 0 0 0 0 6.064453125e-1,9.765625e-4 6.064453125e-1,9.765625e-4 +pow I5F11 138966 5e-1 5e-1 0 0 0 0 0 -2.296875e0,2e0 9.53125e-1,2e0 +pow I4F12 138804 4.999964644600278e-1 4.999964644600278e-1 0 0 0 0 0 7.041015625e-1,3.3076171875e0 7.041015625e-1,3.3076171875e0 +pow I3F13 138905 5e-1 5e-1 0 0 0 0 0 -1.7734375e0,2e0 1.640625e-1,2e0 +pow I32F0 39024 5e-1 5e-1 0 0 0 0 0 -2e0,-1e0 -2e0,-1e0 +pow I31F1 29698 5e-1 5e-1 0 0 0 0 0 -3.25e1,2e0 -5e-1,2e0 +pow I30F2 25158 5e-1 5e-1 0 0 0 0 0 -3.5e0,3e0 -5e-1,3e0 +pow I29F3 23132 5e-1 5e-1 0 0 0 0 0 4.65e1,4e0 -2.5e-1,2e0 +pow I28F4 21941 5e-1 5e-1 0 0 0 0 0 5e-1,5e0 5e-1,5e0 +pow I27F5 21562 5e-1 5e-1 0 0 0 0 0 3.125e0,2e0 3.75e-1,2e0 +pow I26F6 21365 5e-1 5e-1 0 0 0 0 0 2e0,-7e0 2e0,-7e0 +pow I25F7 20946 5e-1 5e-1 0 0 0 0 0 -6.9375e0,2e0 -3.125e-1,2e0 +pow I24F8 21043 4.998882297886434e-1 4.9981200599334447e-1 0 0 0 0 0 5.146727921875e6,5.4296875e-1 7.59375e0,-2.53515625e0 +pow I23F9 21111 5e-1 4.997556206649841e-1 0 0 0 0 0 -2.878125e1,2e0 9.98046875e-1,5e-1 +pow I22F10 21128 4.9998648261511114e-1 4.997782705099778e-1 0 0 0 0 0 8.80778955078125e4,4.990234375e-1 4.404296875e0,-1e0 +pow I21F11 21040 5e-1 5e-1 0 0 0 0 0 8.59375e-1,2e0 8.59375e-1,2e0 +pow I20F12 20924 4.999650401410238e-1 4.999650401410238e-1 0 0 0 0 0 2e0,-2.81884765625e0 2e0,-2.81884765625e0 +pow I19F13 21101 5e-1 5e-1 0 0 0 0 0 2.451796875e2,2e0 -3.515625e-1,2e0 +pow I18F14 21082 4.9997711065219086e-1 4.9996948055911616e-1 0 0 0 0 0 9.9993896484375e-1,-5e-1 1.9998779296875e0,-1e0 +pow I17F15 20943 5e-1 5e-1 0 0 0 0 0 -6.5536e4,-1e0 -6.5536e4,-1e0 +pow I16F16 20989 4.99994277881341e-1 4.999923707218823e-1 0 0 0 0 0 9.999847412109375e-1,-5e-1 -2.000030517578125e0,-1e0 +pow I15F17 20904 5e-1 4.999990463220456e-1 0 0 0 0 0 -1.998046875e0,2e0 9.999923706054688e-1,5e-1 +pow I14F18 20902 4.999995231619323e-1 4.999995231619323e-1 0 0 0 0 0 9.999961853027344e-1,5e-1 9.999961853027344e-1,5e-1 +pow I13F19 20672 4.999992847431258e-1 4.9999880790642237e-1 0 0 0 0 0 9.999980926513672e-1,-5e-1 9.999980926513672e-1,-5e-1 +pow I12F20 20981 4.999998807907673e-1 4.999998807906536e-1 0 0 0 0 0 1.0000009536743164e0,5e-1 9.999990463256836e-1,5e-1 +pow I11F21 21052 4.999995827675008e-1 4.999995827675008e-1 0 0 0 0 0 9.999995231628418e-1,4.999995231628418e-1 9.999995231628418e-1,4.999995231628418e-1 +pow I10F22 21101 4.9999997019767406e-1 4.9999997019767406e-1 0 0 0 0 0 9.999997615814209e-1,5e-1 9.999997615814209e-1,5e-1 +pow I9F23 20890 4.9999989569187253e-1 4.9999989569187253e-1 0 0 0 0 0 9.999998807907104e-1,4.9999988079071045e-1 9.999998807907104e-1,4.9999988079071045e-1 +pow I8F24 20992 4.999999925494192e-1 4.999999925494192e-1 0 0 0 0 0 9.999999403953552e-1,5e-1 9.999999403953552e-1,5e-1 +pow I7F25 21156 4.999999888241288e-1 4.999999813735483e-1 0 0 0 0 0 9.999999701976776e-1,-5e-1 9.999999701976776e-1,-5e-1 +pow I6F26 20841 4.999999944120645e-1 4.9999999068677425e-1 0 0 0 0 0 9.999999850988388e-1,-5e-1 9.999999850988388e-1,-5e-1 +pow I5F27 20980 4.999999990686774e-1 4.999999990686774e-1 0 0 0 0 0 9.999999925494194e-1,5e-1 9.999999925494194e-1,5e-1 +pow I4F28 20832 4.999999995343387e-1 4.999999995343387e-1 0 0 0 0 0 9.999999962747097e-1,5e-1 9.999999962747097e-1,5e-1 +pow I3F29 20811 5e-1 5e-1 0 0 0 0 0 4.99969482421875e-1,2e0 4.99969482421875e-1,2e0 +pow I64F0 39183 5e-1 5e-1 0 0 0 0 0 2e0,-1e0 2e0,-1e0 +pow I63F1 29883 5e-1 5e-1 0 0 0 0 0 5e-1,2e0 5e-1,2e0 +pow I62F2 25350 5e-1 5e-1 0 0 0 0 0 8e0,-1e0 8e0,-1e0 +pow I61F3 23086 5e-1 5e-1 0 0 0 0 0 -4.25e0,2e0 2e0,-4e0 +pow I60F4 22075 5e-1 5e-1 0 0 0 0 0 -5e-1,5e0 -5e-1,5e0 +pow I56F8 21311 5e-1 5e-1 0 0 0 0 0 2.62144e5,-5e-1 2.62144e5,-5e-1 +pow I52F12 21133 4.9999124640733966e-1 4.9996947869601616e-1 0 0 0 0 0 5.578369140625e0,1.5625e-2 9.99755859375e-1,5e-1 +pow I48F16 21174 4.99994277881341e-1 4.999904632131798e-1 0 0 0 0 0 9.999847412109375e-1,-5e-1 9.999847412109375e-1,-5e-1 +pow I44F20 21029 4.999998807906536e-1 4.999998807906536e-1 0 0 0 0 0 9.999990463256836e-1,5e-1 9.999990463256836e-1,5e-1 +pow I40F24 21056 4.999999925494192e-1 4.999999925494192e-1 0 0 0 0 0 9.999999403953552e-1,5e-1 9.999999403953552e-1,5e-1 +pow I36F28 21110 4.999999995343387e-1 4.999999995343387e-1 0 0 0 0 0 9.999999962747097e-1,5e-1 9.999999962747097e-1,5e-1 +pow I33F31 20840 5e-1 5e-1 0 0 0 0 0 4.294967296e9,-1e0 4.294967296e9,-1e0 +pow I32F32 21021 4.9999999997089617e-1 4.9999999997089617e-1 0 0 0 0 0 9.999999997671694e-1,5e-1 9.999999997671694e-1,5e-1 +pow I31F33 20985 4.9999999995634425e-1 4.999999999272404e-1 0 0 0 0 0 9.999999998835847e-1,-5e-1 9.999999998835847e-1,-5e-1 +pow I28F36 21054 4.99999999998181e-1 4.99999999998181e-1 0 0 0 0 0 9.999999999854481e-1,5e-1 9.999999999854481e-1,5e-1 +pow I24F40 20779 4.999999999998863e-1 4.999999999998863e-1 0 0 0 0 0 9.999999999990905e-1,5e-1 9.999999999990905e-1,5e-1 +pow I20F44 20892 4.9999999999992184e-1 4.9999999999990763e-1 0 0 0 0 0 9.999999999999432e-1,-5.000000000000568e-1 9.999999999999432e-1,-5.000000000000568e-1 +pow I16F48 21085 4.9999999999999956e-1 4.9999999999999956e-1 0 0 0 0 0 9.999999999999964e-1,5e-1 9.999999999999964e-1,5e-1 +pow I12F52 21027 4.9999999999999994e-1 4.9999999999999994e-1 0 0 0 0 0 9.999999999999998e-1,5e-1 9.999999999999998e-1,5e-1 +pow I8F56 20963 5e-1 5e-1 0 0 0 0 0 1e0,-5e-1 1e0,-5e-1 +pow I6F58 20902 5e-1 5e-1 0 0 0 0 0 1e0,-5e-1 1e0,-5e-1 +pow I5F59 20751 5e-1 5e-1 0 0 0 0 0 1e0,-5e-1 1e0,-5e-1 +pow I4F60 20896 5e-1 5e-1 0 0 0 0 0 1e0,-5e-1 1e0,-5e-1 +pow I3F61 20942 5e-1 5e-1 0 0 0 0 0 1e0,-5e-1 1e0,-5e-1 +pow I128F0 39335 5e-1 5e-1 0 0 0 0 0 -2e0,-1e0 -2e0,-1e0 +pow I127F1 30195 5e-1 5e-1 0 0 0 0 0 -2e0,-2e0 -2e0,-2e0 +pow I126F2 25464 5e-1 5e-1 0 0 0 0 0 -1.063570890695e11,3e0 -5e-1,3e0 +pow I124F4 22202 5e-1 5e-1 0 0 0 0 0 2e0,-5e0 2e0,-5e0 +pow I120F8 21498 4.9998527085061756e-1 4.9998527085061756e-1 0 0 0 0 0 3.13671875e0,-5.45703125e0 3.13671875e0,-5.45703125e0 +pow I112F16 21377 4.99998092665919e-1 4.999980926368151e-1 0 0 0 0 0 1.0000152587890625e0,5e-1 9.999847412109375e-1,5e-1 +pow I104F24 21397 4.9999996274709946e-1 4.9999996274709946e-1 0 0 0 0 0 1.0000000596046448e0,-5.000000596046448e-1 1.0000000596046448e0,-5.000000596046448e-1 +pow I96F32 21130 4.999999999126885e-1 4.999999998835847e-1 0 0 0 0 0 9.999999997671694e-1,-5e-1 -1.9999999995343387e0,-1e0 +pow I88F40 21125 4.999999999998863e-1 4.999999999998863e-1 0 0 0 0 0 9.999999999990905e-1,5e-1 9.999999999990905e-1,5e-1 +pow I80F48 21301 4.9999999999999867e-1 4.999999999999978e-1 0 0 0 0 0 9.999999999999964e-1,-5e-1 9.999999999999964e-1,-5e-1 +pow I72F56 21055 5e-1 5e-1 0 0 0 0 0 1e0,5e-1 1e0,5e-1 +pow I68F60 21016 5e-1 5e-1 0 0 0 0 0 1e0,-5e-1 1e0,-5e-1 +pow I67F61 21042 5e-1 5e-1 0 0 0 0 0 1e0,-5e-1 1e0,-5e-1 +pow I66F62 21092 5e-1 5e-1 0 0 0 0 0 1e0,5e-1 1e0,5e-1 +pow I65F63 20990 5e-1 5e-1 0 0 0 0 0 -1.8446744073709552e19,-1e0 -1.8446744073709552e19,-1e0 +pow I64F64 21104 5e-1 5e-1 0 0 0 0 0 1e0,5e-1 1e0,5e-1 +pow I63F65 21351 5e-1 5e-1 0 0 0 0 0 1e0,5e-1 1e0,5e-1 +pow I62F66 20905 5e-1 5e-1 0 0 0 0 0 1e0,5e-1 1e0,5e-1 +pow I56F72 21381 5e-1 5e-1 0 0 0 0 0 1e0,5e-1 1e0,5e-1 +pow I48F80 21200 5e-1 5e-1 0 0 0 0 0 1e0,5e-1 1e0,5e-1 +pow I40F88 20990 5e-1 5e-1 0 0 0 0 0 1e0,-5e-1 1e0,-5e-1 +pow I32F96 20754 5e-1 5e-1 0 0 0 0 0 1e0,-5e-1 1e0,-5e-1 +pow I24F104 20871 5e-1 5e-1 0 0 0 0 0 1e0,-5e-1 1e0,-5e-1 +pow I16F112 21100 5e-1 5e-1 0 0 0 0 0 1e0,-5e-1 1e0,-5e-1 +pow I10F118 21070 5e-1 5e-1 0 0 0 0 0 1e0,5e-1 1e0,5e-1 +pow I8F120 20897 5e-1 5e-1 0 0 0 0 0 1e0,-5e-1 1e0,-5e-1 +pow I6F122 20808 5e-1 5e-1 0 0 0 0 0 1e0,-5e-1 1e0,-5e-1 +pow I5F123 20764 5e-1 5e-1 0 0 0 0 0 1e0,-5e-1 1e0,-5e-1 +pow I4F124 21949 5e-1 5e-1 0 0 0 0 0 2e0,-8e0 2e0,-8e0 +pow I3F125 25317 5e-1 5e-1 0 0 0 0 0 1e0,-5e-1 1e0,-5e-1 +sqrt I8F0 128 4.880884817015155e-1 2.928932188134524e-1 0 0 0 127 0 1.1e2 2e0 +sqrt I7F1 128 4.9193338482966753e-1 4.1421356237309503e-1 0 0 0 127 0 6e1 5e-1 +sqrt I6F2 128 4.939015319191968e-1 4.641016151377546e-1 0 0 0 127 0 2.625e1 7.5e-1 +sqrt I5F3 128 4.9603149604724406e-1 4.8331477354788277e-1 0 0 0 127 0 1.55e1 8.75e-1 +sqrt I4F4 128 4.9615361854384143e-1 4.9193338482966753e-1 0 0 0 127 0 4.125e0 9.375e-1 +sqrt I3F5 128 4.980314655501742e-1 4.9603149604724406e-1 0 0 0 127 0 3.9375e0 9.6875e-1 +sqrt I16F0 32768 4.9930747789588126e-1 2.928932188134524e-1 0 0 0 32767 0 3.258e4 2e0 +sqrt I15F1 32768 4.99510762740992e-1 4.1421356237309503e-1 0 0 0 32767 0 1.632e4 5e-1 +sqrt I14F2 32768 4.9965325919524384e-1 4.641016151377546e-1 0 0 0 32767 0 8.1225e3 7.5e-1 +sqrt I13F3 32768 4.997556206649841e-1 4.8331477354788277e-1 0 0 0 32767 0 4.088e3 8.75e-1 +sqrt I12F4 32768 4.9982650934760653e-1 4.9193338482966753e-1 0 0 0 32767 0 2.0278125e3 9.375e-1 +sqrt I11F5 32768 4.9987787004645057e-1 4.9603149604724406e-1 0 0 0 32767 0 1.023e3 9.6875e-1 +sqrt I10F6 32768 4.999112531033454e-1 4.980314655501742e-1 0 0 0 32767 0 4.8434375e2 9.84375e-1 +sqrt I9F7 32768 4.999389499380398e-1 4.9901960407381896e-1 0 0 0 32767 0 2.55875e2 9.921875e-1 +sqrt I8F8 32768 4.9995561867531163e-1 4.99510762740992e-1 0 0 0 32767 0 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0 0 0 0 7.1875e-1 9.375e-2 +ellip_e I26F6 3000 4.9241423561859043e-1 4.787675428593619e-1 0 0 0 0 0 5e-1 1.25e-1 +ellip_e I25F7 3000 4.9775363350509433e-1 4.4579332866540017e-1 0 0 0 0 0 5.78125e-1 2.1875e-1 +ellip_e I24F8 3000 4.9775682699228935e-1 4.7161254803234604e-1 0 0 0 0 0 6.8359375e-1 1.0546875e-1 +ellip_e I23F9 3000 4.996751402091387e-1 4.8441865086038954e-1 0 0 0 0 0 5.41015625e-1 7.03125e-2 +ellip_e I22F10 3000 4.9996162432969815e-1 4.9540965546348636e-1 0 0 0 0 0 8.4765625e-1 1.3671875e-2 +ellip_e I21F11 3000 4.9967762823868583e-1 4.918547202958587e-1 0 0 0 0 0 7.8759765625e-1 3.564453125e-2 +ellip_e I20F12 3000 4.9999886787778997e-1 4.9975252971816614e-1 0 0 0 0 0 8.3447265625e-1 1.171875e-2 +ellip_e I19F13 3000 4.9875469890391055e-1 4.9593953703062277e-1 0 0 0 0 0 5.46875e-2 5.46875e-2 +ellip_e I18F14 3000 4.993672685111597e-1 4.957865796542007e-1 0 0 0 0 0 3.140869140625e-1 3.680419921875e-2 +ellip_e I17F15 3000 4.9983619015065256e-1 4.984483027086628e-1 0 0 0 0 0 5.96649169921875e-1 3.070068359375e-2 +ellip_e I16F16 3000 4.996594198981062e-1 4.9778434168908114e-1 0 0 0 0 0 3.968963623046875e-1 1.263427734375e-2 +ellip_e I15F17 3000 4.988432813457568e-1 4.97267368643199e-1 0 0 0 0 0 8.2177734375e-1 5.3558349609375e-3 +ellip_e I14F18 3000 4.99969277540428e-1 4.996349161226004e-1 0 0 0 0 0 9.399032592773438e-2 6.82830810546875e-4 +ellip_e I13F19 3000 4.998666896731523e-1 4.989314770468042e-1 0 0 0 0 0 6.639804840087891e-1 4.730224609375e-4 +ellip_e I12F20 3000 4.9999704946672935e-1 4.956884103551848e-1 0 0 0 0 0 6.646099090576172e-1 2.8715133666992188e-2 +ellip_e I11F21 3000 4.9974400126306184e-1 4.991867490712003e-1 0 0 0 0 0 4.68538761138916e-1 7.263660430908203e-3 +ellip_e I10F22 3000 4.99853207356714e-1 4.9943823192447395e-1 0 0 0 0 0 6.940207481384277e-1 5.423545837402344e-3 +ellip_e I9F23 3000 4.999498875093283e-1 4.99642118545134e-1 0 0 0 0 0 3.5851383209228516e-1 9.06991958618164e-3 +ellip_e I8F24 3000 4.997199334235105e-1 4.9595678288972894e-1 0 0 0 0 0 5.227957963943481e-1 3.5840272903442383e-4 +ellip_e I7F25 3000 4.999218367858811e-1 4.9885006112314373e-1 0 0 0 0 0 5.698197484016418e-1 4.065662622451782e-3 +ellip_e I6F26 3000 4.9980457413097984e-1 4.975381747629514e-1 0 0 0 0 0 4.5923881232738495e-1 4.187093675136566e-2 +ellip_e I5F27 3000 4.9960682838829007e-1 4.984757989095813e-1 0 0 0 0 0 2.6033685356378555e-1 7.07581639289856e-4 +ellip_e I4F28 3000 4.992341211631416e-1 4.946317182383612e-1 0 0 0 0 0 7.548912540078163e-1 3.920865058898926e-2 +ellip_e I3F29 3000 4.999463219048715e-1 4.999437820081072e-1 0 0 0 0 0 1.1523030698299408e-3 1.1523030698299408e-3 +ellip_e I64F0 3000 4.292036732051034e-1 2.732395447351627e-1 0 0 0 0 0 1e0 1e0 +ellip_e I63F1 3000 4.221120551369191e-1 3.485487421952141e-1 0 0 0 0 0 5e-1 5e-1 +ellip_e I62F2 3000 4.7412695982262376e-1 3.432051054197978e-1 0 0 0 0 0 7.5e-1 7.5e-1 +ellip_e I61F3 3000 4.3362938564082704e-1 3.931522472481445e-1 0 0 0 0 0 1e0 2.5e-1 +ellip_e I60F4 3000 4.118678789971323e-1 3.6400431824230445e-1 0 0 0 0 0 8.75e-1 1.25e-1 +ellip_e I56F8 3000 4.9775682699228935e-1 4.7161254803234604e-1 0 0 0 0 0 6.8359375e-1 1.0546875e-1 +ellip_e I52F12 3000 4.996599816301424e-1 4.912055554173729e-1 0 0 0 0 0 7.26318359375e-1 9.4482421875e-2 +ellip_e I48F16 3000 4.999868109014346e-1 4.9888865076590677e-1 0 0 0 0 0 5.205078125e-1 1.434326171875e-3 +ellip_e I44F20 3000 4.999384985439663e-1 4.979474223194018e-1 0 0 0 0 0 7.710027694702148e-1 3.334808349609375e-2 +ellip_e I40F24 3000 4.995715865667696e-1 4.9735618059985787e-1 0 0 0 0 0 9.97535228729248e-1 1.2850761413574219e-3 +ellip_e I36F28 3000 4.999936723079077e-1 4.953335727524174e-1 0 0 0 0 0 2.2450407221913338e-1 5.027339607477188e-2 +ellip_e I33F31 3000 4.99903750352272e-1 4.9887170255319285e-1 0 0 0 0 0 6.321265543811023e-1 6.013084203004837e-6 +ellip_e I32F32 3000 4.997206690264087e-1 4.97497208108625e-1 0 0 0 0 0 2.9562868759967387e-1 6.664913846179843e-3 +ellip_e I31F33 3000 4.994652599796089e-1 4.9855089689400056e-1 0 0 0 0 0 4.3520263431128114e-1 2.6264853193424642e-2 +ellip_e I28F36 3000 4.995371724668091e-1 4.9661241557775826e-1 0 0 0 0 0 2.3196541953075211e-1 5.401202611392364e-2 +ellip_e I24F40 3000 4.997862569668202e-1 4.9899493199251027e-1 0 0 0 0 0 5.707960883764827e-1 1.4336146050482057e-3 +ellip_e I20F44 3000 4.998011917774265e-1 4.994067097734184e-1 0 0 0 0 0 9.07934307862206e-1 1.8289745330548612e-3 +ellip_e I16F48 3000 4.9993893041795e-1 4.997883625950461e-1 0 0 0 0 0 1.0523574513015888e-2 1.0523574513015888e-2 +ellip_e I12F52 3000 4.9967249073159775e-1 4.9967248839276196e-1 0 0 0 0 0 2.8726956997848774e-5 2.8726956997848774e-5 +ellip_e I8F56 3000 4.997087879633894e-1 4.996608399997709e-1 0 0 0 0 0 6.227039813116296e-1 3.419325210474167e-8 +ellip_e I6F58 3000 4.9991359590637763e-1 4.998356188213884e-1 0 0 0 0 0 4.140359138121857e-1 1.4802671645439622e-3 +ellip_e I5F59 3000 4.998107523801088e-1 4.982672594155748e-1 0 0 0 0 0 1.9580616081094687e-1 2.9703055399689364e-2 +ellip_e I4F60 3000 4.9990987871945297e-1 4.99004645049597e-1 0 0 0 0 0 2.907846318501872e-1 1.9332727346083045e-10 +ellip_e I3F61 3000 4.9985057131635596e-1 4.9923367981629224e-1 0 0 0 0 0 7.229177780134565e-1 1.1247283341280723e-2 +ellip_e I128F0 3000 4.292036732051034e-1 2.732395447351627e-1 0 0 0 0 0 1e0 1e0 +ellip_e I127F1 3000 4.221120551369191e-1 3.485487421952141e-1 0 0 0 0 0 5e-1 5e-1 +ellip_e I126F2 3000 4.7412695982262376e-1 3.432051054197978e-1 0 0 0 0 0 7.5e-1 7.5e-1 +ellip_e I124F4 3000 4.118678789971323e-1 3.6400431824230445e-1 0 0 0 0 0 8.75e-1 1.25e-1 +ellip_e I120F8 3000 4.9775682699228935e-1 4.7161254803234604e-1 0 0 0 0 0 6.8359375e-1 1.0546875e-1 +ellip_e I112F16 3000 4.996940220133325e-1 4.9888865076590677e-1 0 0 0 0 0 5.081024169921875e-1 1.434326171875e-3 +ellip_e I104F24 3000 4.9948784606121643e-1 4.9242875315387885e-1 0 0 0 0 0 9.935958981513977e-1 2.7792632579803467e-2 +ellip_e I96F32 3000 4.9999521001040076e-1 4.9741858061673566e-1 0 0 0 0 0 4.300556373782456e-1 7.7081178314983845e-3 +ellip_e I88F40 3000 4.998217595916948e-1 4.9681160123852236e-1 0 0 0 0 0 7.962379522023184e-1 2.381630110903643e-7 +ellip_e I80F48 3000 4.986654402984678e-1 4.9745595389788744e-1 0 0 0 0 0 2.2951553082466702e-1 3.3239934391460224e-4 +ellip_e I72F56 3000 4.994543506139037e-1 4.9757595973591345e-1 0 0 0 0 0 8.050806826140545e-1 2.4665703183554613e-2 +ellip_e I68F60 3000 4.996883845641723e-1 4.9968825986720716e-1 0 0 0 0 0 2.322348015088658e-4 2.322348015088658e-4 +ellip_e I67F61 3000 4.9989214767704754e-1 4.944992600886316e-1 0 0 0 0 0 8.123694451775896e-1 1.3477230876440816e-10 +ellip_e I66F62 3000 4.99747737054612e-1 4.996844600360618e-1 0 0 0 0 0 6.543142189256966e-3 6.543142189256966e-3 +ellip_e I65F63 3000 4.9987386269253403e-1 4.976076052565664e-1 0 0 0 0 0 5.707963267948968e-1 2.0574634007288116e-2 +ellip_e I64F64 3000 4.9998365526284455e-1 4.9824761218036806e-1 0 0 0 0 0 5.76206202042815e-1 2.2267966812731414e-2 +ellip_e I63F65 3000 4.9979508099691117e-1 4.9956355769185007e-1 0 0 0 0 0 6.067243118874941e-1 8.886407145491404e-3 +ellip_e I62F66 3000 4.9983397664015755e-1 4.944423579970168e-1 0 0 0 0 0 5.70796326794383e-1 1.0529149947540936e-7 +ellip_e I56F72 3000 4.9973836329585e-1 4.968332205861091e-1 0 0 0 0 0 4.786577278390269e-1 1.5626062830408568e-2 +ellip_e I48F80 3000 4.99869399793177e-1 4.97491044316273e-1 0 0 0 0 0 1.2465101038493422e-1 1.5514268579188586e-4 +ellip_e I40F88 3000 4.998311847175187e-1 4.991812572536407e-1 0 0 0 0 0 8.145240861358453e-1 5.831296542534612e-14 +ellip_e I32F96 3000 4.999251338058586e-1 4.999153659554222e-1 0 0 0 0 0 8.23144331267069e-1 1.1920928955078125e-7 +ellip_e I24F104 3000 4.9999617680241704e-1 4.975997116768278e-1 0 0 0 0 0 3.332604823215874e-1 1.578523049426909e-3 +ellip_e I16F112 3000 4.999720711290812e-1 4.988630629819923e-1 0 0 0 0 0 5e-1 6.444293122762323e-17 +ellip_e I10F118 3000 4.995382775517365e-1 4.955660881664161e-1 0 0 0 0 0 1e0 3.910945647002591e-10 +ellip_e I8F120 3000 4.9993471522417376e-1 4.9993471522417177e-1 0 0 0 0 0 2.0610323891836918e-8 2.0610323891836918e-8 +ellip_e I6F122 3000 4.9992052103251955e-1 4.9992052103251955e-1 0 0 0 0 0 1.2489795412100594e-14 1.2489795412100594e-14 +ellip_e I5F123 3000 4.9988019218326574e-1 4.9560448840637306e-1 0 0 0 0 0 1.0657702509353628e-1 6.788313978312296e-2 +ellip_e I4F124 3000 4.999846028467034e-1 4.943417793960827e-1 0 0 0 0 0 1e0 4.414392061569234e-18 +ellip_e I3F125 3000 4.9989120985772634e-1 4.9944141168903305e-1 0 0 0 0 0 5.307983935044178e-1 9.990621113254306e-5 +jacobi_sn I8F0 3000 4.987131538963941e-1 4.987131538963941e-1 0 0 0 0 0 -1.22e2,1e0 -1.22e2,1e0 +jacobi_sn I7F1 3000 4.9854853190591136e-1 4.9854853190591136e-1 0 0 0 0 0 5.7e1,5e-1 5.7e1,5e-1 +jacobi_sn I6F2 3000 4.9889581501676966e-1 4.9889581501676966e-1 0 0 0 0 0 8.75e0,1e0 8.75e0,1e0 +jacobi_sn I5F3 3000 4.996852089872081e-1 4.996852089872081e-1 0 0 0 0 0 1.55e1,8.75e-1 1.55e1,8.75e-1 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4.9945838032723333e-1 0 0 0 0 0 1.7429189e7,1e0 1.7429189e7,1e0 +jacobi_sn I31F1 3000 4.9980337595722507e-1 4.9980337595722507e-1 0 0 0 0 0 8.063040825e8,1e0 8.063040825e8,1e0 +jacobi_sn I30F2 3000 4.9995109833833107e-1 4.9995109833833107e-1 0 0 0 0 0 -4.892666e7,7.5e-1 -4.892666e7,7.5e-1 +jacobi_sn I29F3 3000 4.9940535793292934e-1 4.9940535793292934e-1 0 0 0 0 0 -1.079099845e8,3.75e-1 -1.079099845e8,3.75e-1 +jacobi_sn I28F4 3000 4.998947045157292e-1 4.998947045157292e-1 0 0 0 0 0 5.69545234375e7,9.375e-1 5.69545234375e7,9.375e-1 +jacobi_sn I27F5 3000 4.9974851535302545e-1 4.9974851535302545e-1 0 0 0 0 0 9.6875e-1,4.6875e-1 9.6875e-1,4.6875e-1 +jacobi_sn I26F6 3000 4.9972298635157086e-1 4.9972298635157086e-1 0 0 0 0 0 2.79855309375e6,6.25e-1 2.79855309375e6,6.25e-1 +jacobi_sn I25F7 3000 4.999550169813187e-1 4.999550169813187e-1 0 0 0 0 0 1.8203125e0,8.359375e-1 1.8203125e0,8.359375e-1 +jacobi_sn I24F8 3000 4.999991900404784e-1 4.999991900404784e-1 0 0 0 0 0 -1.06313046875e4,4.609375e-1 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0 0 -2.0000001511070877e0,2.9721297323703766e-5 -2.0000001511070877e0,2.9721297323703766e-5 +jacobi_sn I31F33 3000 4.9967233393367744e-1 4.9967233393367744e-1 0 0 0 0 0 -2.242884515654456e7,2.020973099861294e-1 -2.242884515654456e7,2.020973099861294e-1 +jacobi_sn I28F36 3000 4.9987033683128895e-1 4.9987033683128895e-1 0 0 0 0 0 -1.9897456759354536e3,7.197779234847985e-1 -1.9897456759354536e3,7.197779234847985e-1 +jacobi_sn I24F40 3000 4.998455312758164e-1 4.998455312758164e-1 0 0 0 0 0 5.172651640117692e0,8.750159569863172e-1 5.172651640117692e0,8.750159569863172e-1 +jacobi_sn I20F44 3000 4.999215449359216e-1 4.999215449359216e-1 0 0 0 0 0 -5.080195708894848e0,6.0146671687846265e-2 -5.080195708894848e0,6.0146671687846265e-2 +jacobi_sn I16F48 3000 4.9964607135623584e-1 4.9964607135623584e-1 0 0 0 0 0 -3.9484940920916156e3,9.999999999999964e-1 -3.9484940920916156e3,9.999999999999964e-1 +jacobi_sn I12F52 3000 4.994655460575886e-1 4.994655460575886e-1 0 0 0 0 0 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4.9982802300705287e-1 0 0 0 0 0 -1.28125e0,0e0 -1.28125e0,0e0 +jacobi_cn I16F0 3000 4.799368367374458e-1 4.799368367374458e-1 0 0 0 0 0 -3.1528e4,1e0 -3.1528e4,1e0 +jacobi_cn I15F1 3000 4.9944233388848275e-1 4.9944233388848275e-1 0 0 0 0 0 -1.23585e4,5e-1 -1.23585e4,5e-1 +jacobi_cn I14F2 3000 4.9905236356484206e-1 4.9905236356484206e-1 0 0 0 0 0 9.1075e2,7.5e-1 9.1075e2,7.5e-1 +jacobi_cn I13F3 3000 4.9937469471598034e-1 4.9937469471598034e-1 0 0 0 0 0 3.25e0,7.5e-1 3.25e0,7.5e-1 +jacobi_cn I12F4 3000 4.9993932232792226e-1 4.9993932232792226e-1 0 0 0 0 0 2.4575e2,9.375e-1 2.4575e2,9.375e-1 +jacobi_cn I11F5 3000 4.9999336716766174e-1 4.9999336716766174e-1 0 0 0 0 0 -2.375e0,2.5e-1 -2.375e0,2.5e-1 +jacobi_cn I10F6 3000 4.9967910733862886e-1 4.9967910733862886e-1 0 0 0 0 0 3.36625e2,1.5625e-2 3.36625e2,1.5625e-2 +jacobi_cn I9F7 3000 4.9909640948046313e-1 4.9909640948046313e-1 0 0 0 0 0 -5.1875e0,2.03125e-1 -5.1875e0,2.03125e-1 +jacobi_cn I8F8 3000 4.999388790221244e-1 4.999388790221244e-1 0 0 0 0 0 1e0,1.25e-1 1e0,1.25e-1 +jacobi_cn I7F9 3000 4.9992107742615205e-1 4.9992107742615205e-1 0 0 0 0 0 1.533203125e0,7.34375e-1 1.533203125e0,7.34375e-1 +jacobi_cn I6F10 3000 4.9985473388866125e-1 4.9985473388866125e-1 0 0 0 0 0 -3.125e-2,5.9765625e-1 -3.125e-2,5.9765625e-1 +jacobi_cn I5F11 3000 4.99571066734641e-1 4.99571066734641e-1 0 0 0 0 0 2.04736328125e0,7.34375e-1 2.04736328125e0,7.34375e-1 +jacobi_cn I4F12 3000 4.999627628624151e-1 4.999627628624151e-1 0 0 0 0 0 8.01513671875e-1,9.765625e-4 8.01513671875e-1,9.765625e-4 +jacobi_cn I3F13 3000 4.996713328882823e-1 4.996713328882823e-1 0 0 0 0 0 3.5904541015625e0,9.99755859375e-2 3.5904541015625e0,9.99755859375e-2 +jacobi_cn I32F0 3000 4.9415261869373506e-1 4.9415261869373506e-1 0 0 0 0 0 -8.162825e8,1e0 -8.162825e8,1e0 +jacobi_cn I31F1 3000 4.9966864806111205e-1 4.9966864806111205e-1 0 0 0 0 0 -8.60735e4,5e-1 -8.60735e4,5e-1 +jacobi_cn I30F2 3000 4.9999493029280095e-1 4.9999493029280095e-1 0 0 0 0 0 -3.2497350575e8,7.5e-1 -3.2497350575e8,7.5e-1 +jacobi_cn I29F3 3000 4.9937469471598034e-1 4.9937469471598034e-1 0 0 0 0 0 -3.25e0,7.5e-1 -3.25e0,7.5e-1 +jacobi_cn I28F4 3000 4.999378212648505e-1 4.999378212648505e-1 0 0 0 0 0 4.875e0,2.5e-1 4.875e0,2.5e-1 +jacobi_cn I27F5 3000 4.997528561672728e-1 4.997528561672728e-1 0 0 0 0 0 -5.5e0,4.0625e-1 -5.5e0,4.0625e-1 +jacobi_cn I26F6 3000 4.9995268743566856e-1 4.9995268743566856e-1 0 0 0 0 0 -2.9375e0,9.21875e-1 -2.9375e0,9.21875e-1 +jacobi_cn I25F7 3000 4.996939833554618e-1 4.996939833554618e-1 0 0 0 0 0 1.612127546875e6,1e0 1.612127546875e6,1e0 +jacobi_cn I24F8 3000 4.999430921513428e-1 4.999430921513428e-1 0 0 0 0 0 -4.18581882421875e6,5.6640625e-1 -4.18581882421875e6,5.6640625e-1 +jacobi_cn I23F9 3000 4.9997379796661884e-1 4.9997379796661884e-1 0 0 0 0 0 -1.79951637890625e6,6.42578125e-1 -1.79951637890625e6,6.42578125e-1 +jacobi_cn I22F10 3000 4.998351029203002e-1 4.998351029203002e-1 0 0 0 0 0 -3.125e-2,4.86328125e-1 -3.125e-2,4.86328125e-1 +jacobi_cn I21F11 3000 4.99902838515575e-1 4.99902838515575e-1 0 0 0 0 0 -2.326767578125e4,2.6904296875e-1 -2.326767578125e4,2.6904296875e-1 +jacobi_cn I20F12 3000 4.9985108634825837e-1 4.9985108634825837e-1 0 0 0 0 0 -7.9853515625e0,9.9951171875e-1 -7.9853515625e0,9.9951171875e-1 +jacobi_cn I19F13 3000 4.9998518472880993e-1 4.9998518472880993e-1 0 0 0 0 0 5.221923828125e0,4.44580078125e-1 5.221923828125e0,4.44580078125e-1 +jacobi_cn I18F14 3000 4.9998729163213185e-1 4.9998729163213185e-1 0 0 0 0 0 7.8125e-3,2.618408203125e-2 7.8125e-3,2.618408203125e-2 +jacobi_cn I17F15 3000 4.996353885453893e-1 4.996353885453893e-1 0 0 0 0 0 6.553599996948242e4,8.85009765625e-4 6.553599996948242e4,8.85009765625e-4 +jacobi_cn I16F16 3000 4.999968720861943e-1 4.999968720861943e-1 0 0 0 0 0 -3.90625e-3,1.415863037109375e-1 -3.90625e-3,1.415863037109375e-1 +jacobi_cn I15F17 3000 4.997392193204549e-1 4.997392193204549e-1 0 0 0 0 0 -7.608909606933594e0,4.9999237060546875e-1 -7.608909606933594e0,4.9999237060546875e-1 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0 0 0 0 2.9262394964694977e1,4.108894467353821e-1 2.9262394964694977e1,4.108894467353821e-1 +jacobi_cn I7F25 3000 4.99877956421619e-1 4.99877956421619e-1 0 0 0 0 0 -6.4e1,3.6961734294891357e-3 -6.4e1,3.6961734294891357e-3 +jacobi_cn I6F26 3000 4.9987550995173646e-1 4.9987550995173646e-1 0 0 0 0 0 9.23447671532631e0,3.1134486198425293e-4 9.23447671532631e0,3.1134486198425293e-4 +jacobi_cn I5F27 3000 4.9980830587735775e-1 4.9980830587735775e-1 0 0 0 0 0 1.5707944184541702e0,3.039836883544922e-6 1.5707944184541702e0,3.039836883544922e-6 +jacobi_cn I4F28 3000 4.998157464174239e-1 4.998157464174239e-1 0 0 0 0 0 -1e0,3.3421640843153e-1 -1e0,3.3421640843153e-1 +jacobi_cn I3F29 3000 4.999442271798622e-1 4.999442271798622e-1 0 0 0 0 0 -2.7721257768571377e0,6.889187153428793e-1 -2.7721257768571377e0,6.889187153428793e-1 +jacobi_cn I64F0 3000 4.746780111822703e-1 4.746780111822703e-1 0 0 0 0 0 -4.5e1,1e0 -4.5e1,1e0 +jacobi_cn I63F1 3000 4.9989664540600287e-1 4.9989664540600287e-1 0 0 0 0 0 -1.388389685316967e18,5e-1 -1.388389685316967e18,5e-1 +jacobi_cn I62F2 3000 4.988728939781839e-1 4.988728939781839e-1 0 0 0 0 0 2.30584245945788e18,2.5e-1 2.30584245945788e18,2.5e-1 +jacobi_cn I61F3 3000 4.999495247051906e-1 4.999495247051906e-1 0 0 0 0 0 -4.156210796969305e14,6.25e-1 -4.156210796969305e14,6.25e-1 +jacobi_cn I60F4 3000 4.998005112330545e-1 4.998005112330545e-1 0 0 0 0 0 -7.133470443408918e16,1.25e-1 -7.133470443408918e16,1.25e-1 +jacobi_cn I56F8 3000 4.998432003322074e-1 4.998432003322074e-1 0 0 0 0 0 -7.00662890625e3,5.3515625e-1 -7.00662890625e3,5.3515625e-1 +jacobi_cn I52F12 3000 4.999624092383469e-1 4.999624092383469e-1 0 0 0 0 0 1.5625e-2,5.71044921875e-1 1.5625e-2,5.71044921875e-1 +jacobi_cn I48F16 3000 4.99869421948463e-1 4.99869421948463e-1 0 0 0 0 0 1.40737488355328e14,7.255401611328125e-1 1.40737488355328e14,7.255401611328125e-1 +jacobi_cn I44F20 3000 4.9999260061151213e-1 4.9999260061151213e-1 0 0 0 0 0 -7.813777923583984e0,9.922275543212891e-1 -7.813777923583984e0,9.922275543212891e-1 +jacobi_cn I40F24 3000 4.9974892468356563e-1 4.9974892468356563e-1 0 0 0 0 0 1.5082963109016418e0,5.408734083175659e-1 1.5082963109016418e0,5.408734083175659e-1 +jacobi_cn I36F28 3000 4.9999999984477955e-1 4.9999999984477955e-1 0 0 0 0 0 -6.103515625e-5,9.999999962747097e-1 -6.103515625e-5,9.999999962747097e-1 +jacobi_cn I33F31 3000 4.99991857781725e-1 4.99991857781725e-1 0 0 0 0 0 -1.6777214429203674e7,1.908320840448141e-2 -1.6777214429203674e7,1.908320840448141e-2 +jacobi_cn I32F32 3000 4.9989487601412086e-1 4.9989487601412086e-1 0 0 0 0 0 -5.245311559643596e0,1.4433151111006737e-2 -5.245311559643596e0,1.4433151111006737e-2 +jacobi_cn I31F33 3000 4.9978088697440964e-1 4.9978088697440964e-1 0 0 0 0 0 -1.073741824e9,3.3419207320548594e-2 -1.073741824e9,3.3419207320548594e-2 +jacobi_cn I28F36 3000 4.998717325555828e-1 4.998717325555828e-1 0 0 0 0 0 -3.071437290784687e6,2.0416388439480215e-1 -3.071437290784687e6,2.0416388439480215e-1 +jacobi_cn I24F40 3000 4.9997556151361644e-1 4.9997556151361644e-1 0 0 0 0 0 1.574473243866123e0,5.106282068863948e-1 1.574473243866123e0,5.106282068863948e-1 +jacobi_cn I20F44 3000 4.997692191931575e-1 4.997692191931575e-1 0 0 0 0 0 1.6216012518116226e5,9.999999999999432e-1 1.6216012518116226e5,9.999999999999432e-1 +jacobi_cn I16F48 3000 4.9978672784598627e-1 4.9978672784598627e-1 0 0 0 0 0 -1.3236271872961325e-1,7.817193092615113e-1 -1.3236271872961325e-1,7.817193092615113e-1 +jacobi_cn I12F52 3000 4.9984429401136465e-1 4.9984429401136465e-1 0 0 0 0 0 8.88700306922452e-6,4.9999999999999933e-1 8.88700306922452e-6,4.9999999999999933e-1 +jacobi_cn I8F56 3000 4.9981001005866227e-1 4.9981001005866227e-1 0 0 0 0 0 1.206775351353514e2,8.119873515048133e-1 1.206775351353514e2,8.119873515048133e-1 +jacobi_cn I6F58 3000 4.999912883781411e-1 4.999912883781411e-1 0 0 0 0 0 5.705991231149745e-1,7.53902657810586e-1 5.705991231149745e-1,7.53902657810586e-1 +jacobi_cn I5F59 3000 4.9956529179776127e-1 4.9956529179776127e-1 0 0 0 0 0 -6.514041684213602e0,6.751729487359349e-1 -6.514041684213602e0,6.751729487359349e-1 +jacobi_cn I4F60 3000 5e-1 5e-1 0 0 0 0 0 -9.313225746154785e-10,1e0 -9.313225746154785e-10,1e0 +jacobi_cn I3F61 3000 4.997593458312229e-1 4.997593458312229e-1 0 0 0 0 0 -1.0935669199253029e0,4.5981687717761727e-1 -1.0935669199253029e0,4.5981687717761727e-1 +jacobi_cn I128F0 3000 4.988397698993623e-1 4.988397698993623e-1 0 0 0 0 0 -3.2451855365842673e32,1e0 -3.2451855365842673e32,1e0 +jacobi_cn I127F1 3000 4.9854846993423024e-1 4.9854846993423024e-1 0 0 0 0 0 -8.915832990786027e36,5e-1 -8.915832990786027e36,5e-1 +jacobi_cn I126F2 3000 4.9993482948172674e-1 4.9993482948172674e-1 0 0 0 0 0 2.8171107233111514e37,2.5e-1 2.8171107233111514e37,2.5e-1 +jacobi_cn I124F4 3000 4.998455176356904e-1 4.998455176356904e-1 0 0 0 0 0 -8.537339572004375e11,6.25e-1 -8.537339572004375e11,6.25e-1 +jacobi_cn I120F8 3000 4.9978876065557265e-1 4.9978876065557265e-1 0 0 0 0 0 1.661534994731145e35,9.140625e-1 1.661534994731145e35,9.140625e-1 +jacobi_cn I112F16 3000 4.997497427813081e-1 4.997497427813081e-1 0 0 0 0 0 1.8377444077359847e33,6.3592529296875e-1 1.8377444077359847e33,6.3592529296875e-1 +jacobi_cn I104F24 3000 4.9999999751647195e-1 4.9999999751647195e-1 0 0 0 0 0 -2.44140625e-4,9.999999403953552e-1 -2.44140625e-4,9.999999403953552e-1 +jacobi_cn I96F32 3000 4.9999999996748135e-1 4.9999999996748135e-1 0 0 0 0 0 1.52587890625e-5,6.418725580442697e-1 1.52587890625e-5,6.418725580442697e-1 +jacobi_cn I88F40 3000 4.9996425230347147e-1 4.9996425230347147e-1 0 0 0 0 0 -1.8014398509481982e16,9.622442570871499e-1 -1.8014398509481982e16,9.622442570871499e-1 +jacobi_cn I80F48 3000 4.9999999999999944e-1 4.9999999999999944e-1 0 0 0 0 0 5.960464477539063e-8,5.754082872328503e-1 5.960464477539063e-8,5.754082872328503e-1 +jacobi_cn I72F56 3000 5e-1 5e-1 0 0 0 0 0 -3.725290298461914e-9,1.4159265358979325e-1 -3.725290298461914e-9,1.4159265358979325e-1 +jacobi_cn I68F60 3000 5e-1 5e-1 0 0 0 0 0 9.313225746154785e-10,1.3577832886775842e-1 9.313225746154785e-10,1.3577832886775842e-1 +jacobi_cn I67F61 3000 4.995593739782204e-1 4.995593739782204e-1 0 0 0 0 0 7.378697629483821e19,3.8226797416615066e-1 7.378697629483821e19,3.8226797416615066e-1 +jacobi_cn I66F62 3000 4.9992614640190275e-1 4.9992614640190275e-1 0 0 0 0 0 2.18787918603366e0,5.421406105126702e-1 2.18787918603366e0,5.421406105126702e-1 +jacobi_cn I65F63 3000 4.998184119790692e-1 4.998184119790692e-1 0 0 0 0 0 1.5707963267948966e0,9.999999403953552e-1 1.5707963267948966e0,9.999999403953552e-1 +jacobi_cn I64F64 3000 4.99895798092124e-1 4.99895798092124e-1 0 0 0 0 0 -3.1415926535897967e0,7.674910960397094e-1 -3.1415926535897967e0,7.674910960397094e-1 +jacobi_cn I63F65 3000 4.9998903267628253e-1 4.9998903267628253e-1 0 0 0 0 0 -5e-1,4.457963267948966e-1 -5e-1,4.457963267948966e-1 +jacobi_cn I62F66 3000 4.9995225184392705e-1 4.9995225184392705e-1 0 0 0 0 0 1.2364092336184271e18,5e-1 1.2364092336184271e18,5e-1 +jacobi_cn I56F72 3000 4.999999999756901e-1 4.999999999756901e-1 0 0 0 0 0 -3.141592653575241e0,1e0 -3.141592653575241e0,1e0 +jacobi_cn I48F80 3000 4.996357226273518e-1 4.996357226273518e-1 0 0 0 0 0 -1.40737488355328e14,3.349168264112461e-1 -1.40737488355328e14,3.349168264112461e-1 +jacobi_cn I40F88 3000 5e-1 5e-1 0 0 0 0 0 -5.684341886080802e-14,1e0 -5.684341886080802e-14,1e0 +jacobi_cn I32F96 3000 4.995519011860608e-1 4.995519011860608e-1 0 0 0 0 0 -2.147483648e9,3.9953111000630087e-1 -2.147483648e9,3.9953111000630087e-1 +jacobi_cn I24F104 3000 4.9963197414712723e-1 4.9963197414712723e-1 0 0 0 0 0 -1.025556102134265e-12,6.228075092601316e-1 -1.025556102134265e-12,6.228075092601316e-1 +jacobi_cn I16F112 3000 4.999746061515965e-1 4.999746061515965e-1 0 0 0 0 0 2.0475e3,5.783693601740287e-1 2.0475e3,5.783693601740287e-1 +jacobi_cn I10F118 3000 5e-1 5e-1 0 0 0 0 0 -1.734723475976807e-18,1.4159265358979323e-1 -1.734723475976807e-18,1.4159265358979323e-1 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3000 4.9991125922813917e-1 4.9991125922813917e-1 0 0 0 0 0 5.00719008203125e6,7.6171875e-1 5.00719008203125e6,7.6171875e-1 +jacobi_dn I23F9 3000 4.9998026625712166e-1 4.9998026625712166e-1 0 0 0 0 0 -2.097152e6,5e-1 -2.097152e6,5e-1 +jacobi_dn I22F10 3000 4.9975239524115167e-1 4.9975239524115167e-1 0 0 0 0 0 6.888671875e0,3.525390625e-1 6.888671875e0,3.525390625e-1 +jacobi_dn I21F11 3000 4.9986417001139827e-1 4.9986417001139827e-1 0 0 0 0 0 -7.0224609375e0,5.6494140625e-1 -7.0224609375e0,5.6494140625e-1 +jacobi_dn I20F12 3000 4.999015143382246e-1 4.999015143382246e-1 0 0 0 0 0 -2.001953125e-2,6.25e-1 -2.001953125e-2,6.25e-1 +jacobi_dn I19F13 3000 4.997420703796975e-1 4.997420703796975e-1 0 0 0 0 0 -1.5762618017578125e5,5e-1 -1.5762618017578125e5,5e-1 +jacobi_dn I18F14 3000 4.9970006000311057e-1 4.9970006000311057e-1 0 0 0 0 0 -4.71148681640625e1,6.68212890625e-1 -4.71148681640625e1,6.68212890625e-1 +jacobi_dn I17F15 3000 4.997975478617664e-1 4.997975478617664e-1 0 0 0 0 0 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0 1.346751082938722e-1,9.908898098204633e-1,9.908898098204633e-1,2.909998630951094e-15 1.346751082938722e-1,9.908898098204633e-1,9.908898098204633e-1,2.909998630951094e-15 +jacobi_inverse I128F0 2052 4.292036732051034e-1 2.732395447351627e-1 0 0 0 0 0 -1e0,0e0,1e0,1e0 -1e0,0e0,1e0,1e0 +jacobi_inverse I127F1 2765 4.793080073981073e-1 4.793080073981073e-1 0 0 0 0 0 -5e-1,5e-1,5e-1,0e0 -5e-1,5e-1,5e-1,0e0 +jacobi_inverse I126F2 2941 4.7556543596847756e-1 4.0495909920987555e-1 0 0 0 0 0 1e0,2.5e-1,2.5e-1,0e0 5e-1,7.5e-1,7.5e-1,0e0 +jacobi_inverse I124F4 2994 4.9968462997721164e-1 4.7967607764042863e-1 0 0 0 0 0 9.375e-1,3.125e-1,4.375e-1,3.125e-1 -8.125e-1,5.625e-1,8.125e-1,7.5e-1 +jacobi_inverse I120F8 3000 4.9986800307770013e-1 4.995681213334454e-1 0 0 0 0 0 -8.3984375e-1,5.4296875e-1,7.3046875e-1,5.78125e-1 -7.6171875e-1,6.484375e-1,6.5234375e-1,9.765625e-2 +jacobi_inverse I112F16 3000 4.996307788387483e-1 4.994234255092299e-1 0 0 0 0 0 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0 0 9.21831225201551e-2,-9.957420709814535e-1,9.971313371405577e-1,5.707963267948967e-1 1.4790743767177494e-1,9.890012082304905e-1,9.989088038758984e-1,9.488389054231076e-1 +jacobi_inverse I24F104 3000 4.9938392381515245e-1 4.978615978206001e-1 0 0 0 0 0 -9.649186543602064e-1,2.6254902488428433e-1,8.202152416441116e-1,8.053106133388163e-1 6.725084185649316e-2,9.977360995120869e-1,1e0,1e0 +jacobi_inverse I16F112 3000 4.997082809988517e-1 4.997082809988517e-1 0 0 0 0 0 4.128241442112237e-1,9.108107519986085e-1,9.126844819022433e-1,1.4159265359070272e-1 4.128241442112237e-1,9.108107519986085e-1,9.126844819022433e-1,1.4159265359070272e-1 +jacobi_inverse I10F118 3000 4.9998787309505066e-1 4.998192886875412e-1 0 0 0 0 0 -9.185411445155761e-1,3.953253923440987e-1,5.247646186719401e-1,3.757071823947964e-1 2.6319663770162577e-1,9.647422090395751e-1,9.647422090412535e-1,6.837468380032341e-6 +jacobi_inverse I8F120 3000 4.9998590145426897e-1 4.991982491719953e-1 0 0 0 0 0 8.298779641975856e-1,-5.57944947588264e-1,6.518345906974048e-1,4.0611378248075397e-1 -2.533826379941072e-1,9.673661348027164e-1,9.756269415470036e-1,5e-1 +jacobi_inverse I6F122 3000 4.998297653556378e-1 4.996870272850583e-1 0 0 0 0 0 8.552932975037617e-1,-5.181441645383471e-1,5.865889488170692e-1,3.215020871931399e-1 6.895206251207262e-1,7.24266047480567e-1,7.24266047480567e-1,5.684341886080802e-14 +jacobi_inverse I5F123 3000 4.998911210904352e-1 4.998911210904352e-1 0 0 0 0 0 3.178247280527427e-1,9.481494830659352e-1,9.482015014300175e-1,3.125e-2 3.178247280527427e-1,9.481494830659352e-1,9.482015014300175e-1,3.125e-2 +jacobi_inverse I4F124 3000 4.996137438964504e-1 4.996137438964504e-1 0 0 0 0 0 -2.0258091507450832e-1,9.792655272435433e-1,1e0,1e0 -2.0258091507450832e-1,9.792655272435433e-1,1e0,1e0 +jacobi_inverse I3F125 3000 4.9995445356239804e-1 4.9995445356239804e-1 0 0 0 0 0 7.562569902904006e-1,6.542746859209095e-1,7.838451643685994e-1,5.707963267948967e-1 7.562569902904006e-1,6.542746859209095e-1,7.838451643685994e-1,5.707963267948967e-1 +carlson_rf I8F0 2958 4.9972991769602615e-1 4.9972991769602615e-1 0 0 0 0 0 2e1,1e0,1e0 2e1,1e0,1e0 +carlson_rf I7F1 2978 4.9996700379622644e-1 4.9996700379622644e-1 0 0 0 0 0 4e0,8.5e0,5.75e1 4e0,8.5e0,5.75e1 +carlson_rf I6F2 2983 4.9957598892072996e-1 4.9957598892072996e-1 0 0 0 0 0 2.5e-1,1.5e0,1.15e1 2.5e-1,1.5e0,1.15e1 +carlson_rf I5F3 2991 4.999746505194073e-1 4.999746505194073e-1 0 0 0 0 0 3.125e0,2.875e0,3.5e0 3.125e0,2.875e0,3.5e0 +carlson_rf I4F4 2993 4.9995040462843937e-1 4.9995040462843937e-1 0 0 0 0 0 1e0,6.875e-1,6.125e0 1e0,6.875e-1,6.125e0 +carlson_rf I3F5 2987 4.998366830860317e-1 4.998366830860317e-1 0 0 0 0 0 1.09375e0,1.375e0,1.625e0 1.09375e0,1.375e0,1.625e0 +carlson_rf I16F0 2970 5e-1 5e-1 0 0 0 0 0 4e0,4e0,4e0 4e0,4e0,4e0 +carlson_rf I15F1 2989 4.9940788248251444e-1 4.9940788248251444e-1 0 0 0 0 0 3.5e0,1.815e2,0e0 3.5e0,1.815e2,0e0 +carlson_rf I14F2 2999 4.9936948389990454e-1 4.9936948389990454e-1 0 0 0 0 0 2.9675e2,5.5e0,3.7e1 2.9675e2,5.5e0,3.7e1 +carlson_rf I13F3 2998 4.999688864951705e-1 4.999688864951705e-1 0 0 0 0 0 4.095875e3,8e0,3.5e0 4.095875e3,8e0,3.5e0 +carlson_rf I12F4 2997 4.997802487468177e-1 4.997802487468177e-1 0 0 0 0 0 3.625e0,7.625e0,1.3633125e3 3.625e0,7.625e0,1.3633125e3 +carlson_rf I11F5 2997 4.998708127078801e-1 4.998708127078801e-1 0 0 0 0 0 1.215e2,6.75e0,7.5746875e2 1.215e2,6.75e0,7.5746875e2 +carlson_rf I10F6 2994 4.99860567261879e-1 4.99860567261879e-1 0 0 0 0 0 1.015625e0,3.18125e2,6.25e-2 1.015625e0,3.18125e2,6.25e-2 +carlson_rf I9F7 2997 4.9973775683031435e-1 4.9973775683031435e-1 0 0 0 0 0 4.71875e0,2.20078125e1,6.34375e0 4.71875e0,2.20078125e1,6.34375e0 +carlson_rf I8F8 2994 4.997575938974109e-1 4.997575938974109e-1 0 0 0 0 0 3.9953125e1,7.13671875e0,2.1484375e0 3.9953125e1,7.13671875e0,2.1484375e0 +carlson_rf I7F9 2996 4.9992151361573167e-1 4.9992151361573167e-1 0 0 0 0 0 3.475e1,1.89453125e0,2e0 3.475e1,1.89453125e0,2e0 +carlson_rf I6F10 2994 4.9995322879039983e-1 4.9995322879039983e-1 0 0 0 0 0 2.44091796875e1,8.86328125e0,1.68681640625e1 2.44091796875e1,8.86328125e0,1.68681640625e1 +carlson_rf I5F11 2998 4.9992132348375196e-1 4.9992132348375196e-1 0 0 0 0 0 3.62646484375e0,6.2353515625e-1,1.7060546875e0 3.62646484375e0,6.2353515625e-1,1.7060546875e0 +carlson_rf I4F12 2994 4.996709591732784e-1 4.992860010986221e-1 0 0 0 0 0 2.9296875e-3,3.14111328125e0,5.0048828125e-1 7.999755859375e0,1.6845703125e-1,7.941162109375e0 +carlson_rf I3F13 2997 4.9985985940650934e-1 4.9985985940650934e-1 0 0 0 0 0 1.5709228515625e0,3.1414794921875e0,7.227783203125e-1 1.5709228515625e0,3.1414794921875e0,7.227783203125e-1 +carlson_rf I32F0 2979 4.985231403532998e-1 4.985231403532998e-1 0 0 0 0 0 1e0,2e0,1.5e1 1e0,2e0,1.5e1 +carlson_rf I31F1 2994 4.993563125070062e-1 4.993563125070062e-1 0 0 0 0 0 5e-1,1.76e2,1.5e0 5e-1,1.76e2,1.5e0 +carlson_rf I30F2 2999 4.992510497688701e-1 4.992510497688701e-1 0 0 0 0 0 7.7e2,6.75e0,7.5e-1 7.7e2,6.75e0,7.5e-1 +carlson_rf I29F3 2998 4.993952164718358e-1 4.993952164718358e-1 0 0 0 0 0 2.625e0,0e0,3e0 2.625e0,0e0,3e0 +carlson_rf I28F4 2998 4.9963776648991387e-1 4.9963776648991387e-1 0 0 0 0 0 6.25e-1,6.8125e0,3.0625e0 6.25e-1,6.8125e0,3.0625e0 +carlson_rf I27F5 2997 4.994441740430095e-1 4.994441740430095e-1 0 0 0 0 0 1.96875e0,5.40625e0,1.03125e0 1.96875e0,5.40625e0,1.03125e0 +carlson_rf I26F6 2999 4.9986097872370194e-1 4.9986097872370194e-1 0 0 0 0 0 1.96875e0,1.6875e0,2.51315625e3 1.96875e0,1.6875e0,2.51315625e3 +carlson_rf I25F7 2998 4.9991216952462797e-1 4.9991216952462797e-1 0 0 0 0 0 2.800607109375e5,9.921875e-1,8.48530546875e4 2.800607109375e5,9.921875e-1,8.48530546875e4 +carlson_rf I24F8 2997 4.9994014257343605e-1 4.9994014257343605e-1 0 0 0 0 0 3.171875e0,6.7143633984375e6,4.097e3 3.171875e0,6.7143633984375e6,4.097e3 +carlson_rf I23F9 2998 4.9998245028425425e-1 4.9998245028425425e-1 0 0 0 0 0 3.182247302734375e6,1.474609375e0,2.075347451171875e6 3.182247302734375e6,1.474609375e0,2.075347451171875e6 +carlson_rf I22F10 2998 4.994749629379454e-1 4.994749629379454e-1 0 0 0 0 0 6.25e-1,1.0009765625e0,7.814990234375e2 6.25e-1,1.0009765625e0,7.814990234375e2 +carlson_rf I21F11 3000 4.999772224241277e-1 4.999772224241277e-1 0 0 0 0 0 2.054345703125e1,5.56553662109375e4,7.099147241210938e5 2.054345703125e1,5.56553662109375e4,7.099147241210938e5 +carlson_rf I20F12 2999 4.998018251250978e-1 4.998018251250978e-1 0 0 0 0 0 1.6259765625e-1,5.242879997558594e5,5e-1 1.6259765625e-1,5.242879997558594e5,5e-1 +carlson_rf I19F13 2999 4.998786457850934e-1 4.998786457850934e-1 0 0 0 0 0 1.682373046875e0,2.88525390625e0,1.3187666821289063e5 1.682373046875e0,2.88525390625e0,1.3187666821289063e5 +carlson_rf I18F14 2999 4.9999027460346096e-1 4.9999027460346096e-1 0 0 0 0 0 3.1388021545410156e4,3.2237060546875e1,1.2450294091796875e5 3.1388021545410156e4,3.2237060546875e1,1.2450294091796875e5 +carlson_rf I17F15 3000 4.999215946793705e-1 4.999215946793705e-1 0 0 0 0 0 2e0,3.948492431640625e1,1.0078125e0 2e0,3.948492431640625e1,1.0078125e0 +carlson_rf I16F16 2999 4.9991078307794373e-1 4.9991078307794373e-1 0 0 0 0 0 1.9692346481323242e4,3.14154052734375e0,3.1415863037109375e0 1.9692346481323242e4,3.14154052734375e0,3.1415863037109375e0 +carlson_rf I15F17 3000 4.9989577207486946e-1 4.9989577207486946e-1 0 0 0 0 0 2.416429573059082e3,7.415504455566406e0,1.57086181640625e0 2.416429573059082e3,7.415504455566406e0,1.57086181640625e0 +carlson_rf I14F18 2998 4.9999274769013335e-1 4.9999274769013335e-1 0 0 0 0 0 1.5707931518554688e0,1.6858231964111328e3,8.634567260742188e-2 1.5707931518554688e0,1.6858231964111328e3,8.634567260742188e-2 +carlson_rf I13F19 2998 4.999557037667286e-1 4.999557037667286e-1 0 0 0 0 0 2.333692975997925e3,1.711500249862671e3,2.845592498779297e0 2.333692975997925e3,1.711500249862671e3,2.845592498779297e0 +carlson_rf I12F20 2998 4.998410751490394e-1 4.9975972190879414e-1 0 0 0 0 0 1.7833709716796875e-4,4.525737762451172e-1,8.182525634765625e-4 4.186380386352539e0,5.000009536743164e-1,3.016402244567871e0 +carlson_rf I11F21 2997 4.9977981082834466e-1 4.9977981082834466e-1 0 0 0 0 0 1.5708112716674805e0,4.202661991119385e0,3.018379211425781e-4 1.5708112716674805e0,4.202661991119385e0,3.018379211425781e-4 +carlson_rf I10F22 2999 4.999721662608073e-1 4.999721662608073e-1 0 0 0 0 0 3.616783618927002e0,1.8068657755851746e2,2.5109855341911316e2 3.616783618927002e0,1.8068657755851746e2,2.5109855341911316e2 +carlson_rf I9F23 3000 4.998111118428309e-1 4.998111118428309e-1 0 0 0 0 0 7.359168291091919e0,3.1435457468032837e0,1.0053572738170624e2 7.359168291091919e0,3.1435457468032837e0,1.0053572738170624e2 +carlson_rf I8F24 2998 4.999619887758666e-1 4.999619887758666e-1 0 0 0 0 0 1.0497383517026901e2,1.969919925928116e1,6.774324417114258e0 1.0497383517026901e2,1.969919925928116e1,6.774324417114258e0 +carlson_rf I7F25 2999 4.9990544013704463e-1 4.9990544013704463e-1 0 0 0 0 0 2.086162567138672e-7,6.3999999940395355e1,3.1415927410125732e0 2.086162567138672e-7,6.3999999940395355e1,3.1415927410125732e0 +carlson_rf I6F26 3000 4.9998234467443325e-1 4.9998234467443325e-1 0 0 0 0 0 1.0000001043081284e0,2.0001220703125e0,3.8289095610380173e0 1.0000001043081284e0,2.0001220703125e0,3.8289095610380173e0 +carlson_rf I5F27 2999 4.998250467800235e-1 4.998250467800235e-1 0 0 0 0 0 3.137686401605606e0,2.5472565665841103e0,5.00000037252903e-1 3.137686401605606e0,2.5472565665841103e0,5.00000037252903e-1 +carlson_rf I4F28 2999 4.999857983053982e-1 4.9985520399464334e-1 0 0 0 0 0 1.9999989345669746e0,5.719338357448578e-2,3.107573390007019e-1 2.64462723210454e0,4.138377033174038e0,1.7223030738532543e0 +carlson_rf I3F29 2999 4.9970358052139047e-1 4.9970358052139047e-1 0 0 0 0 0 3.077735338360071e0,9.84375e-1,2e0 3.077735338360071e0,9.84375e-1,2e0 +carlson_rf I64F0 2984 4.983945551802719e-1 4.983945551802719e-1 0 0 0 0 0 8e0,1e0,5e0 8e0,1e0,5e0 +carlson_rf I63F1 2998 4.950569713045055e-1 4.950569713045055e-1 0 0 0 0 0 5.5e0,0e0,3.5e0 5.5e0,0e0,3.5e0 +carlson_rf I62F2 2997 4.99336386594189e-1 4.99336386594189e-1 0 0 0 0 0 3.5e0,2.5e-1,1.022e3 3.5e0,2.5e-1,1.022e3 +carlson_rf I61F3 3000 4.997019510347072e-1 4.997019510347072e-1 0 0 0 0 0 2.625e0,1.125e0,7.5e0 2.625e0,1.125e0,7.5e0 +carlson_rf I60F4 2999 4.990486379218594e-1 4.990486379218594e-1 0 0 0 0 0 4.125e0,1.5625e0,1.76825e3 4.125e0,1.5625e0,1.76825e3 +carlson_rf I56F8 2998 4.9985916359598614e-1 4.9985916359598614e-1 0 0 0 0 0 6.2e1,1.913155078125e4,3.13671875e0 6.2e1,1.913155078125e4,3.13671875e0 +carlson_rf I52F12 3000 4.9964250169819496e-1 4.9964250169819496e-1 0 0 0 0 0 4.59716796875e0,1.1626477783203125e4,1.048578e6 4.59716796875e0,1.1626477783203125e4,1.048578e6 +carlson_rf I48F16 3000 4.9998793141052295e-1 4.9998793141052295e-1 0 0 0 0 0 6.179351806640625e-1,4.83135986328125e0,3.1413421630859375e0 6.179351806640625e-1,4.83135986328125e0,3.1413421630859375e0 +carlson_rf I44F20 2999 4.9975273257250696e-1 4.9975273257250696e-1 0 0 0 0 0 2e0,1.0001087188720703e0,4.436129650525205e12 2e0,1.0001087188720703e0,4.436129650525205e12 +carlson_rf I40F24 2998 4.9969938403845715e-1 4.9969938403845715e-1 0 0 0 0 0 4.627074301242828e0,1.9488494143432227e11,3.9923762327607434e11 4.627074301242828e0,1.9488494143432227e11,3.9923762327607434e11 +carlson_rf I36F28 3000 4.997885983566926e-1 4.997885983566926e-1 0 0 0 0 0 1.0485744292036742e6,1.9343830645084381e-3,6.486710350960493e0 1.0485744292036742e6,1.9343830645084381e-3,6.486710350960493e0 +carlson_rf I33F31 2999 4.9996767773999456e-1 4.9996767773999456e-1 0 0 0 0 0 4.656612873077393e-10,3.8920451025442057e9,4.82291003363207e0 4.656612873077393e-10,3.8920451025442057e9,4.82291003363207e0 +carlson_rf I32F32 2999 4.997450756426803e-1 4.997450756426803e-1 0 0 0 0 0 2.6001155998092145e0,2.147483648e9,1.5707963267341256e0 2.6001155998092145e0,2.147483648e9,1.5707963267341256e0 +carlson_rf I31F33 3000 4.998325081604838e-1 4.998325081604838e-1 0 0 0 0 0 1.5707963266177103e0,1.0489020496606827e-7,8.43138788002245e8 1.5707963266177103e0,1.0489020496606827e-7,8.43138788002245e8 +carlson_rf I28F36 3000 4.9999053285409245e-1 4.9999053285409245e-1 0 0 0 0 0 7.4355815664166585e0,2.4561484635341913e0,1.2271276941279376e8 7.4355815664166585e0,2.4561484635341913e0,1.2271276941279376e8 +carlson_rf I24F40 2999 4.9997756220289086e-1 4.9997756220289086e-1 0 0 0 0 0 5.681607428749885e0,5e-1,1.6448420905882881e6 5.681607428749885e0,5e-1,1.6448420905882881e6 +carlson_rf I20F44 3000 4.999649966337914e-1 4.999649966337914e-1 0 0 0 0 0 8.08826495771138e3,5.24288e5,3.1415926535898393e0 8.08826495771138e3,5.24288e5,3.1415926535898393e0 +carlson_rf I16F48 3000 4.998605925550789e-1 4.998605925550789e-1 0 0 0 0 0 0e0,1.688313786821642e2,2.436569432651261e0 0e0,1.688313786821642e2,2.436569432651261e0 +carlson_rf I12F52 3000 4.9980663663439046e-1 4.9980663663439046e-1 0 0 0 0 0 2.785679067362064e0,7.059162301989899e0,5.1299080131385875e0 2.785679067362064e0,7.059162301989899e0,5.1299080131385875e0 +carlson_rf I8F56 2999 4.997143957474004e-1 4.997143957474004e-1 0 0 0 0 0 7.258107645013102e0,1e0,8.562466615179262e1 7.258107645013102e0,1e0,8.562466615179262e1 +carlson_rf I6F58 2999 4.9968179202067137e-1 4.9968179202067137e-1 0 0 0 0 0 2e0,1.4168730627266271e1,5e-1 2e0,1.4168730627266271e1,5e-1 +carlson_rf I5F59 2999 4.9987577573587066e-1 4.9987577573587066e-1 0 0 0 0 0 2e0,3.1415926545211157e0,1.6332963267948966e0 2e0,3.1415926545211157e0,1.6332963267948966e0 +carlson_rf I4F60 3000 4.998460748807767e-1 4.998460748807767e-1 0 0 0 0 0 5.687174882312138e0,4.432200715505532e0,3.1415926535898095e0 5.687174882312138e0,4.432200715505532e0,3.1415926535898095e0 +carlson_rf I3F61 2999 4.9980589279819937e-1 4.9980589279819937e-1 0 0 0 0 0 3.141592653589793e0,2e0,1.6918931533549397e-13 3.141592653589793e0,2e0,1.6918931533549397e-13 +carlson_rf I128F0 2992 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4.997484056364464e-1 0 0 0 0 0 3.1415926539339125e0,1.7411616457432131e6,1.9940827321261168e0 3.1415926539339125e0,1.7411616457432131e6,1.9940827321261168e0 +carlson_rf I88F40 3000 4.9955263623202384e-1 4.9955263623202384e-1 0 0 0 0 0 2.6614778242733936e23,7.7695353775770855e0,1.3282051262446503e8 2.6614778242733936e23,7.7695353775770855e0,1.3282051262446503e8 +carlson_rf I80F48 3000 4.999626900637425e-1 4.999572196522125e-1 0 0 0 0 0 5.000000000000071e-1,5.70744188029721e-1,1.9223601683594467e0 3.5820805027992275e23,3.141592653589843e0,2.68435455e8 +carlson_rf I72F56 3000 4.99763676584246e-1 4.99763676584246e-1 0 0 0 0 0 1.117543750161685e11,4.491508449381131e19,3.141592653589793e0 1.117543750161685e11,4.491508449381131e19,3.141592653589793e0 +carlson_rf I68F60 3000 4.996868813868521e-1 4.996868813868521e-1 0 0 0 0 0 3.969047313034935e-15,4.294967294e9,7.69288503790618e0 3.969047313034935e-15,4.294967294e9,7.69288503790618e0 +carlson_rf I67F61 3000 4.998840062934761e-1 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3.141592653589793e0,4.639478807383639e-16,2.3642455080740345e6 +carlson_rf I16F112 3000 4.998365365109626e-1 4.998365365109626e-1 0 0 0 0 0 1.3129531169708207e4,4.042838330126295e-1,6.191939284632152e-13 1.3129531169708207e4,4.042838330126295e-1,6.191939284632152e-13 +carlson_rf I10F118 3000 4.9971135314621234e-1 4.9971135314621234e-1 0 0 0 0 0 1e0,9.264095616529322e1,2.3476416093469565e-2 1e0,9.264095616529322e1,2.3476416093469565e-2 +carlson_rf I8F120 3000 4.999750317215469e-1 4.999750317215469e-1 0 0 0 0 0 3.009265538105056e-36,1e0,1.0659546682047483e2 3.009265538105056e-36,1e0,1.0659546682047483e2 +carlson_rf I6F122 3000 4.999592297076516e-1 4.999592297076516e-1 0 0 0 0 0 1.8544122230022566e1,1.3045335513726753e1,6.941457267075526e0 1.8544122230022566e1,1.3045335513726753e1,6.941457267075526e0 +carlson_rf I5F123 2999 4.9997233273162095e-1 4.9982394970799826e-1 0 0 0 0 0 2.022161181180548e0,2e0,3.925913020096984e-4 2.765393352073656e-22,6.545852985446767e0,1.1647141598692182e1 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-2.174072265625e-1,-2.1893310546875e0,-1.4488525390625e0 +carlson_rf_roots I32F0 2976 5e-1 5e-1 0 0 0 0 0 -2e0,2e0,2e0 -2e0,2e0,2e0 +carlson_rf_roots I31F1 2993 4.9947522307507586e-1 4.9947522307507586e-1 0 0 0 0 0 -5e-1,7e0,5e0 -5e-1,7e0,5e0 +carlson_rf_roots I30F2 2998 4.98255494435989e-1 4.98255494435989e-1 0 0 0 0 0 2.925e1,-2.5e0,5e-1 2.925e1,-2.5e0,5e-1 +carlson_rf_roots I29F3 2999 4.9986701819065754e-1 4.9986701819065754e-1 0 0 0 0 0 -5.1125e1,-8e0,5e-1 -5.1125e1,-8e0,5e-1 +carlson_rf_roots I28F4 2999 4.9873541603569327e-1 4.9873541603569327e-1 0 0 0 0 0 -4.6875e0,-5.625e-1,-3.125e-1 -4.6875e0,-5.625e-1,-3.125e-1 +carlson_rf_roots I27F5 2999 4.9882429317992616e-1 4.9882429317992616e-1 0 0 0 0 0 1.5625e0,-3.15625e0,1.5625e0 1.5625e0,-3.15625e0,1.5625e0 +carlson_rf_roots I26F6 2999 4.9955927934287664e-1 4.9955927934287664e-1 0 0 0 0 0 9.375e-1,7.5e-1,-1.90625e0 9.375e-1,7.5e-1,-1.90625e0 +carlson_rf_roots I25F7 3000 4.998123518570109e-1 4.998123518570109e-1 0 0 0 0 0 4.9921875e0,2.405703125e2,-3.78125e0 4.9921875e0,2.405703125e2,-3.78125e0 +carlson_rf_roots I24F8 3000 4.9915798239429776e-1 4.9915798239429776e-1 0 0 0 0 0 3.37890625e0,-3.8203125e0,8.5546875e-1 3.37890625e0,-3.8203125e0,8.5546875e-1 +carlson_rf_roots I23F9 2999 4.99651824492574e-1 4.986508297151552e-1 0 0 0 0 0 3.140625e0,-9.765625e-3,-1.46484375e-1 3.138671875e0,4.98046875e-1,4.0234375e-1 +carlson_rf_roots I22F10 2998 4.99212611965192e-1 4.99212611965192e-1 0 0 0 0 0 2.662861328125e2,4.04296875e-1,-1e0 2.662861328125e2,4.04296875e-1,-1e0 +carlson_rf_roots I21F11 2998 4.989887068704895e-1 4.989887068704895e-1 0 0 0 0 0 1.10205078125e0,1.00048828125e0,2.23046875e0 1.10205078125e0,1.00048828125e0,2.23046875e0 +carlson_rf_roots I20F12 2999 4.9977250666823936e-1 4.9977250666823936e-1 0 0 0 0 0 9.99755859375e-1,4.638671875e-3,1.0617353442382813e5 9.99755859375e-1,4.638671875e-3,1.0617353442382813e5 +carlson_rf_roots I19F13 3000 4.999733376115884e-1 4.999733376115884e-1 0 0 0 0 0 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-3.485006809234619e0,6.552410125732422e-2,-9.978508949279785e-1,9.978528022766113e-1,2.9152631759643555e-2 -7.106362581253052e0,4.322676658630371e-1,9.017453193664551e-1,9.020481109619141e-1,5.40621280670166e-2 +ellip_pi I9F23 2090 4.9956335832932974e-1 4.9925706862367936e-1 0 0 0 0 0 -3.124517560005188e0,-9.144699573516846e-2,-9.958099126815796e-1,9.968594312667847e-1,5.000730752944946e-1 2.9996331930160522e0,-1.0222005844116211e-1,9.947618246078491e-1,9.948909282684326e-1,1.56815767288208e-1 +ellip_pi I8F24 2096 4.995110007889278e-1 4.9941350453321887e-1 0 0 0 0 0 -5.00244140625e-1,-9.742416143417358e-1,-2.2550660371780396e-1,2.643524408340454e-1,1.4159274101257324e-1 -1.3183836996555328e1,-6.52039110660553e-1,-7.581853270530701e-1,8.87270987033844e-1,7.068195343017578e-1 +ellip_pi I7F25 2135 4.9974023140229784e-1 4.987673231750528e-1 0 0 0 0 0 -1.1400580406188965e-2,6.96049839258194e-1,-7.179934680461884e-1,7.179934680461884e-1,7.927417755126953e-6 -6.567962646484375e0,-9.989935457706451e-1,4.485437273979187e-2,5.016281306743622e-1,5.001220703125e-1 +ellip_pi I6F26 2183 4.9984396267467546e-1 4.997726621184999e-1 0 0 0 0 0 -1e0,9.927268475294113e-1,-1.2038859724998474e-1,9.232155829668045e-1,9.220386445522308e-1 1.615232303738594e0,-1.5813376009464264e-1,9.874176979064941e-1,9.991360455751419e-1,9.648474007844925e-1 +ellip_pi I5F27 2219 4.997142085689632e-1 4.997142085689632e-1 0 0 0 0 0 3.7997961044311523e-7,-3.9250805974006653e-1,9.19748567044735e-1,9.888035506010056e-1,9.249131679534912e-1 3.7997961044311523e-7,-3.9250805974006653e-1,9.19748567044735e-1,9.888035506010056e-1,9.249131679534912e-1 +ellip_pi I4F28 2214 4.998471790638445e-1 4.998471790638445e-1 0 0 0 0 0 9.04873013496399e-6,-4.447830952703953e-1,8.95638320595026e-1,9.228358455002308e-1,5.000000037252903e-1 9.04873013496399e-6,-4.447830952703953e-1,8.95638320595026e-1,9.228358455002308e-1,5.000000037252903e-1 +ellip_pi I3F29 2374 4.9954935571401593e-1 4.9954935571401593e-1 0 0 0 0 0 1e0,4.078817442059517e-1,9.130347650498152e-1,9.422507230192423e-1,5.707963239401579e-1 1e0,4.078817442059517e-1,9.130347650498152e-1,9.422507230192423e-1,5.707963239401579e-1 +ellip_pi I64F0 1332 4.8558719900945424e-1 4.8558719900945424e-1 0 0 0 0 0 -5e0,1e0,1e0,1e0,1e0 -5e0,1e0,1e0,1e0,1e0 +ellip_pi I63F1 1609 4.9096537257463074e-1 4.9096537257463074e-1 0 0 0 0 0 -6.25e1,-5e-1,-1e0,1e0,5e-1 -6.25e1,-5e-1,-1e0,1e0,5e-1 +ellip_pi I62F2 1669 4.998445303434407e-1 4.998445303434407e-1 0 0 0 0 0 -6.5e0,7.5e-1,-7.5e-1,7.5e-1,5e-1 -6.5e0,7.5e-1,-7.5e-1,7.5e-1,5e-1 +ellip_pi I61F3 1725 4.987693265681202e-1 4.987693265681202e-1 0 0 0 0 0 2e0,-3.75e-1,8.75e-1,1e0,7.5e-1 2e0,-3.75e-1,8.75e-1,1e0,7.5e-1 +ellip_pi I60F4 1699 4.99688065798457e-1 4.99688065798457e-1 0 0 0 0 0 -2.735e2,7.5e-1,6.25e-1,6.25e-1,6.25e-2 -2.735e2,7.5e-1,6.25e-1,6.25e-1,6.25e-2 +ellip_pi I56F8 1793 4.9987775213586305e-1 4.9987775213586305e-1 0 0 0 0 0 5.859375e-1,-6.171875e-1,7.8515625e-1,7.890625e-1,9.375e-2 5.859375e-1,-6.171875e-1,7.8515625e-1,7.890625e-1,9.375e-2 +ellip_pi I52F12 1802 4.999655206700637e-1 4.999655206700637e-1 0 0 0 0 0 9.765625e-4,-1.9921875e-1,9.7998046875e-1,1e0,9.99755859375e-1 9.765625e-4,-1.9921875e-1,9.7998046875e-1,1e0,9.99755859375e-1 +ellip_pi I48F16 1837 4.9970344040779835e-1 4.9970344040779835e-1 0 0 0 0 0 -8e0,-9.99359130859375e-1,-3.58123779296875e-2,1e0,1e0 -8e0,-9.99359130859375e-1,-3.58123779296875e-2,1e0,1e0 +ellip_pi I44F20 1879 4.990037669185578e-1 4.990037669185578e-1 0 0 0 0 0 2.000011444091797e0,7.716083526611328e-2,9.970188140869141e-1,9.970283508300781e-1,5.773448944091797e-2 2.000011444091797e0,7.716083526611328e-2,9.970188140869141e-1,9.970283508300781e-1,5.773448944091797e-2 +ellip_pi I40F24 1881 4.9911226875889686e-1 4.987531591046641e-1 0 0 0 0 0 4.84375e-1,7.541393041610718e-1,-6.567144989967346e-1,1e0,1e0 -3.3e1,1.7535531520843506e-1,-9.845052361488342e-1,9.848183393478394e-1,1.4161783456802368e-1 +ellip_pi I36F28 1964 4.9999462585108767e-1 4.9999462585108767e-1 0 0 0 0 0 -2.0000000037252903e0,4.953559525310993e-1,8.686900921165943e-1,9.999999962747097e-1,9.999999813735485e-1 -2.0000000037252903e0,4.953559525310993e-1,8.686900921165943e-1,9.999999962747097e-1,9.999999813735485e-1 +ellip_pi I33F31 1931 4.9971907263738524e-1 4.9938270159668036e-1 0 0 0 0 0 2.37921173684299e-2,9.90813591517508e-1,1.3523470889776945e-1,9.999999995343387e-1,9.999999995343387e-1 -2.1397420559134185e8,9.644318772479892e-1,-2.6433152332901955e-1,2.6433152332901955e-1,4.656612873077393e-10 +ellip_pi I32F32 1972 4.9998239800695665e-1 4.9998239800695665e-1 0 0 0 0 0 -1.3190974922974412e9,-1.8265719432383776e-1,-9.831766623537987e-1,9.852259871549904e-1,3.4771672263741493e-1 -1.3190974922974412e9,-1.8265719432383776e-1,-9.831766623537987e-1,9.852259871549904e-1,3.4771672263741493e-1 +ellip_pi I31F33 1980 4.9982946004673223e-1 4.997221084284038e-1 0 0 0 0 0 9.546539664734155e-2,-1.9491926324553788e-1,-9.808192906202748e-1,1e0,9.999999998835847e-1 7.974542154697701e-1,-4.2914402473252267e-1,9.03236074373126e-1,9.99999999650754e-1,9.999999981373549e-1 +ellip_pi I28F36 2016 4.9990117039354415e-1 4.9990117039354415e-1 0 0 0 0 0 -5.287106026589754e0,-9.588365816453006e-1,2.839584647299489e-1,2.839584647299489e-1,8.294591680169106e-10 -5.287106026589754e0,-9.588365816453006e-1,2.839584647299489e-1,2.839584647299489e-1,8.294591680169106e-10 +ellip_pi I24F40 1984 4.999146260637362e-1 4.999146260637362e-1 0 0 0 0 0 -5.045225430644289e5,8.062442058890156e-1,-5.915828601891917e-1,6.145728729388793e-1,2.065402470771005e-1 -5.045225430644289e5,8.062442058890156e-1,-5.915828601891917e-1,6.145728729388793e-1,2.065402470771005e-1 +ellip_pi I20F44 2032 4.99972592844543e-1 4.99972592844543e-1 0 0 0 0 0 -5.24288e5,-2.395031391809539e-1,9.70895589815143e-1,9.782528736179756e-1,5e-1 -5.24288e5,-2.395031391809539e-1,9.70895589815143e-1,9.782528736179756e-1,5e-1 +ellip_pi I16F48 2069 4.9935418383187385e-1 4.9935418383187385e-1 0 0 0 0 0 -3.3518868011148655e0,7.046555851305669e-2,9.975142129631251e-1,9.975142129631251e-1,7.105427357601002e-15 -3.3518868011148655e0,7.046555851305669e-2,9.975142129631251e-1,9.975142129631251e-1,7.105427357601002e-15 +ellip_pi I12F52 2079 4.9956108460079807e-1 4.990685788430886e-1 0 0 0 0 0 6.8833827526759706e-15,-5.95048434747778e-1,-8.036898408616469e-1,8.036898408616469e-1,2.4886623872788505e-9 -1.7e1,6.973056780880384e-1,-7.167738773882464e-1,7.224723651187113e-1,1.2987458094790427e-1 +ellip_pi I8F56 2154 4.999475422504715e-1 4.999475422504715e-1 0 0 0 0 0 -7.485325694408982e0,1.0580236299298454e-2,9.999440277334782e-1,9.999440277334782e-1,1.862645149230957e-9 -7.485325694408982e0,1.0580236299298454e-2,9.999440277334782e-1,9.999440277334782e-1,1.862645149230957e-9 +ellip_pi I6F58 2146 4.9960257233162736e-1 4.9844416632608546e-1 0 0 0 0 0 -7.533797222394112e-1,-3.215576522631944e-1,-9.468900022024641e-1,9.480476522903596e-1,1.456556274107189e-1 1.570781068005834e0,2.623987846120818e-1,9.649595213448605e-1,9.68867432616301e-1,3.312983956400065e-1 +ellip_pi I5F59 2163 4.999892514981251e-1 4.999892514981251e-1 0 0 0 0 0 -9.431861541786546e-12,-6.141788440175229e-1,7.891668692749965e-1,9.192196750855826e-1,7.674641213894897e-1 -9.431861541786546e-12,-6.141788440175229e-1,7.891668692749965e-1,9.192196750855826e-1,7.674641213894897e-1 +ellip_pi I4F60 2236 4.9978036403412723e-1 4.9978036403412723e-1 0 0 0 0 0 -6.0211959878666965e0,9.332024230991743e-1,3.593511340205144e-1,4.6381132692459504e-1,3.14226155838603e-1 -6.0211959878666965e0,9.332024230991743e-1,3.593511340205144e-1,4.6381132692459504e-1,3.14226155838603e-1 +ellip_pi I3F61 2349 4.9999860687937314e-1 4.9999860687937314e-1 0 0 0 0 0 7.591781994720215e-10,2.9635692021874507e-2,9.995607664161217e-1,9.9970389362291e-1,5.707963267948967e-1 7.591781994720215e-10,2.9635692021874507e-2,9.995607664161217e-1,9.9970389362291e-1,5.707963267948967e-1 +ellip_pi I128F0 1291 4.8558719900945424e-1 4.8558719900945424e-1 0 0 0 0 0 -5e0,-1e0,1e0,1e0,0e0 -5e0,-1e0,1e0,1e0,0e0 +ellip_pi I127F1 1603 4.8946983986779563e-1 4.8922713169859755e-1 0 0 0 0 0 -2.5e0,1e0,-5e-1,5e-1,5e-1 -7e0,-1e0,-5e-1,5e-1,5e-1 +ellip_pi I126F2 1655 4.9747235720985256e-1 4.94890485785077e-1 0 0 0 0 0 -1e0,-1e0,-2.5e-1,2.5e-1,2.5e-1 -7.25e0,-2.5e-1,-1e0,1e0,1e0 +ellip_pi I124F4 1661 4.983251236765089e-1 4.983251236765089e-1 0 0 0 0 0 -2.0625e0,5e-1,8.75e-1,8.75e-1,5e-1 -2.0625e0,5e-1,8.75e-1,8.75e-1,5e-1 +ellip_pi I120F8 1741 4.991492525326033e-1 4.96693751590094e-1 0 0 0 0 0 -1e0,2.5e-1,-9.6875e-1,9.765625e-1,5.0390625e-1 -7.93256640625e3,8.3203125e-1,5.546875e-1,9.21875e-1,8.8671875e-1 +ellip_pi I112F16 1710 4.9876565654438615e-1 4.9876565654438615e-1 0 0 0 0 0 2e0,3.505096435546875e-1,9.36553955078125e-1,9.37774658203125e-1,1.357879638671875e-1 2e0,3.505096435546875e-1,9.36553955078125e-1,9.37774658203125e-1,1.357879638671875e-1 +ellip_pi I104F24 1818 4.999001345275441e-1 4.9956252247450905e-1 0 0 0 0 0 -4.500688314437866e-1,9.211909770965576e-1,3.89110803604126e-1,6.857868432998657e-1,6.13020658493042e-1 5.002129077911377e-1,-3.073263168334961e-1,9.516041874885559e-1,9.51606273651123e-1,6.50399923324585e-3 +ellip_pi I96F32 1791 4.9999637182961015e-1 4.9999637182961015e-1 0 0 0 0 0 5.000016591511667e-1,-6.901599511038512e-1,7.236568536609411e-1,9.999886734876782e-1,9.999762205407023e-1 5.000016591511667e-1,-6.901599511038512e-1,7.236568536609411e-1,9.999886734876782e-1,9.999762205407023e-1 +ellip_pi I88F40 1836 4.998381244244722e-1 4.998381244244722e-1 0 0 0 0 0 -2.57e2,-5.089009701569012e-1,8.608250708321066e-1,9.276784879848492e-1,6.794749195505574e-1 -2.57e2,-5.089009701569012e-1,8.608250708321066e-1,9.276784879848492e-1,6.794749195505574e-1 +ellip_pi I80F48 1854 4.9980875947492903e-1 4.9949124074511686e-1 0 0 0 0 0 -5.000000000000107e-1,-9.959039737056408e-1,9.041722821073961e-2,1.7263575139967102e-1,1.476688808050035e-1 5.278482424565389e-1,1.4912926421769868e-1,9.888177094661508e-1,9.895583039400044e-1,2.566739401114333e-1 +ellip_pi I72F56 1899 4.995900413489014e-1 4.995900413489014e-1 0 0 0 0 0 -2.0464292036732052e3,7.78668928380271e-2,9.969637641357639e-1,9.982075564066976e-1,6.397513019930244e-1 -2.0464292036732052e3,7.78668928380271e-2,9.969637641357639e-1,9.982075564066976e-1,6.397513019930244e-1 +ellip_pi I68F60 1913 4.9912060687582877e-1 4.9912060687582877e-1 0 0 0 0 0 -5.151415926535898e2,-2.062127854893159e-1,-9.785071727385228e-1,9.855611964498302e-1,5.707963267948967e-1 -5.151415926535898e2,-2.062127854893159e-1,-9.785071727385228e-1,9.855611964498302e-1,5.707963267948967e-1 +ellip_pi I67F61 1914 4.997966701361423e-1 4.997966701361423e-1 0 0 0 0 0 1.5707963267930776e0,-1.9808306566687658e-1,9.801852371342938e-1,9.801852371342938e-1,3.552713678800501e-15 1.5707963267930776e0,-1.9808306566687658e-1,9.801852371342938e-1,9.801852371342938e-1,3.552713678800501e-15 +ellip_pi I66F62 1939 4.9987658821473296e-1 4.995136935517123e-1 0 0 0 0 0 -1e0,-9.72385578604351e-1,2.3338013309251807e-1,6.021039286634077e-1,5.707963118937355e-1 -3.754336189542669e17,-9.760160568219362e-1,-2.1769854575940373e-1,5.877001649132159e-1,5.59307077346636e-1 +ellip_pi I65F63 1970 4.99506555237103e-1 4.99506555237103e-1 0 0 0 0 0 -4.21810554538496e0,-9.44986213986953e-1,3.27110157859099e-1,4.0647569032777386e-1,2.553383023917188e-1 -4.21810554538496e0,-9.44986213986953e-1,3.27110157859099e-1,4.0647569032777386e-1,2.553383023917188e-1 +ellip_pi I64F64 1957 4.9951992758319547e-1 4.994682431178067e-1 0 0 0 0 0 5.551115123125783e-17,9.529464574515437e-1,3.031386633713244e-1,3.031386633713244e-1,0e0 -2e0,-6.809686124874507e-1,7.323126032009255e-1,9.999999999999737e-1,9.999999999999432e-1 +ellip_pi I63F65 1994 4.999922538050393e-1 4.999922538050393e-1 0 0 0 0 0 -5.242911415926536e5,-9.452814854244032e-1,3.2625590157701945e-1,3.262560075002909e-1,2.7811778517011185e-4 -5.242911415926536e5,-9.452814854244032e-1,3.2625590157701945e-1,3.262560075002909e-1,2.7811778517011185e-4 +ellip_pi I62F66 1962 4.9973840071561026e-1 4.991374962909843e-1 0 0 0 0 0 9.631751610562118e-1,9.9555296105627e-1,9.420351231293277e-2,8.15114072679102e-1,8.132688163495404e-1 -6.659330100365333e0,3.8890266686965963e-1,9.21278847961716e-1,9.26836768620102e-1,2.606032445824013e-1 +ellip_pi I56F72 1948 4.9993228854158306e-1 4.9993228854158306e-1 0 0 0 0 0 -1.0995116277791416e12,9.658549798539638e-1,-2.5908330299596516e-1,9.524186509184502e-1,9.489029727236303e-1 -1.0995116277791416e12,9.658549798539638e-1,-2.5908330299596516e-1,9.524186509184502e-1,9.489029727236303e-1 +ellip_pi I48F80 1994 4.998202608362677e-1 4.998202608362677e-1 0 0 0 0 0 -1.31073e5,-5.378145307140849e-1,-8.430631830134611e-1,1e0,1e0 -1.31073e5,-5.378145307140849e-1,-8.430631830134611e-1,1e0,1e0 +ellip_pi I40F88 2006 4.998965409078107e-1 4.998965409078107e-1 0 0 0 0 0 3.141592653589793e0,2.3296893370922808e-1,9.724841777254709e-1,9.909969251798971e-1,8.183720516867069e-1 3.141592653589793e0,2.3296893370922808e-1,9.724841777254709e-1,9.909969251798971e-1,8.183720516867069e-1 +ellip_pi I32F96 2051 4.994517832286591e-1 4.990972509935114e-1 0 0 0 0 0 1.0788532260054734e-18,8.976459573070596e-1,-4.4071729638203727e-1,5.021549053903646e-1,2.681257284802529e-1 1.0656210392036232e0,2.553308096732164e-1,9.668537519354309e-1,9.668537519354309e-1,1.262177448353619e-29 +ellip_pi I24F104 2097 4.9970569804575105e-1 4.9940383970290164e-1 0 0 0 0 0 -6.367406546234121e-16,-7.466539168435024e-1,-6.652126941529726e-1,6.969905296920963e-1,2.7865302417308185e-1 -2.0539950318401216e-17,-9.207442058632553e-2,9.957521283299837e-1,1e0,1e0 +ellip_pi I16F112 2082 4.998822802545169e-1 4.9839644865756383e-1 0 0 0 0 0 -1.4650254523595392e-22,-9.964135890608606e-1,8.461654409661516e-2,8.461654409661516e-2,6.112260122524238e-20 -1.2308478087160667e4,9.84695740925152e-1,-1.7428223605366663e-1,6.302211451097398e-1,6.150567728929971e-1 +ellip_pi I10F118 2074 4.996942884017847e-1 4.996942884017847e-1 0 0 0 0 0 -1.7516333591877705e2,-6.572306211600036e-1,-7.536895319756244e-1,7.536912659543159e-1,2.459889381965623e-3 -1.7516333591877705e2,-6.572306211600036e-1,-7.536895319756244e-1,7.536912659543159e-1,2.459889381965623e-3 +ellip_pi I8F120 2120 4.998362218511982e-1 4.998362218511982e-1 0 0 0 0 0 1.3574561957435468e-8,-8.281064554094404e-1,5.605708684093498e-1,9.489236845847969e-1,9.24576465107191e-1 1.3574561957435468e-8,-8.281064554094404e-1,5.605708684093498e-1,9.489236845847969e-1,9.24576465107191e-1 +ellip_pi I6F122 2142 4.999246665238786e-1 4.999246665238786e-1 0 0 0 0 0 2.0000000000000036e0,-3.010409022332981e-1,9.536112285321319e-1,9.654174130327883e-1,5e-1 2.0000000000000036e0,-3.010409022332981e-1,9.536112285321319e-1,9.654174130327883e-1,5e-1 +ellip_pi I5F123 2240 4.9985062575862976e-1 4.985274153932832e-1 0 0 0 0 0 2.6767202690907793e-2,8.034426846455732e-1,-5.953821062893258e-1,6.282938898451498e-1,2.4977791078454542e-1 -8.981275522548565e0,-5.736772568984674e-1,8.190814397344441e-1,9.905923711937075e-1,9.711323482384057e-1 +ellip_pi I4F124 2468 4.998691378829337e-1 4.998691378829337e-1 0 0 0 0 0 -8e0,8.271939736110715e-1,-5.619164795781717e-1,1e0,1e0 -8e0,8.271939736110715e-1,-5.619164795781717e-1,1e0,1e0 +ellip_pi I3F125 2538 4.999526806934367e-1 4.999526806934367e-1 0 0 0 0 0 -9.999999999708962e-1,-7.636578864263396e-1,6.456211214782674e-1,7.020675702920016e-1,3.6117215733264485e-1 -9.999999999708962e-1,-7.636578864263396e-1,6.456211214782674e-1,7.020675702920016e-1,3.6117215733264485e-1 +ellip_pi_complete I8F0 205 4.7640122440170113e-1 4.7640122440170113e-1 0 0 0 0 0 -8e0,1e0 -8e0,1e0 +ellip_pi_complete I7F1 959 4.9913851115228675e-1 4.9913851115228675e-1 0 0 0 0 0 -4.5e1,5e-1 -4.5e1,5e-1 +ellip_pi_complete I6F2 1401 4.968395842542096e-1 4.968395842542096e-1 0 0 0 0 0 -2.425e1,2.5e-1 -2.425e1,2.5e-1 +ellip_pi_complete I5F3 1612 4.9983123620301284e-1 4.9983123620301284e-1 0 0 0 0 0 -1.2625e1,8.75e-1 -1.2625e1,8.75e-1 +ellip_pi_complete I4F4 1710 4.9985703107285356e-1 4.9767830944238617e-1 0 0 0 0 0 -1.625e0,4.375e-1 -3.8125e0,5.625e-1 +ellip_pi_complete I3F5 1923 4.9975106438440553e-1 4.9888306199418997e-1 0 0 0 0 0 -2.5e-1,2.8125e-1 -2.75e0,7.5e-1 +ellip_pi_complete I16F0 137 4.7640122440170113e-1 4.7640122440170113e-1 0 0 0 0 0 -8e0,1e0 -8e0,1e0 +ellip_pi_complete I15F1 908 4.962040002580569e-1 4.962040002580569e-1 0 0 0 0 0 -4.45e1,5e-1 -4.45e1,5e-1 +ellip_pi_complete I14F2 1305 4.984056789923573e-1 4.984056789923573e-1 0 0 0 0 0 -1.6425e2,7.5e-1 -1.6425e2,7.5e-1 +ellip_pi_complete I13F3 1445 4.9995721494993534e-1 4.9995721494993534e-1 0 0 0 0 0 -6.51125e2,6.25e-1 -6.51125e2,6.25e-1 +ellip_pi_complete I12F4 1591 4.9985703107285356e-1 4.988084848451859e-1 0 0 0 0 0 -1.625e0,4.375e-1 -5.125e0,4.375e-1 +ellip_pi_complete I11F5 1635 4.996902047207378e-1 4.996902047207378e-1 0 0 0 0 0 -6.25e0,5.625e-1 -6.25e0,5.625e-1 +ellip_pi_complete I10F6 1740 4.999346251640232e-1 4.998069206456182e-1 0 0 0 0 0 -5.15625e-1,2.5e-1 -5.12e2,7.03125e-1 +ellip_pi_complete I9F7 1778 4.998276889060627e-1 4.998276889060627e-1 0 0 0 0 0 -3.71875e0,3.90625e-1 -3.71875e0,3.90625e-1 +ellip_pi_complete I8F8 1784 4.9997396761841495e-1 4.9997396761841495e-1 0 0 0 0 0 -7.18359375e0,1.015625e-1 -7.18359375e0,1.015625e-1 +ellip_pi_complete I7F9 1816 4.99634816306874e-1 4.994496916205812e-1 0 0 0 0 0 -4.3359375e-1,4.4921875e-2 -3.9181640625e1,3.125e-1 +ellip_pi_complete I6F10 1889 4.9996064625373093e-1 4.9889453455787147e-1 0 0 0 0 0 -3.671875e-1,2.890625e-1 -1.1169921875e1,1.142578125e-1 +ellip_pi_complete I5F11 1864 4.999386088682982e-1 4.9959279961264097e-1 0 0 0 0 0 0e0,6.9287109375e-1 -1.26572265625e1,1e0 +ellip_pi_complete I4F12 1981 4.9967501465267244e-1 4.994483759983469e-1 0 0 0 0 0 5e-1,2.44140625e-1 -6.86279296875e0,9.3408203125e-1 +ellip_pi_complete I3F13 2078 4.9992091869648575e-1 4.969908885291913e-1 0 0 0 0 0 -1.220703125e-4,4.99755859375e-1 -2.0001220703125e0,9.998779296875e-1 +ellip_pi_complete I32F0 118 4.7640122440170113e-1 4.7640122440170113e-1 0 0 0 0 0 -8e0,1e0 -8e0,1e0 +ellip_pi_complete I31F1 889 4.901813623675646e-1 4.901813623675646e-1 0 0 0 0 0 -4.35e1,5e-1 -4.35e1,5e-1 +ellip_pi_complete I30F2 1216 4.968395842542096e-1 4.968395842542096e-1 0 0 0 0 0 -2.425e1,2.5e-1 -2.425e1,2.5e-1 +ellip_pi_complete I29F3 1399 4.937709117906958e-1 4.933666924379287e-1 0 0 0 0 0 -5e-1,6.25e-1 -7.9e1,5e-1 +ellip_pi_complete I28F4 1534 4.9985703107285356e-1 4.9928329378980757e-1 0 0 0 0 0 -1.625e0,4.375e-1 -6e0,1e0 +ellip_pi_complete I27F5 1602 4.9989726435521403e-1 4.9989726435521403e-1 0 0 0 0 0 -1.3834375e2,4.375e-1 -1.3834375e2,4.375e-1 +ellip_pi_complete I26F6 1641 4.9962894196807456e-1 4.992565307420273e-1 0 0 0 0 0 -9.21875e-1,4.53125e-1 -5.578125e0,6.71875e-1 +ellip_pi_complete I25F7 1599 4.9971849999695656e-1 4.9928910920204744e-1 0 0 0 0 0 -1.9921875e0,6.25e-2 -6.2890625e0,8.59375e-1 +ellip_pi_complete I24F8 1685 4.9908164419887635e-1 4.9821873032838243e-1 0 0 0 0 0 -6.578125e0,2.34375e-2 -2.3125e0,8.203125e-1 +ellip_pi_complete I23F9 1707 4.99802318381496e-1 4.996598697348113e-1 0 0 0 0 0 -2.001953125e0,9.375e-2 -2.588508337890625e6,1.40625e-1 +ellip_pi_complete I22F10 1700 4.9977812898032925e-1 4.9977812898032925e-1 0 0 0 0 0 -4.404296875e0,7.75390625e-1 -4.404296875e0,7.75390625e-1 +ellip_pi_complete I21F11 1731 4.997018238743259e-1 4.997018238743259e-1 0 0 0 0 0 -2.25777783203125e3,1.9091796875e-1 -2.25777783203125e3,1.9091796875e-1 +ellip_pi_complete I20F12 1703 4.996625978676365e-1 4.99216928429829e-1 0 0 0 0 0 -5e-1,1.3427734375e-2 -4.592235012207031e5,6.875e-1 +ellip_pi_complete I19F13 1779 4.999041497179053e-1 4.999041497179053e-1 0 0 0 0 0 -6.2e1,4.576416015625e-1 -6.2e1,4.576416015625e-1 +ellip_pi_complete I18F14 1802 4.99971830368737e-1 4.99971830368737e-1 0 0 0 0 0 -4.76275634765625e0,1.6595458984375e-1 -4.76275634765625e0,1.6595458984375e-1 +ellip_pi_complete I17F15 1797 4.999949301089274e-1 4.999949301089274e-1 0 0 0 0 0 -7.48773193359375e0,7.786865234375e-1 -7.48773193359375e0,7.786865234375e-1 +ellip_pi_complete I16F16 1745 4.995397505006549e-1 4.981986759542926e-1 0 0 0 0 0 -2.13623046875e-4,9.999847412109375e-1 -7.098236083984375e0,5.93597412109375e-1 +ellip_pi_complete I15F17 1811 4.9999682844305426e-1 4.9991067407552997e-1 0 0 0 0 0 -6.7138671875e-4,3.3160400390625e-1 -4.092506408691406e0,4.455108642578125e-1 +ellip_pi_complete I14F18 1780 4.9987095790483854e-1 4.998422763448703e-1 0 0 0 0 0 0e0,7.782554626464844e-1 -2.5216486473083496e3,3.814697265625e-6 +ellip_pi_complete I13F19 1819 4.97354741231745e-1 4.97354741231745e-1 0 0 0 0 0 -2.8031321029663086e3,6.666049957275391e-1 -2.8031321029663086e3,6.666049957275391e-1 +ellip_pi_complete I12F20 1825 4.9980110663619515e-1 4.9980110663619515e-1 0 0 0 0 0 -5.795724868774414e0,7.899971008300781e-1 -5.795724868774414e0,7.899971008300781e-1 +ellip_pi_complete I11F21 1884 4.999031116078873e-1 4.996571226082123e-1 0 0 0 0 0 4.9988365173339844e-1,4.4727325439453125e-4 -5.221722602844238e0,8.21713924407959e-1 +ellip_pi_complete I10F22 1855 4.999214422289486e-1 4.999214422289486e-1 0 0 0 0 0 -5.12e2,5.327868461608887e-1 -5.12e2,5.327868461608887e-1 +ellip_pi_complete I9F23 1855 4.99913266292631e-1 4.99913266292631e-1 0 0 0 0 0 -3.1415927410125732e0,4.7839319705963135e-1 -3.1415927410125732e0,4.7839319705963135e-1 +ellip_pi_complete I8F24 1901 4.9989407152444393e-1 4.993508427995374e-1 0 0 0 0 0 1.5684521198272705e-1,1.9880825281143188e-1 -5.569103837013245e0,9.999983310699463e-1 +ellip_pi_complete I7F25 1902 4.9978980524919503e-1 4.9978980524919503e-1 0 0 0 0 0 -6.3986877501010895e0,9.999999701976776e-1 -6.3986877501010895e0,9.999999701976776e-1 +ellip_pi_complete I6F26 1921 4.99902530214896e-1 4.9957354543707577e-1 0 0 0 0 0 -3.839565262198448e0,3.7550926208496094e-6 -5.730417683720589e0,2.911106050014496e-1 +ellip_pi_complete I5F27 1966 4.999047921707538e-1 4.998889935413335e-1 0 0 0 0 0 -5e-1,2.0235255360603333e-2 -3.5393421426415443e0,9.999999478459358e-1 +ellip_pi_complete I4F28 1944 4.996600739729896e-1 4.996600739729896e-1 0 0 0 0 0 -5.8783409632742405e0,1.3478536531329155e-1 -5.8783409632742405e0,1.3478536531329155e-1 +ellip_pi_complete I3F29 2111 4.9962422393523015e-1 4.9598652406985133e-1 0 0 0 0 0 6.413668487221003e-1,9.999999832361937e-1 -2.123175583779812e0,7.230544313788414e-1 +ellip_pi_complete I64F0 106 4.7640122440170113e-1 4.7640122440170113e-1 0 0 0 0 0 -8e0,1e0 -8e0,1e0 +ellip_pi_complete I63F1 867 4.809609793861221e-1 4.809609793861221e-1 0 0 0 0 0 -3.5e0,1e0 -3.5e0,1e0 +ellip_pi_complete I62F2 1174 4.9927453573556857e-1 4.9927453573556857e-1 0 0 0 0 0 -2.075e1,5e-1 -2.075e1,5e-1 +ellip_pi_complete I61F3 1382 4.974631668059057e-1 4.974631668059057e-1 0 0 0 0 0 -6.77375e2,2.5e-1 -6.77375e2,2.5e-1 +ellip_pi_complete I60F4 1480 4.9985703107285356e-1 4.988084848451859e-1 0 0 0 0 0 -1.625e0,4.375e-1 -5.125e0,4.375e-1 +ellip_pi_complete I56F8 1549 4.9894880050967755e-1 4.9894880050967755e-1 0 0 0 0 0 -2e0,9.9609375e-1 -2e0,9.9609375e-1 +ellip_pi_complete I52F12 1566 4.9992853470804105e-1 4.9992853470804105e-1 0 0 0 0 0 -1.63835e4,5e-1 -1.63835e4,5e-1 +ellip_pi_complete I48F16 1609 4.9973649716634977e-1 4.9973649716634977e-1 0 0 0 0 0 -6.21820068359375e0,4.64630126953125e-1 -6.21820068359375e0,4.64630126953125e-1 +ellip_pi_complete I44F20 1702 4.996114237488898e-1 4.996114237488898e-1 0 0 0 0 0 -5.7222089767456055e0,2.681417465209961e-1 -5.7222089767456055e0,2.681417465209961e-1 +ellip_pi_complete I40F24 1670 4.9985946212140736e-1 4.9985946212140736e-1 0 0 0 0 0 -3.0781597069163336e11,3.3877235651016235e-1 -3.0781597069163336e11,3.3877235651016235e-1 +ellip_pi_complete I36F28 1740 4.9983450620303893e-1 4.9983450620303893e-1 0 0 0 0 0 -2.283854073310271e4,1.2105267867445946e-1 -2.283854073310271e4,1.2105267867445946e-1 +ellip_pi_complete I33F31 1788 4.999801558435053e-1 4.999801558435053e-1 0 0 0 0 0 -2.09715e6,5e-1 -2.09715e6,5e-1 +ellip_pi_complete I32F32 1716 4.9943504407039346e-1 4.9914825536251833e-1 0 0 0 0 0 -1.3521592156030238e0,4.6772918407805264e-1 -7.29637166942487e8,8.388719873037189e-1 +ellip_pi_complete I31F33 1741 4.9997272545128557e-1 4.9997272545128557e-1 0 0 0 0 0 -1.695482874836976e7,8.809835700085387e-1 -1.695482874836976e7,8.809835700085387e-1 +ellip_pi_complete I28F36 1770 4.992502899835616e-1 4.992502899835616e-1 0 0 0 0 0 -1.7606176941361908e7,8.774004313163459e-1 -1.7606176941361908e7,8.774004313163459e-1 +ellip_pi_complete I24F40 1846 4.9968457305279035e-1 4.9878891147143617e-1 0 0 0 0 0 4.999999999990905e-1,1.6548611434700433e-2 -5.181126656902052e0,8.284256789429492e-1 +ellip_pi_complete I20F44 1826 4.997359189829743e-1 4.9882263634542634e-1 0 0 0 0 0 -5.785433865627169e0,1.5502198795957156e-2 -9.132238171539103e2,4.099821573844906e-1 +ellip_pi_complete I16F48 1902 4.999908441018702e-1 4.99341690117358e-1 0 0 0 0 0 -1.0730617596465208e-2,1.824075655640911e-4 -2.6029983207027726e1,3.516280946591088e-1 +ellip_pi_complete I12F52 1916 4.995409831767018e-1 4.994659569985249e-1 0 0 0 0 0 1.2710873789800292e-1,2.5983234777633424e-3 -8.258829446361601e2,5.000000001058076e-1 +ellip_pi_complete I8F56 1975 4.999185490921177e-1 4.991839921240786e-1 0 0 0 0 0 -3.1415926535897363e0,1.2067572899455072e-1 -5.521997956680405e0,1.81274871613263e-1 +ellip_pi_complete I6F58 1950 4.997887392974721e-1 4.974174429706546e-1 0 0 0 0 0 2.2447321779139884e-14,4.371138940389985e-9 -1.7115920966781488e1,6.94678149205329e-1 +ellip_pi_complete I5F59 2036 4.9983341194888503e-1 4.982150301181739e-1 0 0 0 0 0 -1.8393047146939265e0,5.796673809376311e-1 -4.809638712955597e0,6.146903711375497e-1 +ellip_pi_complete I4F60 2007 4.9998469246167443e-1 4.9857873699019856e-1 0 0 0 0 0 -6.798610333387998e0,5.0654792860260756e-14 -1.999755859375e0,9.999999999106653e-1 +ellip_pi_complete I3F61 2138 4.99606389292305e-1 4.9719326934199143e-1 0 0 0 0 0 -3.1598793531414926e-8,9.13455999825741e-1 -2.8975494910771125e0,6.882450940227555e-1 +ellip_pi_complete I128F0 88 4.4463963273020424e-1 4.4463963273020424e-1 0 0 0 0 0 -7e0,1e0 -7e0,1e0 +ellip_pi_complete I127F1 822 4.9234847555220046e-1 4.9234847555220046e-1 0 0 0 0 0 -4.65e1,5e-1 -4.65e1,5e-1 +ellip_pi_complete I126F2 1151 4.9673559934049455e-1 4.9673559934049455e-1 0 0 0 0 0 -3.75e0,5e-1 -3.75e0,5e-1 +ellip_pi_complete I124F4 1435 4.9985703107285356e-1 4.9669441743128645e-1 0 0 0 0 0 -1.625e0,4.375e-1 -5.81875e1,5e-1 +ellip_pi_complete I120F8 1448 4.983624263248521e-1 4.978117332560813e-1 0 0 0 0 0 -2.578125e0,4.0234375e-1 -1.60546875e0,9.9609375e-1 +ellip_pi_complete I112F16 1506 4.9804144404173045e-1 4.9804144404173045e-1 0 0 0 0 0 -1.6380858413696289e4,3.67431640625e-1 -1.6380858413696289e4,3.67431640625e-1 +ellip_pi_complete I104F24 1513 4.996216027677127e-1 4.9753484223306926e-1 0 0 0 0 0 -1.7272575497627258e0,4.4956129789352417e-1 -1.1134135332280473e14,4.1533130407333374e-1 +ellip_pi_complete I96F32 1586 4.996904064511333e-1 4.996904064511333e-1 0 0 0 0 0 -2.8797476531472057e0,6.66343754157424e-1 -2.8797476531472057e0,6.66343754157424e-1 +ellip_pi_complete I88F40 1643 4.994756461239187e-1 4.994756461239187e-1 0 0 0 0 0 -1.5889243501554396e16,1.386114317438114e-1 -1.5889243501554396e16,1.386114317438114e-1 +ellip_pi_complete I80F48 1648 4.995443505026103e-1 4.995443505026103e-1 0 0 0 0 0 -1.5131101914261142e7,6.166393468180509e-1 -1.5131101914261142e7,6.166393468180509e-1 +ellip_pi_complete I72F56 1692 4.998297699696785e-1 4.998297699696785e-1 0 0 0 0 0 -1.323201667495727e5,6.331929115235895e-1 -1.323201667495727e5,6.331929115235895e-1 +ellip_pi_complete I68F60 1640 4.996044227903779e-1 4.996044227903779e-1 0 0 0 0 0 -1.4757395258967641e20,2.44140625e-4 -1.4757395258967641e20,2.44140625e-4 +ellip_pi_complete I67F61 1644 4.9993550095658296e-1 4.99837716601567e-1 0 0 0 0 0 -3.141592651727148e0,1.155294793411918e-1 -7.289020551802719e15,4.363810283252749e-1 +ellip_pi_complete I66F62 1687 4.9981431522978725e-1 4.9958136968333855e-1 0 0 0 0 0 9.9609375e-1,7.731447621137359e-3 -2.8147497671065284e14,6.581672770611152e-1 +ellip_pi_complete I65F63 1696 4.999016911899303e-1 4.999016911899303e-1 0 0 0 0 0 -2e0,9.036027427800983e-1 -2e0,9.036027427800983e-1 +ellip_pi_complete I64F64 1740 4.9997717248061446e-1 4.9997717248061446e-1 0 0 0 0 0 -9.223372036854743e18,5.133925498697136e-1 -9.223372036854743e18,5.133925498697136e-1 +ellip_pi_complete I63F65 1712 4.9998696679425597e-1 4.9998696679425597e-1 0 0 0 0 0 -4.659091886850993e5,4.3729524405045256e-1 -4.659091886850993e5,4.3729524405045256e-1 +ellip_pi_complete I62F66 1713 4.9984935120878005e-1 4.9890805920090514e-1 0 0 0 0 0 -2.206435260812946e-8,1e0 -3.1719180478873892e4,1e0 +ellip_pi_complete I56F72 1751 4.997783189270748e-1 4.997783189270748e-1 0 0 0 0 0 -3.6501386508071927e0,1e0 -3.6501386508071927e0,1e0 +ellip_pi_complete I48F80 1829 4.999149408302587e-1 4.999149408302587e-1 0 0 0 0 0 -7.268355890411737e4,3.3330900916899636e-1 -7.268355890411737e4,3.3330900916899636e-1 +ellip_pi_complete I40F88 1796 4.99945602504143e-1 4.987424614417103e-1 0 0 0 0 0 -4.858407346410207e0,9.348666508749902e-3 -5.1844274527142426e11,2.7041545943591055e-1 +ellip_pi_complete I32F96 1837 4.9965473086697987e-1 4.9965473086697987e-1 0 0 0 0 0 -3.598633474510807e0,7.097113222486481e-1 -3.598633474510807e0,7.097113222486481e-1 +ellip_pi_complete I24F104 1891 4.9985781246180255e-1 4.9985781246180255e-1 0 0 0 0 0 -1.5822655709760378e6,1.115096072891065e-20 -1.5822655709760378e6,1.115096072891065e-20 +ellip_pi_complete I16F112 1919 4.992095082837292e-1 4.9672357497432873e-1 0 0 0 0 0 -6.612213844100361e-20,1.4159265358979323e-1 -2.0934290519576454e4,1.4676079303611408e-1 +ellip_pi_complete I10F118 1978 4.9973576918945994e-1 4.9973576918945994e-1 0 0 0 0 0 -5.12e2,2.1221450794061586e-1 -5.12e2,2.1221450794061586e-1 +ellip_pi_complete I8F120 1996 4.999614727419004e-1 4.994388056430743e-1 0 0 0 0 0 0e0,7.851824877417154e-1 -1.28e2,5e-1 +ellip_pi_complete I6F122 2006 4.9996078489043644e-1 4.9853497705541805e-1 0 0 0 0 0 -3.141592653589793e0,3.9248166928134526e-1 -5.7255528608179596e0,4.09543987899447e-1 +ellip_pi_complete I5F123 2001 4.99808101197547e-1 4.99808101197547e-1 0 0 0 0 0 -5.273433465455354e0,1e0 -5.273433465455354e0,1e0 +ellip_pi_complete I4F124 2230 4.9985799262463665e-1 4.9985799262463665e-1 0 0 0 0 0 -8e0,5.717728892948967e-1 -8e0,5.717728892948967e-1 +ellip_pi_complete I3F125 1827 4.9942738775370066e-1 4.982631312917623e-1 0 0 0 0 0 0e0,5.707963267948967e-1 -3.165378512395861e0,1e0 +sin_turns I8F0 256 8.828405110119728e-287 8.828405110119728e-287 0 255 0 0 0 -1.18e2 -1.18e2 +sin_turns I7F1 256 8.828405110119728e-287 8.828405110119728e-287 0 255 0 0 0 -5.9e1 -5.9e1 +sin_turns I6F2 256 8.828405110119728e-287 8.828405110119728e-287 0 255 0 0 0 -2.95e1 -2.95e1 +sin_turns I5F3 256 3.431457505076198e-1 3.431457505076198e-1 0 255 0 2 0 -1.5875e1 -1.5875e1 +sin_turns I4F4 256 3.137084989847604e-1 3.137084989847604e-1 0 255 0 6 0 -7.875e0 -7.875e0 +sin_turns I3F5 256 4.358549596388238e-1 4.358549596388238e-1 0 255 0 14 0 -3.8125e0 -3.8125e0 +sin_turns I16F0 65536 2.417101491465897e-284 2.417101491465897e-284 0 65535 0 0 0 -3.2702e4 -3.2702e4 +sin_turns I15F1 65536 2.417101491465897e-284 2.417101491465897e-284 0 65535 0 0 0 -1.6351e4 -1.6351e4 +sin_turns I14F2 65536 2.417101491465897e-284 2.417101491465897e-284 0 65535 0 0 0 -8.1755e3 -8.1755e3 +sin_turns I13F3 65536 3.431457505076198e-1 3.431457505076198e-1 0 65535 0 2 0 -4.095875e3 -4.095875e3 +sin_turns I12F4 65536 3.137084989847604e-1 3.137084989847604e-1 0 65535 0 6 0 -2.047875e3 -2.047875e3 +sin_turns I11F5 65536 4.358549596388238e-1 4.358549596388238e-1 0 65535 0 14 0 -1.0238125e3 -1.0238125e3 +sin_turns I10F6 65536 4.9173967136574537e-1 4.9173967136574537e-1 0 65535 0 30 0 -5.119375e2 -5.119375e2 +sin_turns I9F7 65536 4.9033200812191685e-1 4.9033200812191685e-1 0 65535 0 62 0 -2.55875e2 -2.55875e2 +sin_turns I8F8 65536 4.9928718864047245e-1 4.9928718864047245e-1 0 65535 0 126 0 -1.2789453125e2 -1.2789453125e2 +sin_turns I7F9 65536 4.9832609682349177e-1 4.9832609682349177e-1 0 65535 0 254 0 -6.3775390625e1 -6.3775390625e1 +sin_turns I6F10 65536 4.998334695732479e-1 4.998334695732479e-1 0 65535 0 510 0 -3.18681640625e1 -3.18681640625e1 +sin_turns I5F11 65536 4.991548770762951e-1 4.991548770762951e-1 0 65535 0 1022 0 -1.581201171875e1 -1.581201171875e1 +sin_turns I4F12 65536 4.994907075050886e-1 4.994907075050886e-1 0 65535 0 2046 0 -7.84228515625e0 -7.84228515625e0 +sin_turns I3F13 65536 4.9973087984044495e-1 4.9973087984044495e-1 0 65535 0 4094 0 -3.8341064453125e0 -3.8341064453125e0 +sin_turns I32F0 40897 1.5845054321458144e-279 1.5845054321458144e-279 0 40290 0 0 0 -2.147051995e9 -2.147051995e9 +sin_turns I31F1 40897 1.5916507726592276e-279 1.5916507726592276e-279 0 40312 0 0 0 1.0737293315e9 1.0737293315e9 +sin_turns I30F2 40897 1.5837855152069769e-279 1.5837855152069769e-279 0 40301 0 0 0 5.368695765e8 5.368695765e8 +sin_turns I29F3 40897 3.431457505076198e-1 3.431457505076198e-1 0 40288 0 2 0 -3.75e-1 -3.75e-1 +sin_turns I28F4 40897 3.137084989847604e-1 3.137084989847604e-1 0 40277 0 6 0 -1.25e-1 -1.25e-1 +sin_turns I27F5 40897 4.358549596388238e-1 4.358549596388238e-1 0 40259 0 14 0 3.1875e0 3.1875e0 +sin_turns I26F6 40897 4.9173967136574537e-1 4.9173967136574537e-1 0 40273 0 30 0 6.25e-2 6.25e-2 +sin_turns I25F7 40897 4.9033200812191685e-1 4.9033200812191685e-1 0 40284 0 62 0 1.25e-1 1.25e-1 +sin_turns I24F8 40897 4.9928718864047245e-1 4.9928718864047245e-1 0 40267 0 126 0 7.10546875e0 7.10546875e0 +sin_turns I23F9 40897 4.9832609682349177e-1 4.9832609682349177e-1 0 40298 0 254 0 -2.0775390625e1 -2.0775390625e1 +sin_turns I22F10 40897 4.998334695732479e-1 4.998334695732479e-1 0 40242 0 510 0 1.0028348681640625e6 1.0028348681640625e6 +sin_turns I21F11 40897 4.991548770762951e-1 4.991548770762951e-1 0 40304 0 992 0 3.31201171875e0 3.31201171875e0 +sin_turns I20F12 40897 4.994907075050886e-1 4.994907075050886e-1 0 40284 0 1789 0 6.15771484375e0 6.15771484375e0 +sin_turns I19F13 40897 4.9973087984044495e-1 4.9973087984044495e-1 0 40294 0 2720 0 -7.759916589355469e4 -7.759916589355469e4 +sin_turns I18F14 40897 4.99767253679565e-1 4.99767253679565e-1 0 40297 0 3915 0 -7.135474487304688e4 -7.135474487304688e4 +sin_turns I17F15 40897 4.9996974986071113e-1 4.9996974986071113e-1 0 40241 0 4961 0 5.22308349609375e0 5.22308349609375e0 +sin_turns I16F16 40897 4.9999452422173607e-1 4.9999452422173607e-1 0 40327 0 6099 0 2.130194386291504e4 2.130194386291504e4 +sin_turns I15F17 40897 4.9995441294132037e-1 4.9995441294132037e-1 0 40277 0 7054 0 1.739630126953125e1 1.739630126953125e1 +sin_turns I14F18 40897 4.999995558842456e-1 4.999995558842456e-1 0 40280 0 8107 0 2.1290962600708008e2 2.1290962600708008e2 +sin_turns I13F19 40897 4.9999200318981757e-1 4.9999200318981757e-1 0 40291 0 9041 0 -1.4892456092834473e3 -1.4892456092834473e3 +sin_turns I12F20 40897 4.9995753482137023e-1 4.9995753482137023e-1 0 40228 0 9999 0 -5.749187469482422e-1 -5.749187469482422e-1 +sin_turns I11F21 40897 4.999884227282955e-1 4.999884227282955e-1 0 40296 0 10932 0 1.0123261022567749e3 1.0123261022567749e3 +sin_turns I10F22 40897 4.999967043294491e-1 4.999967043294491e-1 0 40285 0 11842 0 -2.025153636932373e-1 -2.025153636932373e-1 +sin_turns I9F23 40897 4.9999055438226087e-1 4.9999055438226087e-1 0 40298 0 13025 0 6.242431402206421e0 6.242431402206421e0 +sin_turns I8F24 40897 4.9995513911949735e-1 4.9995513911949735e-1 0 40294 0 14136 0 -3.231026828289032e0 -3.231026828289032e0 +sin_turns I7F25 40897 4.9999785956441284e-1 4.9999785956441284e-1 0 40272 0 14890 0 -3.092972582578659e1 -3.092972582578659e1 +sin_turns I6F26 40897 4.999948781290128e-1 4.999948781290128e-1 0 40271 0 16221 0 -5.883255168795586e0 -5.883255168795586e0 +sin_turns I5F27 40897 4.999728269055447e-1 4.999728269055447e-1 0 40337 0 17382 0 -5.250453345477581e0 -5.250453345477581e0 +sin_turns I4F28 40897 4.999992513913124e-1 4.999992513913124e-1 0 40296 0 18698 0 -1.1761896759271622e0 -1.1761896759271622e0 +sin_turns I3F29 40897 4.999828020527509e-1 4.999828020527509e-1 0 40283 0 21404 0 -6.1003586277365685e-2 -6.1003586277365685e-2 +sin_turns I64F0 41729 6.8186831664834035e-270 6.8186831664834035e-270 0 41073 0 0 0 -9.207181266424741e18 -9.207181266424741e18 +sin_turns I63F1 41729 6.785935678856601e-270 6.785935678856601e-270 0 41063 0 0 0 4.601787327990929e18 4.601787327990929e18 +sin_turns I62F2 41729 6.803134293545554e-270 6.803134293545554e-270 0 41069 0 0 0 2.3058430092035443e18 2.3058430092035443e18 +sin_turns I61F3 41729 3.431457505076198e-1 3.431457505076198e-1 0 41113 0 2 0 -3.75e-1 -3.75e-1 +sin_turns I60F4 41729 3.137084989847604e-1 3.137084989847604e-1 0 41076 0 6 0 -1.25e-1 -1.25e-1 +sin_turns I56F8 41729 4.9928718864047245e-1 4.9928718864047245e-1 0 41068 0 126 0 3.367833761810952e16 3.367833761810952e16 +sin_turns I52F12 41729 4.994907075050886e-1 4.994907075050886e-1 0 41085 0 1551 0 6.5771484375e-1 6.5771484375e-1 +sin_turns I48F16 41729 4.9999452422173607e-1 4.9999452422173607e-1 0 41094 0 3894 0 9.438629150390625e-1 9.438629150390625e-1 +sin_turns I44F20 41729 4.999496127918272e-1 4.999496127918272e-1 0 41040 0 6207 0 5.004005393868009e12 5.004005393868009e12 +sin_turns I40F24 41729 4.9999714823538133e-1 4.9999714823538133e-1 0 41092 0 8217 0 -5.2555832862854e0 -5.2555832862854e0 +sin_turns I36F28 41729 4.999969616439336e-1 4.999969616439336e-1 0 41114 0 9973 0 1.3820895233277231e4 1.3820895233277231e4 +sin_turns I33F31 41729 4.9999784269266134e-1 4.9999784269266134e-1 0 41085 0 11773 0 -4.2949672668515315e9 -4.2949672668515315e9 +sin_turns I32F32 41729 4.99949845851839e-1 4.99949845851839e-1 0 41106 0 11937 0 -3.3511155052110553e-1 -3.3511155052110553e-1 +sin_turns I31F33 41729 4.999940181193858e-1 4.999940181193858e-1 0 41123 0 12571 0 2.4435389952966943e1 2.4435389952966943e1 +sin_turns I28F36 41729 4.999940324169864e-1 4.999940324169864e-1 0 41038 0 13949 0 -7.351474694418988e7 -7.351474694418988e7 +sin_turns I24F40 41729 4.9999851500285847e-1 4.9999851500285847e-1 0 41098 0 16073 0 -4.961810991517268e-2 -4.961810991517268e-2 +sin_turns I20F44 41729 4.9998672322309073e-1 4.9998672322309073e-1 0 41058 0 18223 0 -3.540749685950004e-1 -3.540749685950004e-1 +sin_turns I16F48 41729 4.999639097874324e-1 4.999639097874324e-1 0 41084 0 19812 0 1.2976597247056255e1 1.2976597247056255e1 +sin_turns I12F52 41729 4.999946372269447e-1 4.999946372269447e-1 0 41108 0 21846 0 1.386589824438493e0 1.386589824438493e0 +sin_turns I8F56 41729 4.9997869654206833e-1 4.9997869654206833e-1 0 41086 0 23828 0 1.036904679453796e2 1.036904679453796e2 +sin_turns I6F58 41729 4.999795510423495e-1 4.999795510423495e-1 0 41048 0 24815 0 1.3015046576577067e1 1.3015046576577067e1 +sin_turns I5F59 41729 4.9999979431195046e-1 4.9999979431195046e-1 0 41108 0 25369 0 -6.305994594011416e0 -6.305994594011416e0 +sin_turns I4F60 41729 4.9998990773677915e-1 4.9998990773677915e-1 0 41069 0 26589 0 4.72414536391276e0 4.72414536391276e0 +sin_turns I3F61 41729 4.999299850493199e-1 4.999299850493199e-1 0 41074 0 29080 0 -3.2944000864916303e0 -3.2944000864916303e0 +sin_turns I128F0 43393 1.2576554481867623e-250 1.2576554481867623e-250 0 42674 0 0 0 1.7014118346046923e38 1.7014118346046923e38 +sin_turns I127F1 43393 1.2610384272035988e-250 1.2610384272035988e-250 0 42700 0 0 0 8.507058896438976e37 8.507058896438976e37 +sin_turns I126F2 43393 1.2610766775851151e-250 1.2610766775851151e-250 0 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-2.856757392883301e2,-7.21233606338501e0 -2.856757392883301e2,-7.21233606338501e0 +faddeeva_re I10F22 3000 4.998033199481134e-1 4.998033199481134e-1 0 0 0 0 0 -3.0829315543174744e2,1.4258570671081543e0 -3.0829315543174744e2,1.4258570671081543e0 +faddeeva_re I9F23 3000 4.999559476175563e-1 4.999559476175563e-1 0 0 0 0 0 -6.79786205291748e-2,3.981590270996094e-5 -6.79786205291748e-2,3.981590270996094e-5 +faddeeva_re I8F24 3000 4.99657786211735e-1 4.99657786211735e-1 0 0 0 0 0 1.7435503005981445e-1,1.9999998211860657e0 1.7435503005981445e-1,1.9999998211860657e0 +faddeeva_re I7F25 3000 4.9980829883039396e-1 4.9980829883039396e-1 0 0 0 0 0 -9.99998539686203e-1,3.144291430711746e0 -9.99998539686203e-1,3.144291430711746e0 +faddeeva_re I6F26 3000 4.9974239960289435e-1 4.9974239960289435e-1 0 0 0 0 0 5.406767085194588e0,7.174580633640289e0 5.406767085194588e0,7.174580633640289e0 +faddeeva_re I5F27 3000 4.999987811271838e-1 4.999987811271838e-1 0 0 0 0 0 1.2615986809134483e1,-2.77722030878067e-2 1.2615986809134483e1,-2.77722030878067e-2 +faddeeva_re I4F28 3000 4.998733624428438e-1 4.998733624428438e-1 0 0 0 0 0 -3.1415926590561867e0,4.81664152815938e0 -3.1415926590561867e0,4.81664152815938e0 +faddeeva_re I3F29 3000 4.999397603299244e-1 4.999397603299244e-1 0 0 0 0 0 -3.1415926292538643e0,1.3659459631890059e0 -3.1415926292538643e0,1.3659459631890059e0 +faddeeva_re I64F0 3000 4.7605877819920683e-1 4.389528271292429e-1 0 0 0 0 0 -1e0,-2e0 -2e0,-2e0 +faddeeva_re I63F1 3000 4.9055198045271703e-1 4.9055198045271703e-1 0 0 0 0 0 -5e-1,2e0 -5e-1,2e0 +faddeeva_re I62F2 3000 4.9945660825264354e-1 4.9945660825264354e-1 0 0 0 0 0 2.25e0,2.75e0 2.25e0,2.75e0 +faddeeva_re I61F3 3000 4.9927316484038464e-1 4.9927316484038464e-1 0 0 0 0 0 -2.5e0,-5e-1 -2.5e0,-5e-1 +faddeeva_re I60F4 3000 4.99988105518751e-1 4.99988105518751e-1 0 0 0 0 0 -2.5e0,1e0 -2.5e0,1e0 +faddeeva_re I56F8 3000 4.995222280993087e-1 4.9804851886644635e-1 0 0 0 0 0 -5.15625e-1,-1.8828125e0 1.5703125e0,3.125e0 +faddeeva_re I52F12 3000 4.9961329623270656e-1 4.9961329623270656e-1 0 0 0 0 0 1.172607421875e0,2.20071044921875e2 1.172607421875e0,2.20071044921875e2 +faddeeva_re I48F16 3000 4.993909620161383e-1 4.993909620161383e-1 0 0 0 0 0 2.7016143798828125e0,1.478628872680664e4 2.7016143798828125e0,1.478628872680664e4 +faddeeva_re I44F20 3000 4.9973942865339815e-1 4.9973942865339815e-1 0 0 0 0 0 -3.141594886779785e0,5.645751953125e-4 -3.141594886779785e0,5.645751953125e-4 +faddeeva_re I40F24 3000 4.9962920169625646e-1 4.992435423663497e-1 0 0 0 0 0 -1.1920928955078125e-7,-9.99755859375e-1 -4.760424017906189e0,-5e-1 +faddeeva_re I36F28 3000 4.9907397351338634e-1 4.9907397351338634e-1 0 0 0 0 0 -1.1846232116222382e0,-3.562464192509651e-2 -1.1846232116222382e0,-3.562464192509651e-2 +faddeeva_re I33F31 3000 4.998416473580479e-1 4.998416473580479e-1 0 0 0 0 0 -1.161391622061806e9,-8.654966272204057e8 -1.161391622061806e9,-8.654966272204057e8 +faddeeva_re I32F32 3000 4.99764196467667e-1 4.99764196467667e-1 0 0 0 0 0 -2.1741396047873423e1,5.058623994700611e-1 -2.1741396047873423e1,5.058623994700611e-1 +faddeeva_re I31F33 3000 4.9980752021506475e-1 4.9980752021506475e-1 0 0 0 0 0 -3.141592653701082e0,-1.9999999998835847e0 -3.141592653701082e0,-1.9999999998835847e0 +faddeeva_re I28F36 3000 4.999787943727269e-1 4.999787943727269e-1 0 0 0 0 0 -2.5403498492232757e0,0e0 -2.5403498492232757e0,0e0 +faddeeva_re I24F40 3000 4.995170717173235e-1 4.995170717173235e-1 0 0 0 0 0 -7.173986919907915e6,-3.1568852090614644e4 -7.173986919907915e6,-3.1568852090614644e4 +faddeeva_re I20F44 3000 4.998679553806465e-1 4.998679553806465e-1 0 0 0 0 0 -1.0255707963267948e3,4.740082188267991e0 -1.0255707963267948e3,4.740082188267991e0 +faddeeva_re I16F48 3000 4.9984537831099773e-1 4.9984537831099773e-1 0 0 0 0 0 1.5707963267948948e0,1.999999999999119e0 1.5707963267948948e0,1.999999999999119e0 +faddeeva_re I12F52 3000 4.997238824792017e-1 4.996324385859832e-1 0 0 0 0 0 7.363347622932892e-4,-1.5707963267948966e0 1e0,1e0 +faddeeva_re I8F56 3000 4.9988596979167876e-1 4.9988596979167876e-1 0 0 0 0 0 -1.28e2,-1.1081788264810655e-11 -1.28e2,-1.1081788264810655e-11 +faddeeva_re I6F58 3000 4.9975410455564623e-1 4.9975410455564623e-1 0 0 0 0 0 -1.547373340571312e-15,5.834772512379647e0 -1.547373340571312e-15,5.834772512379647e0 +faddeeva_re I5F59 3000 4.9957830658916264e-1 4.9957830658916264e-1 0 0 0 0 0 5.980768737604227e0,1.696992060762348e-10 5.980768737604227e0,1.696992060762348e-10 +faddeeva_re I4F60 3000 4.9990605949229305e-1 4.9990605949229305e-1 0 0 0 0 0 -2.306449265913891e0,-1.726432705272714e0 -2.306449265913891e0,-1.726432705272714e0 +faddeeva_re I3F61 3000 4.999252992313635e-1 4.999252992313635e-1 0 0 0 0 0 1.5707960883763175e0,2.7914825747304426e0 1.5707960883763175e0,2.7914825747304426e0 +faddeeva_re I128F0 3000 4.7605877819920683e-1 4.4333582267624455e-1 0 0 0 0 0 -1e0,-2e0 -1.7014118346046923e38,-1.7014118346046923e38 +faddeeva_re I127F1 3000 4.9319041168769845e-1 4.9055198045271703e-1 0 0 0 0 0 -3e0,-4e0 -5e-1,2e0 +faddeeva_re I126F2 3000 4.997039674072391e-1 4.9945660825264354e-1 0 0 0 0 0 -4.5e0,-5e0 2.25e0,2.75e0 +faddeeva_re I124F4 3000 4.995078913447666e-1 4.995078913447666e-1 0 0 0 0 0 4.0625e0,8.75e-1 4.0625e0,8.75e-1 +faddeeva_re I120F8 3000 4.979845065758191e-1 4.979845065758191e-1 0 0 0 0 0 -3.14453125e0,3.13671875e0 -3.14453125e0,3.13671875e0 +faddeeva_re I112F16 3000 4.994300012166118e-1 4.994300012166118e-1 0 0 0 0 0 -2.4512176513671875e0,-7.923126220703125e-1 -2.4512176513671875e0,-7.923126220703125e-1 +faddeeva_re I104F24 3000 4.999556749481769e-1 4.9942163366429243e-1 0 0 0 0 0 -5e-1,-1.4085004329681396e0 -2.555e2,-5e-1 +faddeeva_re I96F32 3000 4.976295272144194e-1 4.976295272144194e-1 0 0 0 0 0 9.00995070880442e3,3.2010845695156604e1 9.00995070880442e3,3.2010845695156604e1 +faddeeva_re I88F40 3000 4.9866076063504416e-1 4.9866076063504416e-1 0 0 0 0 0 -1.99609375e0,6.83940015733242e-10 -1.99609375e0,6.83940015733242e-10 +faddeeva_re I80F48 3000 4.9781372509263555e-1 4.9781372509263555e-1 0 0 0 0 0 1.5707963267948948e0,8.972884582529933e8 1.5707963267948948e0,8.972884582529933e8 +faddeeva_re I72F56 3000 4.9917387547134745e-1 4.9917387547134745e-1 0 0 0 0 0 4.999999999446159e-1,5.410843805320911e0 4.999999999446159e-1,5.410843805320911e0 +faddeeva_re I68F60 3000 4.9975116308604856e-1 4.9975116308604856e-1 0 0 0 0 0 -2.6480026437467402e8,-4.6541930843125294e2 -2.6480026437467402e8,-4.6541930843125294e2 +faddeeva_re I67F61 3000 4.998410312583211e-1 4.998410312583211e-1 0 0 0 0 0 9.999999701976776e-1,5e-1 9.999999701976776e-1,5e-1 +faddeeva_re I66F62 3000 4.99711178721301e-1 4.99711178721301e-1 0 0 0 0 0 -2.0461253788480103e0,1.000000238418579e0 -2.0461253788480103e0,1.000000238418579e0 +faddeeva_re I65F63 3000 4.999701382316752e-1 4.999701382316752e-1 0 0 0 0 0 -6.518829548973618e0,1.9999998251962672e0 -6.518829548973618e0,1.9999998251962672e0 +faddeeva_re I64F64 3000 4.9994237433307176e-1 4.9994237433307176e-1 0 0 0 0 0 4.317215938734662e6,-1.5710404674198966e0 4.317215938734662e6,-1.5710404674198966e0 +faddeeva_re I63F65 3000 4.998658536740944e-1 4.998658536740944e-1 0 0 0 0 0 6.32698134721795e0,-8.538163594651489e-10 6.32698134721795e0,-8.538163594651489e-10 +faddeeva_re I62F66 3000 4.984507590583954e-1 4.984507590583954e-1 0 0 0 0 0 -1.4809605611895722e0,2e0 -1.4809605611895722e0,2e0 +faddeeva_re I56F72 3000 4.993579033660181e-1 4.9853630393470044e-1 0 0 0 0 0 -3.602879701896397e16,-3.602879701896397e16 3.2528902230214404e16,-3.270489951797018e15 +faddeeva_re I48F80 3000 4.99561646961338e-1 4.99561646961338e-1 0 0 0 0 0 -1e0,-7.718413488534888e-1 -1e0,-7.718413488534888e-1 +faddeeva_re I40F88 3000 4.995147178514358e-1 4.995147178514358e-1 0 0 0 0 0 1.0925031958591003e0,1e0 1.0925031958591003e0,1e0 +faddeeva_re I32F96 3000 4.998360357141207e-1 4.9965512454579997e-1 0 0 0 0 0 -4.9999999999999795e-1,-4.571066778760415e0 4.29433071858848e0,7.64948741559516e-1 +faddeeva_re I24F104 3000 4.997266179452541e-1 4.997266179452541e-1 0 0 0 0 0 1.022429203673205e3,-2e0 1.022429203673205e3,-2e0 +faddeeva_re I16F112 3000 4.998911232615427e-1 4.998911232615427e-1 0 0 0 0 0 -2.3534991213090444e0,1.863862516871236e4 -2.3534991213090444e0,1.863862516871236e4 +faddeeva_re I10F118 3000 4.9966771991220554e-1 4.9966771991220554e-1 0 0 0 0 0 7.428335115051762e0,-3.141592654184371e0 7.428335115051762e0,-3.141592654184371e0 +faddeeva_re I8F120 3000 4.997097191670448e-1 4.997097191670448e-1 0 0 0 0 0 9.235711169370379e-22,3.745015354492555e0 9.235711169370379e-22,3.745015354492555e0 +faddeeva_re I6F122 3000 4.9968283784876083e-1 4.9968283784876083e-1 0 0 0 0 0 4.501027379553711e0,2e0 4.501027379553711e0,2e0 +faddeeva_re I5F123 3000 4.998164104766752e-1 4.998164104766752e-1 0 0 0 0 0 -1.5707963267948966e0,2e0 -1.5707963267948966e0,2e0 +faddeeva_re I4F124 3000 4.9994775543558295e-1 4.9994775543558295e-1 0 0 0 0 0 -8e0,1.1900301534019197e-5 -8e0,1.1900301534019197e-5 +faddeeva_re I3F125 3000 4.9988433425621864e-1 4.9988433425621864e-1 0 0 0 0 0 -1.999999999999999e0,0e0 -1.999999999999999e0,0e0 +faddeeva_im I8F0 3000 4.7719794657537057e-1 4.6891096463246656e-1 0 0 0 0 0 -8e0,-8e0 -5e0,-5e0 +faddeeva_im I7F1 3000 4.9655395077628484e-1 4.9655395077628484e-1 0 0 0 0 0 -2.5e0,0e0 -2.5e0,0e0 +faddeeva_im I6F2 3000 4.997875793280315e-1 4.997875793280315e-1 0 0 0 0 0 3.25e0,2e0 3.25e0,2e0 +faddeeva_im I5F3 3000 4.995647612723898e-1 4.995647612723898e-1 0 0 0 0 0 7.375e0,-3.5e0 7.375e0,-3.5e0 +faddeeva_im I4F4 3000 4.9952004257985866e-1 4.9952004257985866e-1 0 0 0 0 0 -5.875e0,-1.125e0 -5.875e0,-1.125e0 +faddeeva_im I3F5 3000 4.9956177460323736e-1 4.9956177460323736e-1 0 0 0 0 0 -1.5e0,-5.3125e-1 -1.5e0,-5.3125e-1 +faddeeva_im I16F0 3000 4.8302003725386455e-1 4.6891096463246656e-1 0 0 0 0 0 -4e0,-5e0 -5e0,-5e0 +faddeeva_im I15F1 3000 4.9655395077628484e-1 4.9655395077628484e-1 0 0 0 0 0 -2.5e0,0e0 -2.5e0,0e0 +faddeeva_im I14F2 3000 4.997875793280315e-1 4.997875793280315e-1 0 0 0 0 0 -3.25e0,2e0 -3.25e0,2e0 +faddeeva_im I13F3 3000 4.9947684124092e-1 4.9947684124092e-1 0 0 0 0 0 -8.875e0,1.375e0 -8.875e0,1.375e0 +faddeeva_im I12F4 3000 4.995554342305062e-1 4.995554342305062e-1 0 0 0 0 0 5.3125e0,-2e0 5.3125e0,-2e0 +faddeeva_im I11F5 3000 4.993617789245652e-1 4.993617789245652e-1 0 0 0 0 0 1.65625e0,-5.3125e-1 1.65625e0,-5.3125e-1 +faddeeva_im I10F6 3000 4.9920700667911694e-1 4.9920700667911694e-1 0 0 0 0 0 1.71875e-1,3.3125e0 1.71875e-1,3.3125e0 +faddeeva_im I9F7 3000 4.994241799441277e-1 4.994241799441277e-1 0 0 0 0 0 -4.859375e0,-1.0078125e0 -4.859375e0,-1.0078125e0 +faddeeva_im I8F8 3000 4.994165687164754e-1 4.994165687164754e-1 0 0 0 0 0 3.46484375e0,2.0703125e-1 3.46484375e0,2.0703125e-1 +faddeeva_im I7F9 3000 4.9996562476419976e-1 4.9996562476419976e-1 0 0 0 0 0 -3.0412109375e1,-5.80078125e-1 -3.0412109375e1,-5.80078125e-1 +faddeeva_im I6F10 3000 4.999159276746494e-1 4.999159276746494e-1 0 0 0 0 0 8.556640625e0,3.19990234375e1 8.556640625e0,3.19990234375e1 +faddeeva_im I5F11 3000 4.9990054714504606e-1 4.9990054714504606e-1 0 0 0 0 0 5.48779296875e0,6.03125e0 5.48779296875e0,6.03125e0 +faddeeva_im I4F12 3000 4.99551643720203e-1 4.99551643720203e-1 0 0 0 0 0 -8e0,1.105712890625e0 -8e0,1.105712890625e0 +faddeeva_im I3F13 3000 4.9973662255341533e-1 4.9973662255341533e-1 0 0 0 0 0 -3.819580078125e-1,1.18701171875e0 -3.819580078125e-1,1.18701171875e0 +faddeeva_im I32F0 3000 4.9707677709741943e-1 4.6891096463246656e-1 0 0 0 0 0 5e0,-6e0 -5e0,-5e0 +faddeeva_im I31F1 3000 4.9655395077628484e-1 4.9655395077628484e-1 0 0 0 0 0 2.5e0,0e0 2.5e0,0e0 +faddeeva_im I30F2 3000 4.9948903340160844e-1 4.9948903340160844e-1 0 0 0 0 0 4.75e0,-4.5e0 4.75e0,-4.5e0 +faddeeva_im I29F3 3000 4.996022398873934e-1 4.99567877270632e-1 0 0 0 0 0 2e0,-3.25e0 -3.75e0,4.375e0 +faddeeva_im I28F4 3000 4.9964287222734527e-1 4.9964287222734527e-1 0 0 0 0 0 3.0625e0,6.6875e0 3.0625e0,6.6875e0 +faddeeva_im I27F5 3000 4.9956014903624235e-1 4.9956014903624235e-1 0 0 0 0 0 3.60625e1,1e0 3.60625e1,1e0 +faddeeva_im I26F6 3000 4.9979895741633235e-1 4.9979895741633235e-1 0 0 0 0 0 -2.3671875e1,-3.140625e0 -2.3671875e1,-3.140625e0 +faddeeva_im I25F7 3000 4.999561404903751e-1 4.999561404903751e-1 0 0 0 0 0 -1.4440625e2,-1.5703125e0 -1.4440625e2,-1.5703125e0 +faddeeva_im I24F8 3000 4.999693783888515e-1 4.999693783888515e-1 0 0 0 0 0 3.96484375e0,2.41015625e0 3.96484375e0,2.41015625e0 +faddeeva_im I23F9 3000 4.9957459802551263e-1 4.980728971547768e-1 0 0 0 0 0 -2.5390625e-2,-3.908203125e0 1.5703125e0,-1.369140625e0 +faddeeva_im I22F10 3000 4.9895933800042036e-1 4.9895933800042036e-1 0 0 0 0 0 -4.64453125e0,1.953125e-3 -4.64453125e0,1.953125e-3 +faddeeva_im I21F11 3000 4.9968760959247016e-1 4.9968760959247016e-1 0 0 0 0 0 -2.312357421875e3,4.314453125e0 -2.312357421875e3,4.314453125e0 +faddeeva_im I20F12 3000 4.9939783309361085e-1 4.9939783309361085e-1 0 0 0 0 0 3.451416015625e0,1.632568359375e0 3.451416015625e0,1.632568359375e0 +faddeeva_im I19F13 3000 4.9966233244133645e-1 4.9966233244133645e-1 0 0 0 0 0 1.982421875e0,3.90283203125e0 1.982421875e0,3.90283203125e0 +faddeeva_im I18F14 3000 4.9931657935156354e-1 4.9931657935156354e-1 0 0 0 0 0 -3.14166259765625e0,-1.632080078125e-1 -3.14166259765625e0,-1.632080078125e-1 +faddeeva_im I17F15 3000 4.9968945011717436e-1 4.9968945011717436e-1 0 0 0 0 0 9.99969482421875e-1,4.99969482421875e-1 9.99969482421875e-1,4.99969482421875e-1 +faddeeva_im I16F16 3000 4.9996830436469575e-1 4.9996830436469575e-1 0 0 0 0 0 -3.1415863037109375e0,3.0517578125e-4 -3.1415863037109375e0,3.0517578125e-4 +faddeeva_im I15F17 3000 4.999932200798461e-1 4.999932200798461e-1 0 0 0 0 0 1.1376826393127441e4,-3.9002456665039063e0 1.1376826393127441e4,-3.9002456665039063e0 +faddeeva_im I14F18 3000 4.999327216998194e-1 4.999327216998194e-1 0 0 0 0 0 2.6145050048828125e1,2.58e2 2.6145050048828125e1,2.58e2 +faddeeva_im I13F19 3000 4.998522581987673e-1 4.998522581987673e-1 0 0 0 0 0 5e-1,-1.9073486328125e-6 5e-1,-1.9073486328125e-6 +faddeeva_im I12F20 3000 4.998768778758506e-1 4.998768778758506e-1 0 0 0 0 0 1.570796012878418e0,4.900853157043457e0 1.570796012878418e0,4.900853157043457e0 +faddeeva_im I11F21 3000 4.996705136322607e-1 4.996705136322607e-1 0 0 0 0 0 -1.7142810821533203e1,-2.0000009536743164e0 -1.7142810821533203e1,-2.0000009536743164e0 +faddeeva_im I10F22 3000 4.9984350701915886e-1 4.9984350701915886e-1 0 0 0 0 0 -4.970797140598297e2,-2.03125e0 -4.970797140598297e2,-2.03125e0 +faddeeva_im I9F23 3000 4.999503957226411e-1 4.999503957226411e-1 0 0 0 0 0 -2.5174749183654785e2,-1.5306730377674103e2 -2.5174749183654785e2,-1.5306730377674103e2 +faddeeva_im I8F24 3000 4.9901936145799836e-1 4.9901936145799836e-1 0 0 0 0 0 2.6235580444335938e-3,1.1920928955078125e-7 2.6235580444335938e-3,1.1920928955078125e-7 +faddeeva_im I7F25 3000 4.9967673277231645e-1 4.994264131935927e-1 0 0 0 0 0 -4.7392210364341736e-1,-1e0 9.332410335540771e0,-3.141592651605606e0 +faddeeva_im I6F26 3000 4.999446257702116e-1 4.999446257702116e-1 0 0 0 0 0 7.374078020453453e0,3.199218748509884e1 7.374078020453453e0,3.199218748509884e1 +faddeeva_im I5F27 3000 4.998179401948262e-1 4.998179401948262e-1 0 0 0 0 0 -9.758010640740395e0,-1.7065541595220566e0 -9.758010640740395e0,-1.7065541595220566e0 +faddeeva_im I4F28 3000 4.9993714727356753e-1 4.9993714727356753e-1 0 0 0 0 0 1.5708039551973343e0,1.000002235174179e0 1.5708039551973343e0,1.000002235174179e0 +faddeeva_im I3F29 3000 4.9960153433186855e-1 4.9960153433186855e-1 0 0 0 0 0 -3.743630548939109e0,-1.4230608940124512e-6 -3.743630548939109e0,-1.4230608940124512e-6 +faddeeva_im I64F0 3000 4.8302003725386455e-1 3.928422941586063e-1 0 0 0 0 0 -4e0,-5e0 -1e0,0e0 +faddeeva_im I63F1 3000 4.979749775152583e-1 4.7077472868976006e-1 0 0 0 0 0 -3.5e0,-5.5e0 2.5e0,-5e-1 +faddeeva_im I62F2 3000 4.997875793280315e-1 4.997875793280315e-1 0 0 0 0 0 -3.25e0,2e0 -3.25e0,2e0 +faddeeva_im I61F3 3000 4.984643558205023e-1 4.984643558205023e-1 0 0 0 0 0 5.625e0,-5.125e0 5.625e0,-5.125e0 +faddeeva_im I60F4 3000 4.995200425799291e-1 4.995200425799291e-1 0 0 0 0 0 5.875e0,1.125e0 5.875e0,1.125e0 +faddeeva_im I56F8 3000 4.9856862186430206e-1 4.9856862186430206e-1 0 0 0 0 0 3.390625e0,3.104296875e1 3.390625e0,3.104296875e1 +faddeeva_im I52F12 3000 4.992147874395357e-1 4.992147874395357e-1 0 0 0 0 0 -7.431640625e0,-1.86279296875e0 -7.431640625e0,-1.86279296875e0 +faddeeva_im I48F16 3000 4.9948940288428156e-1 4.9948940288428156e-1 0 0 0 0 0 -6.4544677734375e-3,3.452911376953125e-1 -6.4544677734375e-3,3.452911376953125e-1 +faddeeva_im I44F20 3000 4.990934718747413e-1 4.990934718747413e-1 0 0 0 0 0 -2.1311721801757813e0,3.6892452239990234e0 -2.1311721801757813e0,3.6892452239990234e0 +faddeeva_im I40F24 3000 4.995062817594221e-1 4.995062817594221e-1 0 0 0 0 0 -7.752674698829651e0,5.417325372815132e3 -7.752674698829651e0,5.417325372815132e3 +faddeeva_im I36F28 3000 4.988811214957579e-1 4.988811214957579e-1 0 0 0 0 0 -6.556510925292969e-7,-3.2523920461535454e0 -6.556510925292969e-7,-3.2523920461535454e0 +faddeeva_im I33F31 3000 4.999757690537599e-1 4.999757690537599e-1 0 0 0 0 0 2.4230583840865903e9,-1.0000000172294676e0 2.4230583840865903e9,-1.0000000172294676e0 +faddeeva_im I32F32 3000 4.994413736580858e-1 4.994413736580858e-1 0 0 0 0 0 9.690537884434067e8,0e0 9.690537884434067e8,0e0 +faddeeva_im I31F33 3000 4.99984978421589e-1 4.99984978421589e-1 0 0 0 0 0 -9.5367431640625e-7,1.323568996740505e-1 -9.5367431640625e-7,1.323568996740505e-1 +faddeeva_im I28F36 3000 4.999484420049593e-1 4.999484420049593e-1 0 0 0 0 0 -1.1424254216629507e8,8.55168686369975e7 -1.1424254216629507e8,8.55168686369975e7 +faddeeva_im I24F40 3000 4.994412553511464e-1 4.994412553511464e-1 0 0 0 0 0 4.002436036898871e-3,-2.684177852688663e-1 4.002436036898871e-3,-2.684177852688663e-1 +faddeeva_im I20F44 3000 4.995982911536366e-1 4.995982911536366e-1 0 0 0 0 0 8.151329912857591e0,-2.3931079340400174e-10 8.151329912857591e0,-2.3931079340400174e-10 +faddeeva_im I16F48 3000 4.9965532883417624e-1 4.9965532883417624e-1 0 0 0 0 0 3.1415926518182538e0,7.317496188022449e0 3.1415926518182538e0,7.317496188022449e0 +faddeeva_im I12F52 3000 4.9996182039816656e-1 4.9996182039816656e-1 0 0 0 0 0 -7.102561918185795e0,-1.5707963268531042e0 -7.102561918185795e0,-1.5707963268531042e0 +faddeeva_im I8F56 3000 4.999584258310573e-1 4.999584258310573e-1 0 0 0 0 0 -2.1454286648701915e-4,-1.0661587971061426e-4 -2.1454286648701915e-4,-1.0661587971061426e-4 +faddeeva_im I6F58 3000 4.994612577070355e-1 4.994612577070355e-1 0 0 0 0 0 -3.2e1,-8.804333614968026e0 -3.2e1,-8.804333614968026e0 +faddeeva_im I5F59 3000 4.999854380642548e-1 4.999854380642548e-1 0 0 0 0 0 -3.7703804179614084e0,-2e0 -3.7703804179614084e0,-2e0 +faddeeva_im I4F60 3000 4.996957807731293e-1 4.996957807731293e-1 0 0 0 0 0 -5.136784555655943e0,7.329442955339438e-11 -5.136784555655943e0,7.329442955339438e-11 +faddeeva_im I3F61 3000 4.999858093221881e-1 4.999858093221881e-1 0 0 0 0 0 1.2867675279238906e0,3.1415926495050237e0 1.2867675279238906e0,3.1415926495050237e0 +faddeeva_im I128F0 3000 4.8302003725386455e-1 4.4567052960619546e-1 0 0 0 0 0 4e0,-5e0 -6e0,-6e0 +faddeeva_im I127F1 3000 4.9903375631157243e-1 4.9655395077628484e-1 0 0 0 0 0 -3.5e0,-5e0 2.5e0,0e0 +faddeeva_im I126F2 3000 4.989344549902777e-1 4.989344549902777e-1 0 0 0 0 0 -4e0,-1.5e0 -4e0,-1.5e0 +faddeeva_im I124F4 3000 4.9240887168610564e-1 4.9240887168610564e-1 0 0 0 0 0 -7.3125e0,8.9375e0 -7.3125e0,8.9375e0 +faddeeva_im I120F8 3000 4.9706204168621326e-1 4.9706204168621326e-1 0 0 0 0 0 4.9609375e-1,3.78125e0 4.9609375e-1,3.78125e0 +faddeeva_im I112F16 3000 4.987300535410562e-1 4.987300535410562e-1 0 0 0 0 0 2.4628966674804688e4,-4.2415618896484375e0 2.4628966674804688e4,-4.2415618896484375e0 +faddeeva_im I104F24 3000 4.987541021806748e-1 4.987541021806748e-1 0 0 0 0 0 1.570796251296997e0,9.850475788116455e-1 1.570796251296997e0,9.850475788116455e-1 +faddeeva_im I96F32 3000 4.999190581371862e-1 4.999190581371862e-1 0 0 0 0 0 -4.460891222581267e0,-4.656612873077393e-10 -4.460891222581267e0,-4.656612873077393e-10 +faddeeva_im I88F40 3000 4.9831549345153187e-1 4.9831549345153187e-1 0 0 0 0 0 8.192e3,5.658640128878687e0 8.192e3,5.658640128878687e0 +faddeeva_im I80F48 3000 4.9999802267663257e-1 4.9999802267663257e-1 0 0 0 0 0 -5.8212620378768065e-3,1.000000000007276e0 -5.8212620378768065e-3,1.000000000007276e0 +faddeeva_im I72F56 3000 4.99768800942645e-1 4.99768800942645e-1 0 0 0 0 0 6.119536649821054e10,1.34217726e8 6.119536649821054e10,1.34217726e8 +faddeeva_im I68F60 3000 4.996850646370678e-1 4.996850646370678e-1 0 0 0 0 0 6.386477955891597e-5,-6.678685382510707e-17 6.386477955891597e-5,-6.678685382510707e-17 +faddeeva_im I67F61 3000 4.997127826341242e-1 4.997127826341242e-1 0 0 0 0 0 3.5038660320295643e0,-3.303347179128835e-15 3.5038660320295643e0,-3.303347179128835e-15 +faddeeva_im I66F62 3000 4.990630382642826e-1 4.990630382642826e-1 0 0 0 0 0 5.640449251403164e0,4.2176907907066804e8 5.640449251403164e0,4.2176907907066804e8 +faddeeva_im I65F63 3000 4.999534591391158e-1 4.999534591391158e-1 0 0 0 0 0 -1.0408429691314807e19,-1.0842021724855044e-19 -1.0408429691314807e19,-1.0842021724855044e-19 +faddeeva_im I64F64 3000 4.99762141055983e-1 4.99762141055983e-1 0 0 0 0 0 -6.937207186654926e18,-2e0 -6.937207186654926e18,-2e0 +faddeeva_im I63F65 3000 4.998094460553476e-1 4.998094460553476e-1 0 0 0 0 0 2.4897084384052016e2,4.6875e-1 2.4897084384052016e2,4.6875e-1 +faddeeva_im I62F66 3000 4.9966724688180775e-1 4.9966724688180775e-1 0 0 0 0 0 -4.02804613501672e16,-9.887638185199335e11 -4.02804613501672e16,-9.887638185199335e11 +faddeeva_im I56F72 3000 4.9996005972709595e-1 4.9996005972709595e-1 0 0 0 0 0 -2.094097656983485e16,1.776121331328001e10 -2.094097656983485e16,1.776121331328001e10 +faddeeva_im I48F80 3000 4.994233245889141e-1 4.994233245889141e-1 0 0 0 0 0 2.0511170073173635e-7,-2.0155253878219614e0 2.0511170073173635e-7,-2.0155253878219614e0 +faddeeva_im I40F88 3000 4.987835676070243e-1 4.987835676070243e-1 0 0 0 0 0 -2.364447777564489e10,-5.000000596046448e-1 -2.364447777564489e10,-5.000000596046448e-1 +faddeeva_im I32F96 3000 4.998180323409173e-1 4.998180323409173e-1 0 0 0 0 0 -2e0,1.2308670379537185e0 -2e0,1.2308670379537185e0 +faddeeva_im I24F104 3000 4.9988177898976e-1 4.9988177898976e-1 0 0 0 0 0 1.0423400762335376e6,-8.043216914948253e-25 1.0423400762335376e6,-8.043216914948253e-25 +faddeeva_im I16F112 3000 4.9968139181702304e-1 4.9968139181702304e-1 0 0 0 0 0 2.914045656498551e0,1.5707963267948966e0 2.914045656498551e0,1.5707963267948966e0 +faddeeva_im I10F118 3000 4.999114680961581e-1 4.999114680961581e-1 0 0 0 0 0 -2e0,2e0 -2e0,2e0 +faddeeva_im I8F120 3000 4.9978067300076207e-1 4.9970562768122057e-1 0 0 0 0 0 -2.692352687734916e-1,-1.8197727078015202e0 1.0000000000002274e0,2.140051324226569e-16 +faddeeva_im I6F122 3000 4.993735957090628e-1 4.993735957090628e-1 0 0 0 0 0 1.3479484446929723e1,1.648356701047021e-19 1.3479484446929723e1,1.648356701047021e-19 +faddeeva_im I5F123 3000 4.999491029243723e-1 4.999491029243723e-1 0 0 0 0 0 4.161573062884809e0,-1.5707963267948966e0 4.161573062884809e0,-1.5707963267948966e0 +faddeeva_im I4F124 3000 4.999254204972771e-1 4.999254204972771e-1 0 0 0 0 0 -8e0,3.141592653589793e0 -8e0,3.141592653589793e0 +faddeeva_im I3F125 3000 4.997386892022773e-1 4.997386892022773e-1 0 0 0 0 0 -2e0,-5e-1 -2e0,-5e-1 +c_erf_re I8F0 3000 4.6335643422143497e-1 4.6335643422143497e-1 0 0 0 0 0 1e0,-2e0 1e0,-2e0 +c_erf_re I7F1 3000 4.965715359682441e-1 3.9780276976975065e-1 0 0 0 0 0 -2e0,2.5e0 -2e0,2.5e0 +c_erf_re I6F2 3000 4.988284191536342e-1 4.945472185983147e-1 0 0 0 0 0 5.25e0,-5.75e0 2.5e0,2.5e0 +c_erf_re I5F3 3000 4.9606643943635215e-1 4.9606643943635215e-1 0 0 0 0 0 8.75e-1,-2.125e0 8.75e-1,-2.125e0 +c_erf_re I4F4 3000 4.9962997726451575e-1 4.9962997726451575e-1 0 0 0 0 0 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I8F8 3000 4.9980883353449856e-1 4.9980883353449856e-1 0 0 0 0 0 1.6328125e0,-3.0078125e-1 1.6328125e0,-3.0078125e-1 +c_erf_re I7F9 3000 4.9991751126309253e-1 4.9991751126309253e-1 0 0 0 0 0 1.89453125e0,-1.552734375e0 1.89453125e0,-1.552734375e0 +c_erf_re I6F10 3000 4.998454656240211e-1 4.998454656240211e-1 0 0 0 0 0 -4.98046875e-2,-9.765625e-4 -4.98046875e-2,-9.765625e-4 +c_erf_re I5F11 3000 4.996454837318896e-1 4.996454837318896e-1 0 0 0 0 0 4.8828125e-4,-1.61865234375e0 4.8828125e-4,-1.61865234375e0 +c_erf_re I4F12 3000 4.998396115606149e-1 4.998396115606149e-1 0 0 0 0 0 -3.14013671875e0,2e0 -3.14013671875e0,2e0 +c_erf_re I3F13 3000 4.998713390106565e-1 4.9974546082773513e-1 0 0 0 0 0 1.147705078125e0,9.990234375e-1 1.0001220703125e0,2.818603515625e-1 +c_erf_re I32F0 3000 4.7796882750497516e-1 4.6335643422143497e-1 0 0 0 0 0 -6e0,-7e0 1e0,2e0 +c_erf_re I31F1 3000 4.965715359682441e-1 3.9780276976975065e-1 0 0 0 0 0 -2e0,2.5e0 -2e0,2.5e0 +c_erf_re I30F2 3000 4.959140732376958e-1 4.7565625476447354e-1 0 0 0 0 0 -4.5e0,-5e0 1e0,2.5e-1 +c_erf_re I29F3 3000 4.9853211914651036e-1 4.942911443540362e-1 0 0 0 0 0 5e-1,3e0 8.75e-1,-2.5e-1 +c_erf_re I28F4 3000 4.929958994414081e-1 4.9082620786491554e-1 0 0 0 0 0 6.25e-1,-2.125e0 1e0,-6.25e-2 +c_erf_re I27F5 3000 4.999522113130013e-1 4.790995313024172e-1 0 0 0 0 0 1.5625e0,-3.15625e0 -1.71875e0,-3.125e-2 +c_erf_re I26F6 3000 4.960382156389952e-1 4.926125057885674e-1 0 0 0 0 0 -2e0,-3.4375e0 5e-1,3.75e-1 +c_erf_re I25F7 3000 4.955460675147742e-1 4.9499257911739275e-1 0 0 0 0 0 -8.59375e-1,-1.5703125e0 4.921875e-1,-5.3125e-1 +c_erf_re I24F8 3000 4.9985990569113226e-1 4.9859906947652083e-1 0 0 0 0 0 -1.01171875e0,-1.00390625e0 -1.55859375e0,-1.9921875e0 +c_erf_re I23F9 3000 4.9942245660685325e-1 4.9937020655368636e-1 0 0 0 0 0 1.865234375e0,-3.138671875e0 5e-1,-3.90625e-3 +c_erf_re I22F10 3000 4.9797281644332797e-1 4.947689695551648e-1 0 0 0 0 0 -4.615234375e0,-6.234375e0 -9.765625e-4,5.302734375e-1 +c_erf_re I21F11 3000 4.9834015246570634e-1 4.898284626822947e-1 0 0 0 0 0 6.1572265625e-1,-1.33056640625e0 -1.7080078125e0,4.21875e-1 +c_erf_re I20F12 3000 4.9976518543638776e-1 4.941435300088607e-1 0 0 0 0 0 1.708984375e-3,2.726806640625e0 -2.923095703125e0,-2.000244140625e0 +c_erf_re I19F13 3000 4.986486028758178e-1 4.9861825188769865e-1 0 0 0 0 0 3.1416015625e0,-1.5709228515625e0 3.1416015625e0,-1.5709228515625e0 +c_erf_re I18F14 3000 4.990694540561073e-1 4.990694540561073e-1 0 0 0 0 0 -2.55694580078125e0,9.9993896484375e-1 -2.55694580078125e0,9.9993896484375e-1 +c_erf_re I17F15 3000 4.999943478383531e-1 4.990571754387069e-1 0 0 0 0 0 -2.3297119140625e-1,2e0 2.791534423828125e0,3.141571044921875e0 +c_erf_re I16F16 3000 4.976308048430527e-1 4.976308048430527e-1 0 0 0 0 0 1.2481536865234375e0,0e0 1.2481536865234375e0,0e0 +c_erf_re I15F17 3000 4.992599792273508e-1 4.992599792273508e-1 0 0 0 0 0 -9.990234375e-1,-5e-1 -9.990234375e-1,-5e-1 +c_erf_re I14F18 3000 4.995477291618104e-1 4.9627403634911293e-1 0 0 0 0 0 -7.413482666015625e-2,2.3116378784179688e0 -2.7887115478515625e0,-2.0000038146972656e0 +c_erf_re I13F19 3000 4.993803729116629e-1 4.993803729116629e-1 0 0 0 0 0 2.9073715209960938e-2,-1.0107364654541016e0 2.9073715209960938e-2,-1.0107364654541016e0 +c_erf_re I12F20 3000 4.993121272348385e-1 4.993121272348385e-1 0 0 0 0 0 4.999980926513672e-1,5e-1 4.999980926513672e-1,5e-1 +c_erf_re I11F21 3000 4.9775868766768167e-1 4.9775868766768167e-1 0 0 0 0 0 -9.99995231628418e-1,0e0 -9.99995231628418e-1,0e0 +c_erf_re I10F22 3000 4.9994239194119316e-1 4.9994239194119316e-1 0 0 0 0 0 3.3939833641052246e0,1.570796251296997e0 3.3939833641052246e0,1.570796251296997e0 +c_erf_re I9F23 3000 4.997105331688577e-1 4.997105331688577e-1 0 0 0 0 0 -1.1996757984161377e-1,0e0 -1.1996757984161377e-1,0e0 +c_erf_re I8F24 3000 4.993279007774074e-1 4.993279007774074e-1 0 0 0 0 0 5.313754081726074e-4,-2.086162567138672e-6 5.313754081726074e-4,-2.086162567138672e-6 +c_erf_re I7F25 3000 4.996288485288869e-1 4.97541577739827e-1 0 0 0 0 0 -3.772258758544922e-3,-3.0396655201911926e0 3.5856133699417114e-1,0e0 +c_erf_re I6F26 3000 4.994152415973407e-1 4.994152415973407e-1 0 0 0 0 0 -1.054021805524826e0,-1.924536406993866e0 -1.054021805524826e0,-1.924536406993866e0 +c_erf_re I5F27 3000 4.9962211224388814e-1 4.9962211224388814e-1 0 0 0 0 0 6.386706113815308e0,-6.213774681091309e0 6.386706113815308e0,-6.213774681091309e0 +c_erf_re I4F28 3000 4.998313343204349e-1 4.998313343204349e-1 0 0 0 0 0 1.3481825590133667e-5,1.9835054576396942e0 1.3481825590133667e-5,1.9835054576396942e0 +c_erf_re I3F29 3000 4.999394465852112e-1 4.999394465852112e-1 0 0 0 0 0 3.7103883419185877e0,1.9941039755940437e-1 3.7103883419185877e0,1.9941039755940437e-1 +c_erf_re I64F0 3000 4.7796882750497516e-1 4.6335643422143497e-1 0 0 0 0 0 -6e0,-7e0 -1e0,-2e0 +c_erf_re I63F1 3000 4.965715359682441e-1 3.9780276976975065e-1 0 0 0 0 0 -2e0,2.5e0 -2e0,2.5e0 +c_erf_re I62F2 3000 4.951768938884897e-1 4.730458643550011e-1 0 0 0 0 0 -1.75e0,-2.25e0 1.5e0,-1.5e0 +c_erf_re I61F3 3000 4.8144840129840444e-1 4.8144840129840444e-1 0 0 0 0 0 3.75e-1,-8.75e-1 3.75e-1,-8.75e-1 +c_erf_re I60F4 3000 4.909427146235658e-1 4.9082620786491554e-1 0 0 0 0 0 -8.125e-1,3.125e0 -1e0,-6.25e-2 +c_erf_re I56F8 3000 4.9965366936347244e-1 4.9965366936347244e-1 0 0 0 0 0 -7.4609375e-1,9.375e-2 -7.4609375e-1,9.375e-2 +c_erf_re I52F12 3000 4.9966076231827683e-1 4.972395624541511e-1 0 0 0 0 0 -5.3125e-1,-1.540771484375e0 -2.000244140625e0,1.85302734375e-1 +c_erf_re I48F16 3000 4.999482354008035e-1 4.995429137180403e-1 0 0 0 0 0 -2e0,3.1493988037109375e0 5e-1,-8.23974609375e-4 +c_erf_re I44F20 3000 4.989501195755432e-1 4.952124638200242e-1 0 0 0 0 0 -5.00244140625e-1,-3.1415929794311523e0 -4.999980926513672e-1,1.1243820190429688e-3 +c_erf_re I40F24 3000 4.9957124615957e-1 4.9942882254406273e-1 0 0 0 0 0 -1.7146522402763367e0,4.083234906196594e0 -1.713375449180603e-1,-6.30222737789154e-1 +c_erf_re I36F28 3000 4.995515187832319e-1 4.995515187832319e-1 0 0 0 0 0 -3.300780948251486e0,9.999999962747097e-1 -3.300780948251486e0,9.999999962747097e-1 +c_erf_re I33F31 3000 4.9872552869235176e-1 4.9872552869235176e-1 0 0 0 0 0 3.141591699793935e0,4.889443516731262e-8 3.141591699793935e0,4.889443516731262e-8 +c_erf_re I32F32 3000 4.994076832238426e-1 4.977437735856333e-1 0 0 0 0 0 5.000000922009349e-1,-1.5708971612621099e0 -3.172605416737497e0,-2.384185791015625e-7 +c_erf_re I31F33 3000 4.988097717900688e-1 4.988097717900688e-1 0 0 0 0 0 -2.1184868589043617e0,-1.5707963267341256e0 -2.1184868589043617e0,-1.5707963267341256e0 +c_erf_re I28F36 3000 4.995433061867047e-1 4.9954158282417593e-1 0 0 0 0 0 -4.478618834764347e0,-3.1415902315929998e0 -4.478618834764347e0,-3.1415902315929998e0 +c_erf_re I24F40 3000 4.9992985678795143e-1 4.9992985678795143e-1 0 0 0 0 0 2.837432198024544e0,-1.862645149230957e-9 2.837432198024544e0,-1.862645149230957e-9 +c_erf_re I20F44 3000 4.985060911445717e-1 4.985060911445717e-1 0 0 0 0 0 2.686931566131534e-9,3.8323123546426245e0 2.686931566131534e-9,3.8323123546426245e0 +c_erf_re I16F48 3000 4.9950620385563976e-1 4.9950620385563976e-1 0 0 0 0 0 -3e0,3.14159265358985e0 -3e0,3.14159265358985e0 +c_erf_re I12F52 3000 4.9990384292919204e-1 4.9990384292919204e-1 0 0 0 0 0 -7.105427357601002e-15,3.1415924431414846e0 -7.105427357601002e-15,3.1415924431414846e0 +c_erf_re I8F56 3000 4.998432762162152e-1 4.998432762162152e-1 0 0 0 0 0 -4.238694194247383e-8,-1.3739009929736312e-15 -4.238694194247383e-8,-1.3739009929736312e-15 +c_erf_re I6F58 3000 4.983300959584658e-1 4.983300959584658e-1 0 0 0 0 0 8.125497271554982e-9,3.141592653589793e0 8.125497271554982e-9,3.141592653589793e0 +c_erf_re I5F59 3000 4.9971517181855746e-1 4.9875924672695743e-1 0 0 0 0 0 -1.5707963263292353e0,3.1721187229893264e0 -1.0581571209533625e-9,1.2192370662921566e0 +c_erf_re I4F60 3000 4.9985562069808237e-1 4.9985562069808237e-1 0 0 0 0 0 3.620239611382739e0,1.0000000000002085e0 3.620239611382739e0,1.0000000000002085e0 +c_erf_re I3F61 3000 4.9987369898152983e-1 4.9987368188808223e-1 0 0 0 0 0 -4e0,1.2792163236399827e0 -4e0,1.2792163236399827e0 +c_erf_re I128F0 3000 4.9029548142799484e-1 4.6335643422143497e-1 0 0 0 0 0 4e0,-8e0 -1e0,-2e0 +c_erf_re I127F1 3000 4.965715359682441e-1 3.9780276976975065e-1 0 0 0 0 0 -2e0,2.5e0 -2e0,2.5e0 +c_erf_re I126F2 3000 4.9158876946294594e-1 4.730458643550011e-1 0 0 0 0 0 -7.5e-1,-1e0 1.5e0,1.5e0 +c_erf_re I124F4 3000 4.994992063688514e-1 4.994992063688514e-1 0 0 0 0 0 -5.5e0,5.5625e0 -5.5e0,5.5625e0 +c_erf_re I120F8 3000 4.9582073044698616e-1 4.935616896904379e-1 0 0 0 0 0 -2.00390625e0,9.296875e-1 -5.46875e-2,1.578125e0 +c_erf_re I112F16 3000 4.989883270681628e-1 4.964067926463175e-1 0 0 0 0 0 -2.768035888671875e0,-3.3723297119140625e0 -2.732421875e0,2.190216064453125e0 +c_erf_re I104F24 3000 4.983285274937807e-1 4.96953471653057e-1 0 0 0 0 0 5e-1,3.1415926218032837e0 2.900471091270447e-1,1e0 +c_erf_re I96F32 3000 4.9825278937722334e-1 4.9741064087209735e-1 0 0 0 0 0 -2.0000000002328306e0,3.5476613119244576e0 -2.3256456037051976e0,-5.0006103515625e-1 +c_erf_re I88F40 3000 4.986644576173381e-1 4.980033660277105e-1 0 0 0 0 0 1.8823115413442792e0,3.141591699915807e0 4.999999999990905e-1,7.790295323138707e-2 +c_erf_re I80F48 3000 7.954018473006532e-1 4.995322085191004e-1 0 0 0 0 0 -1.170949760664719e-1,-5.469643837406263e0 -3.474467185582931e0,-2.5934809855243657e-13 +c_erf_re I72F56 3000 4.980566426160191e-1 4.9804267738677094e-1 0 0 0 0 0 3e0,4.990234375e-1 3e0,4.990234375e-1 +c_erf_re I68F60 3000 4.975999394634301e-1 4.9696520840706926e-1 0 0 0 0 0 5e-1,3.882894946352354e0 6.0763178584427004e0,1.9990234375e0 +c_erf_re I67F61 3000 4.9902496269023805e-1 4.9902496269023805e-1 0 0 0 0 0 -4.25137355875016e-15,-6.823972940536716e-11 -4.25137355875016e-15,-6.823972940536716e-11 +c_erf_re I66F62 3000 4.9965734138076573e-1 4.9965734138076573e-1 0 0 0 0 0 1.9081958235744878e-17,-5e-1 1.9081958235744878e-17,-5e-1 +c_erf_re I65F63 3000 4.998181794160293e-1 4.998181794160293e-1 0 0 0 0 0 -6.844504281880567e-7,-1.4203048459560108e-17 -6.844504281880567e-7,-1.4203048459560108e-17 +c_erf_re I64F64 3000 4.9886690786475696e-1 4.9886690786475696e-1 0 0 0 0 0 -5.000000000023392e-1,-2.168404344971009e-19 -5.000000000023392e-1,-2.168404344971009e-19 +c_erf_re I63F65 3000 4.984211252294003e-1 4.984211252294003e-1 0 0 0 0 0 3.6576346793900067e0,-1.0000000000000036e0 3.6576346793900067e0,-1.0000000000000036e0 +c_erf_re I62F66 3000 4.9981794801859014e-1 4.9981794801859014e-1 0 0 0 0 0 -8.803914217211202e-3,-2.5340310313094533e-1 -8.803914217211202e-3,-2.5340310313094533e-1 +c_erf_re I56F72 3000 4.9981483370086505e-1 4.996182083539496e-1 0 0 0 0 0 -5e-1,-3.139639528589793e0 -5e-1,2.257810379325089e-13 +c_erf_re I48F80 3000 4.9963541955573243e-1 4.9963541955573243e-1 0 0 0 0 0 -2.141592653589793e0,-4.961929888947131e-9 -2.141592653589793e0,-4.961929888947131e-9 +c_erf_re I40F88 3000 4.99730197393017e-1 4.99730197393017e-1 0 0 0 0 0 9.999999962747097e-1,2.305272395527487e-14 9.999999962747097e-1,2.305272395527487e-14 +c_erf_re I32F96 3000 4.9915541223083965e-1 4.9915541223083965e-1 0 0 0 0 0 8.835242138475332e-29,-6.227814767381501e0 8.835242138475332e-29,-6.227814767381501e0 +c_erf_re I24F104 3000 4.998037191010879e-1 4.998037191010879e-1 0 0 0 0 0 -4.26634586912596e-19,-1e0 -4.26634586912596e-19,-1e0 +c_erf_re I16F112 3000 4.988070545795131e-1 4.988070545795131e-1 0 0 0 0 0 -3.01526074021091e-9,5.708595900384364e-1 -3.01526074021091e-9,5.708595900384364e-1 +c_erf_re I10F118 3000 4.999443491773863e-1 4.999443491773863e-1 0 0 0 0 0 3.141592653589793e0,-2e0 3.141592653589793e0,-2e0 +c_erf_re I8F120 3000 4.991436948714165e-1 4.991436948714165e-1 0 0 0 0 0 -6.157587045532268e0,1e0 -6.157587045532268e0,1e0 +c_erf_re I6F122 3000 4.9956467093329054e-1 4.9956467093329054e-1 0 0 0 0 0 -9.358739434192492e-17,3.2900432639861896e0 -9.358739434192492e-17,3.2900432639861896e0 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0 0 0 0 1.90625e0,-1e0 1.90625e0,-1e0 +c_erf_im I16F0 3000 4.822311449732651e-1 2.1180852233980388e-1 0 0 0 0 0 -5e0,6e0 0e0,1e0 +c_erf_im I15F1 3000 4.99325412752195e-1 4.6248015018260413e-1 0 0 0 0 0 3e0,-4e0 1.5e0,1.5e0 +c_erf_im I14F2 3000 4.966598110212652e-1 4.9083348214743683e-1 0 0 0 0 0 -5e-1,1.5e0 2e0,-2e0 +c_erf_im I13F3 3000 4.9854617176130517e-1 4.9037853029886175e-1 0 0 0 0 0 2e0,3e0 -1.25e-1,3.75e-1 +c_erf_im I12F4 3000 4.975219771400018e-1 4.975219771400018e-1 0 0 0 0 0 2.5e-1,5.625e-1 2.5e-1,5.625e-1 +c_erf_im I11F5 3000 4.997316010358363e-1 4.997316010358363e-1 0 0 0 0 0 -1.75e0,4.6875e-1 -1.75e0,4.6875e-1 +c_erf_im I10F6 3000 4.9734802826469665e-1 4.9734802826469665e-1 0 0 0 0 0 -1.671875e0,1.546875e0 -1.671875e0,1.546875e0 +c_erf_im I9F7 3000 4.9842508442055855e-1 4.9842508442055855e-1 0 0 0 0 0 1.625e0,-1.2890625e0 1.625e0,-1.2890625e0 +c_erf_im I8F8 3000 4.9967353840544393e-1 4.9956675273812096e-1 0 0 0 0 0 -4.296875e-2,7.734375e-1 -6.6796875e-1,-3.515625e-2 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4.9621226221293346e-1 0 0 0 0 0 1.25e-1,3.5e0 1e0,-5e-1 +c_erf_im I28F4 3000 4.972991241353482e-1 4.972991241353482e-1 0 0 0 0 0 -1.0625e0,-1.0625e0 -1.0625e0,-1.0625e0 +c_erf_im I27F5 3000 4.995704434933838e-1 4.908114749359103e-1 0 0 0 0 0 -1.09375e0,-4.625e0 1.125e0,1.4375e0 +c_erf_im I26F6 3000 4.98931992765013e-1 4.98931992765013e-1 0 0 0 0 0 -7.8125e-2,-4.375e-1 -7.8125e-2,-4.375e-1 +c_erf_im I25F7 3000 4.989156382387421e-1 4.989156382387421e-1 0 0 0 0 0 2e0,-2.0625e0 2e0,-2.0625e0 +c_erf_im I24F8 3000 4.9874062259959995e-1 4.9874062259959995e-1 0 0 0 0 0 9.921875e-1,-3.75e-1 9.921875e-1,-3.75e-1 +c_erf_im I23F9 3000 4.997713116136738e-1 4.9696522323590653e-1 0 0 0 0 0 -1.27734375e0,-2.1796875e0 3.140625e0,2.2734375e0 +c_erf_im I22F10 3000 4.993787310413188e-1 4.984734061671343e-1 0 0 0 0 0 3.1416015625e0,4.6357421875e0 -2e0,-1.9111328125e0 +c_erf_im I21F11 3000 4.991100801449937e-1 4.9818290540928073e-1 0 0 0 0 0 -5.37109375e-3,-1.70263671875e0 0e0,6.6796875e-1 +c_erf_im I20F12 3000 4.995098514419296e-1 4.995098514419296e-1 0 0 0 0 0 -4.6923828125e-1,9.033203125e-3 -4.6923828125e-1,9.033203125e-3 +c_erf_im I19F13 3000 4.9706083154736597e-1 4.9682061740688455e-1 0 0 0 0 0 5.001220703125e-1,1.569580078125e0 3.1416015625e0,3.126953125e0 +c_erf_im I18F14 3000 4.989757875042046e-1 4.943874366734991e-1 0 0 0 0 0 1.57073974609375e0,-2.40557861328125e0 -1.2939453125e-2,1.224365234375e-1 +c_erf_im I17F15 3000 4.971008157390664e-1 4.9595916517874256e-1 0 0 0 0 0 -4.2724609375e-4,3.524749755859375e0 -1.463409423828125e0,9.393310546875e-1 +c_erf_im I16F16 3000 4.999363854286814e-1 4.999363854286814e-1 0 0 0 0 0 1.068115234375e-4,-5e-1 1.068115234375e-4,-5e-1 +c_erf_im I15F17 3000 4.9983538457946447e-1 4.995485867300276e-1 0 0 0 0 0 -9.800796508789063e-1,-1.57080078125e0 -1.6384e4,1.6383999992370605e4 +c_erf_im I14F18 3000 4.9929039679285775e-1 4.9929039679285775e-1 0 0 0 0 0 1.5707931518554688e0,-1.5668907165527344e0 1.5707931518554688e0,-1.5668907165527344e0 +c_erf_im I13F19 3000 4.988202391163274e-1 4.967816078264063e-1 0 0 0 0 0 9.999980926513672e-1,-3.1415939331054688e0 -4.861745834350586e0,-4.192272186279297e0 +c_erf_im I12F20 3000 4.9940468687739986e-1 4.9887175305181036e-1 0 0 0 0 0 5.7220458984375e-6,-9.835042953491211e-1 -9.999990463256836e-1,4.8722076416015625e-1 +c_erf_im I11F21 3000 4.9894914578903987e-1 4.9894914578903987e-1 0 0 0 0 0 -1e0,4.99969482421875e-1 -1e0,4.99969482421875e-1 +c_erf_im I10F22 3000 4.9900694025872055e-1 4.9900694025872055e-1 0 0 0 0 0 -9.999992847442627e-1,-4.243946075439453e-2 -9.999992847442627e-1,-4.243946075439453e-2 +c_erf_im I9F23 3000 4.9994982704927815e-1 4.9994982704927815e-1 0 0 0 0 0 -1.8497705459594727e-3,-2.5435805320739746e-2 -1.8497705459594727e-3,-2.5435805320739746e-2 +c_erf_im I8F24 3000 4.9924787908821816e-1 4.979626634231654e-1 0 0 0 0 0 0e0,1.9999999403953552e0 2.7627354860305786e-1,-5e-1 +c_erf_im I7F25 3000 4.993916400661836e-1 4.993916400661836e-1 0 0 0 0 0 0e0,-4.8065185546875e-4 0e0,-4.8065185546875e-4 +c_erf_im I6F26 3000 4.999258720076702e-1 4.999258720076702e-1 0 0 0 0 0 -3.137631118297577e0,1.4039994180202484e0 -3.137631118297577e0,1.4039994180202484e0 +c_erf_im I5F27 3000 4.9954105805144144e-1 4.9954105805144144e-1 0 0 0 0 0 8.245211094617844e-1,-1.1079758405685425e-4 8.245211094617844e-1,-1.1079758405685425e-4 +c_erf_im I4F28 3000 4.999138028109154e-1 4.999138028109154e-1 0 0 0 0 0 1.6104299575090408e-1,7.320195436477661e-6 1.6104299575090408e-1,7.320195436477661e-6 +c_erf_im I3F29 3000 4.9989273460115763e-1 4.9989273460115763e-1 0 0 0 0 0 -1.327074933797121e0,-5.3551048040390015e-6 -1.327074933797121e0,-5.3551048040390015e-6 +c_erf_im I64F0 3000 4.837393368580561e-1 2.1180852233980388e-1 0 0 0 0 0 -5e0,7e0 0e0,-1e0 +c_erf_im I63F1 3000 4.99325412752195e-1 4.6248015018260413e-1 0 0 0 0 0 3e0,-4e0 1.5e0,1.5e0 +c_erf_im I62F2 3000 4.966598110212652e-1 4.9083348214743683e-1 0 0 0 0 0 5e-1,-1.5e0 2e0,-2e0 +c_erf_im I61F3 3000 4.992926737922128e-1 4.949644180353265e-1 0 0 0 0 0 1.25e-1,3.5e0 2.125e0,-2.125e0 +c_erf_im I60F4 3000 4.9499611776143065e-1 4.8815940722475387e-1 0 0 0 0 0 1.8125e0,3.125e0 3.125e0,3.4375e0 +c_erf_im I56F8 3000 4.9943524168857895e-1 4.993411356437088e-1 0 0 0 0 0 5e-1,4.0859375e0 1.5703125e0,-2e0 +c_erf_im I52F12 3000 4.978517075376998e-1 4.954837665124047e-1 0 0 0 0 0 1.998046875e0,-4.7265625e0 -2.000244140625e0,-5.859375e-3 +c_erf_im I48F16 3000 4.996631650945246e-1 4.9804501489526776e-1 0 0 0 0 0 0e0,-2.0000152587890625e0 1.9999847412109375e0,1.7519073486328125e0 +c_erf_im I44F20 3000 4.985208332529185e-1 4.9374493688793847e-1 0 0 0 0 0 3.3343915939331055e0,3.7899932861328125e0 -1e0,1.4059057235717773e0 +c_erf_im I40F24 3000 4.993984517134919e-1 4.949229329408811e-1 0 0 0 0 0 3.1415926218032837e0,4.542162835597992e0 -3.1415927410125732e0,-1.5717728734016418e0 +c_erf_im I36F28 3000 4.99768229815732e-1 4.99768229815732e-1 0 0 0 0 0 3.692880917340517e0,1.5707964450120926e0 3.692880917340517e0,1.5707964450120926e0 +c_erf_im I33F31 3000 4.9665803001953474e-1 4.956876791857674e-1 0 0 0 0 0 -2.126265410333872e-2,-1.9999999995343387e0 -4.710275139193982e0,4.48998504364863e0 +c_erf_im I32F32 3000 4.9871290890900255e-1 4.9707149573139797e-1 0 0 0 0 0 -2.3283064365386963e-10,-3.1415926835034043e0 1.5258067632094026e0,-1.681077170651406e0 +c_erf_im I31F33 3000 4.9994163540310177e-1 4.9994163540310177e-1 0 0 0 0 0 -2.375105395913124e-6,-8.614733815193176e-9 -2.375105395913124e-6,-8.614733815193176e-9 +c_erf_im I28F36 3000 4.979143426322027e-1 4.979143426322027e-1 0 0 0 0 0 8.019762016483583e-1,-9.528689245489659e-1 8.019762016483583e-1,-9.528689245489659e-1 +c_erf_im I24F40 3000 4.988980944680328e-1 4.988980944680328e-1 0 0 0 0 0 -4.475938831516032e0,2.0184834102110472e-4 -4.475938831516032e0,2.0184834102110472e-4 +c_erf_im I20F44 3000 4.9994163493277105e-1 4.9994163493277105e-1 0 0 0 0 0 9.165756864604191e-8,-4.206412995699793e-12 9.165756864604191e-8,-4.206412995699793e-12 +c_erf_im I16F48 3000 4.989978142276961e-1 4.989978142276961e-1 0 0 0 0 0 3.141592653586155e0,5.000019073486328e-1 3.141592653586155e0,5.000019073486328e-1 +c_erf_im I12F52 3000 4.9940351057957943e-1 4.9811391157565477e-1 0 0 0 0 0 1.3322676295501878e-15,2.0869945621840786e0 -3.793740647757282e-4,1.4758371669811732e-1 +c_erf_im I8F56 3000 4.9994163796929675e-1 4.9994163796929675e-1 0 0 0 0 0 -6.031082278479261e-6,1.0269562977782698e-15 -6.031082278479261e-6,1.0269562977782698e-15 +c_erf_im I6F58 3000 4.998249047962086e-1 4.998249047962086e-1 0 0 0 0 0 1.0408340855860843e-17,7.7021722333370235e-16 1.0408340855860843e-17,7.7021722333370235e-16 +c_erf_im I5F59 3000 4.999703487340284e-1 4.999703487340284e-1 0 0 0 0 0 -2e0,-1.5717728892948966e0 -2e0,-1.5717728892948966e0 +c_erf_im I4F60 3000 4.9988476325152087e-1 4.9988476325152087e-1 0 0 0 0 0 -1.7010283356546745e0,-9.999999999999984e-1 -1.7010283356546745e0,-9.999999999999984e-1 +c_erf_im I3F61 3000 4.999876177832816e-1 4.999876177832816e-1 0 0 0 0 0 -3.9999999999999525e0,5.803738879645669e-1 -3.9999999999999525e0,5.803738879645669e-1 +c_erf_im I128F0 3000 4.837393368580561e-1 2.1180852233980388e-1 0 0 0 0 0 -5e0,7e0 0e0,1e0 +c_erf_im I127F1 3000 4.99325412752195e-1 4.6248015018260413e-1 0 0 0 0 0 -3e0,4e0 -1.5e0,1.5e0 +c_erf_im I126F2 3000 4.966598110212652e-1 4.598083787860439e-1 0 0 0 0 0 5e-1,-1.5e0 0e0,5e-1 +c_erf_im I124F4 3000 4.845925360874747e-1 4.845925360874747e-1 0 0 0 0 0 -1.5625e0,-4.375e-1 -1.5625e0,-4.375e-1 +c_erf_im I120F8 3000 4.9899071876496404e-1 4.832205011244352e-1 0 0 0 0 0 9.9609375e-1,7.609375e0 -5.0390625e-1,-1.00390625e0 +c_erf_im I112F16 3000 4.984510384649005e-1 4.8491500106238444e-1 0 0 0 0 0 -4.978790283203125e-1,4.111846923828125e0 2e0,-2.220977783203125e0 +c_erf_im I104F24 3000 4.9924053491400777e-1 4.9628805167492734e-1 0 0 0 0 0 -5.960464477539063e-8,1.9999999403953552e0 -4.0199220180511475e-2,1.9538640975952148e-1 +c_erf_im I96F32 3000 4.998957155648867e-1 4.998957155648867e-1 0 0 0 0 0 3.7483812116552144e0,-3.1415926532354206e0 3.7483812116552144e0,-3.1415926532354206e0 +c_erf_im I88F40 3000 4.99453688082199e-1 4.99453688082199e-1 0 0 0 0 0 -4.7587182246888915e0,-1.4916481614309305e0 -4.7587182246888915e0,-1.4916481614309305e0 +c_erf_im I80F48 3000 4.9951609265316954e-1 4.9951609265316954e-1 0 0 0 0 0 2.802295454397715e0,1.8040770445158856e0 2.802295454397715e0,1.8040770445158856e0 +c_erf_im I72F56 3000 4.9980683056753344e-1 4.979364536023855e-1 0 0 0 0 0 -4.990822060772937e0,-5.373575369540984e0 1.010805601745897e0,-1.6136642022601677e-10 +c_erf_im I68F60 3000 4.9952599677448317e-1 4.9952599677448317e-1 0 0 0 0 0 7.670097042433434e-1,-1.5707963267948966e0 7.670097042433434e-1,-1.5707963267948966e0 +c_erf_im I67F61 3000 4.999455930116097e-1 4.999455930116097e-1 0 0 0 0 0 -3.141470583277293e0,3.141592653589793e0 -3.141470583277293e0,3.141592653589793e0 +c_erf_im I66F62 3000 4.9939233929946647e-1 4.990688415564277e-1 0 0 0 0 0 9.6875e-1,-3.141592653589793e0 -1.730611628407996e0,1.849569339778829e-7 +c_erf_im I65F63 3000 4.993954302984744e-1 4.993954302984744e-1 0 0 0 0 0 -9.375e-1,-5.000000000000071e-1 -9.375e-1,-5.000000000000071e-1 +c_erf_im I64F64 3000 4.9995110814032445e-1 4.9540703752781107e-1 0 0 0 0 0 -8.565197162635485e-18,1.3357333687321615e0 1.5707963267948966e0,9.999999999999929e-1 +c_erf_im I63F65 3000 4.994034996751832e-1 4.994034996751832e-1 0 0 0 0 0 -1.388259679056208e0,9.50030370000674e-1 -1.388259679056208e0,9.50030370000674e-1 +c_erf_im I62F66 3000 4.993907255945824e-1 4.993907255945824e-1 0 0 0 0 0 3.142080934839793e0,-5.618785146037509e-1 3.142080934839793e0,-5.618785146037509e-1 +c_erf_im I56F72 3000 4.996069680705993e-1 4.996069680705993e-1 0 0 0 0 0 0e0,-1.193902884827317e-8 0e0,-1.193902884827317e-8 +c_erf_im I48F80 3000 4.989686249669322e-1 4.989686249669322e-1 0 0 0 0 0 -6.9879119646709675e0,9.999999999998863e-1 -6.9879119646709675e0,9.999999999998863e-1 +c_erf_im I40F88 3000 4.990892014457924e-1 4.986903842442767e-1 0 0 0 0 0 2.125e0,4.587614515823855e0 1e0,-1e0 +c_erf_im I32F96 3000 4.9985037495303003e-1 4.991906019097895e-1 0 0 0 0 0 1.298611306733095e-5,-3.141592653589793e0 1.4580171988893568e-12,5e-1 +c_erf_im I24F104 3000 4.992947721548615e-1 4.992947721548615e-1 0 0 0 0 0 2e0,5.965760595733902e-30 2e0,5.965760595733902e-30 +c_erf_im I16F112 3000 4.9978930527753845e-1 4.9978930527753845e-1 0 0 0 0 0 1.5707963267948966e0,1.5707963267948966e0 1.5707963267948966e0,1.5707963267948966e0 +c_erf_im I10F118 3000 4.996079636086305e-1 4.983091429173214e-1 0 0 0 0 0 3.308722450212111e-24,1e0 -2.8137150005506752e-6,5.169878828456423e-26 +c_erf_im I8F120 3000 4.998013674540604e-1 4.998013674540604e-1 0 0 0 0 0 2.3705495284217756e-1,7.888609052210118e-31 2.3705495284217756e-1,7.888609052210118e-31 +c_erf_im I6F122 3000 4.9987875621907585e-1 4.9987875621907585e-1 0 0 0 0 0 -5.41455263257712e0,-6.52745531334726e-1 -5.41455263257712e0,-6.52745531334726e-1 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4.99775423551403e-1 4.99775423551403e-1 0 0 0 0 0 -4.9993896484375e-1,-5.7280540466308594e-2 -4.9993896484375e-1,-5.7280540466308594e-2 +e1_re I11F21 2999 4.998428782997581e-1 4.993941504275097e-1 0 0 0 0 0 7.486343383789063e-5,-1.9073486328125e-6 3.814697265625e-6,5.539233207702637e0 +e1_re I10F22 3000 4.998453304328206e-1 4.998453304328206e-1 0 0 0 0 0 1.5067696571350098e-1,5.000007152557373e-1 1.5067696571350098e-1,5.000007152557373e-1 +e1_re I9F23 2999 4.99962156327006e-1 4.99962156327006e-1 0 0 0 0 0 1.1173689365386963e-1,3.131144881248474e0 1.1173689365386963e-1,3.131144881248474e0 +e1_re I8F24 2999 4.99916814001889e-1 4.99916814001889e-1 0 0 0 0 0 -1.5727494359016418e0,-6.515205174684525e1 -1.5727494359016418e0,-6.515205174684525e1 +e1_re I7F25 3000 4.999936429683835e-1 4.999936429683835e-1 0 0 0 0 0 7.545385360717773e-1,-9.999984204769135e-1 7.545385360717773e-1,-9.999984204769135e-1 +e1_re I6F26 2999 4.998707934575456e-1 4.998707934575456e-1 0 0 0 0 0 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2999 4.9824846974980924e-1 4.9824846974980924e-1 0 0 0 0 0 4.9609375e-1,-6.23046875e0 4.9609375e-1,-6.23046875e0 +e1_re I52F12 3000 4.9914779873939863e-1 4.989165231813081e-1 0 0 0 0 0 -2.08349609375e0,1.570556640625e0 3.5400390625e-2,3.493408203125e0 +e1_re I48F16 3000 4.9991658705316383e-1 4.9991658705316383e-1 0 0 0 0 0 1.5707855224609375e0,-1.7848052978515625e0 1.5707855224609375e0,-1.7848052978515625e0 +e1_re I44F20 3000 4.994036495605188e-1 4.994036495605188e-1 0 0 0 0 0 2.86102294921875e-6,-3.069173936843872e2 2.86102294921875e-6,-3.069173936843872e2 +e1_re I40F24 3000 4.9985073861069496e-1 4.9985073861069496e-1 0 0 0 0 0 3.4545262455940247e0,-2e0 3.4545262455940247e0,-2e0 +e1_re I36F28 3000 4.997256780815219e-1 4.997256780815219e-1 0 0 0 0 0 1.1117271706461906e0,3.1004217453300953e0 1.1117271706461906e0,3.1004217453300953e0 +e1_re I33F31 3000 4.998383599711249e-1 4.99763649403302e-1 0 0 0 0 0 -6.777260718867183e0,2.5746412575244904e-4 5.000038146972656e-1,2.6007197307132e9 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1.9257928585147965e-12,-5.131706130858068e2 1.9257928585147965e-12,-5.131706130858068e2 +e1_re I8F56 3000 4.999304594614356e-1 4.999304594614356e-1 0 0 0 0 0 -5.000038146972656e-1,4.136778009602993e-6 -5.000038146972656e-1,4.136778009602993e-6 +e1_re I6F58 3000 4.9979463103038885e-1 4.997712668758173e-1 0 0 0 0 0 -5.620504062164855e-16,-2.888661532196579e-14 4.166690823793895e-7,-2.6532754374689123e1 +e1_re I5F59 3000 4.9989697039135556e-1 4.9989697039135556e-1 0 0 0 0 0 1.9999999999901146e0,1e0 1.9999999999901146e0,1e0 +e1_re I4F60 3000 4.998608605548817e-1 4.996457908180438e-1 0 0 0 0 0 -9.999999955854871e-1,-9.99999999999995e-1 3.5707657329036757e0,1.0000000000028673e0 +e1_re I3F61 3000 4.999419621313801e-1 4.999419621313801e-1 0 0 0 0 0 1.2813713229364865e-8,1.5707963267948968e0 1.2813713229364865e-8,1.5707963267948968e0 +e1_re I128F0 2999 4.985961813721072e-1 4.973904728306047e-1 0 0 0 0 0 -1.24e2,-1.89352668441315e37 -1e0,-4e0 +e1_re I127F1 3000 4.977525436584792e-1 4.9669491544850924e-1 0 0 0 0 0 -3.5e0,-1.5e0 -1e0,-8.5e0 +e1_re I126F2 3000 4.942213666489753e-1 4.8974001156316677e-1 0 0 0 0 0 -4.5e0,4.5e0 1e0,-2.25e0 +e1_re I124F4 3000 4.958827611324776e-1 4.9588055057636155e-1 0 0 0 0 0 -6.0625e0,7.5625e0 2.1875e0,4.375e-1 +e1_re I120F8 3000 4.9762305461779066e-1 4.9762305461779066e-1 0 0 0 0 0 -3.90625e-3,-7.06640625e0 -3.90625e-3,-7.06640625e0 +e1_re I112F16 3000 4.9774235707566983e-1 4.9774235707566983e-1 0 0 0 0 0 0e0,-5.000152587890625e-1 0e0,-5.000152587890625e-1 +e1_re I104F24 3000 4.9677898089057054e-1 4.9677898089057054e-1 0 0 0 0 0 -3.1415621638298035e0,2.0115251211035252e5 -3.1415621638298035e0,2.0115251211035252e5 +e1_re I96F32 2999 4.9991889366447845e-1 4.9991889366447845e-1 0 0 0 0 0 5.6710706383455545e0,-2.0000152587890625e0 5.6710706383455545e0,-2.0000152587890625e0 +e1_re I88F40 2999 4.9996842683154097e-1 4.9996842683154097e-1 0 0 0 0 0 2.8969873063160776e0,-5e-1 2.8969873063160776e0,-5e-1 +e1_re I80F48 3000 4.992307570152323e-1 4.992307570152323e-1 0 0 0 0 0 -3.248814550715906e-10,-9.99999999998181e-1 -3.248814550715906e-10,-9.99999999998181e-1 +e1_re I72F56 3000 4.9981458996983136e-1 4.9981458996983136e-1 0 0 0 0 0 4.5645159882656445e0,4.824007859838275e-11 4.5645159882656445e0,4.824007859838275e-11 +e1_re I68F60 3000 4.9933445189347025e-1 4.9927366299686843e-1 0 0 0 0 0 -1e0,5.0006103515625e-1 -6.147868699840029e0,-1.8012805081474012e18 +e1_re I67F61 3000 4.979280267091585e-1 4.979280267091585e-1 0 0 0 0 0 2.1981815894432655e-9,-4.294967297e9 2.1981815894432655e-9,-4.294967297e9 +e1_re I66F62 3000 4.999521395476124e-1 4.999521395476124e-1 0 0 0 0 0 9.999999999999432e-1,3.3541122012574577e9 9.999999999999432e-1,3.3541122012574577e9 +e1_re I65F63 3000 4.996717216786356e-1 4.996717216786356e-1 0 0 0 0 0 5.000000596046448e-1,1.1173595076359707e19 5.000000596046448e-1,1.1173595076359707e19 +e1_re I64F64 3000 4.999523853927525e-1 4.999523853927525e-1 0 0 0 0 0 2.1006830748415917e0,4.9999999999999867e-1 2.1006830748415917e0,4.9999999999999867e-1 +e1_re I63F65 3000 4.993842584671496e-1 4.993842584671496e-1 0 0 0 0 0 4.374348692718396e-1,1.5707963267948966e0 4.374348692718396e-1,1.5707963267948966e0 +e1_re I62F66 3000 4.998277392239757e-1 4.998277392239757e-1 0 0 0 0 0 2.999639901875592e0,4.3685216034872187e-16 2.999639901875592e0,4.3685216034872187e-16 +e1_re I56F72 3000 4.992507240039927e-1 4.990591332185813e-1 0 0 0 0 0 -1.6141750907163948e0,-2.9193829089737243e-15 -3.141592653589793e0,7.949924567981759e7 +e1_re I48F80 3000 4.999619203355781e-1 4.999619203355781e-1 0 0 0 0 0 5.79086470688179e-8,2.9115363941009256e0 5.79086470688179e-8,2.9115363941009256e0 +e1_re I40F88 2999 4.998144705420972e-1 4.998144705420972e-1 0 0 0 0 0 -3.554182515296832e0,2.0885563447882025e11 -3.554182515296832e0,2.0885563447882025e11 +e1_re I32F96 3000 4.997298072133111e-1 4.997298072133111e-1 0 0 0 0 0 -6.77657846574324e-1,-4.313174266724625e0 -6.77657846574324e-1,-4.313174266724625e0 +e1_re I24F104 3000 4.9998813187647007e-1 4.9941603502994625e-1 0 0 0 0 0 -6.821663125942307e0,1.570794321891414e0 -5.000000000291038e-1,3.3195839049556505e-21 +e1_re I16F112 3000 4.9990900502172403e-1 4.9990900502172403e-1 0 0 0 0 0 1e0,-3.705302820238868e0 1e0,-3.705302820238868e0 +e1_re I10F118 3000 4.99926302720446e-1 4.99926302720446e-1 0 0 0 0 0 -1e0,5.1076022318714234e2 -1e0,5.1076022318714234e2 +e1_re I8F120 3000 4.9987369021380923e-1 4.987593158211082e-1 0 0 0 0 0 -3.079092653589793e0,-3.141592653589793e0 -1.118611297050147e-2,-2e0 +e1_re I6F122 3000 4.999876264282997e-1 4.9996430170204387e-1 0 0 0 0 0 -1.5707963267948966e0,1.3702283554079429e0 4.93940174240463e0,4.4642078099596243e-32 +e1_re I5F123 3000 4.998252492290094e-1 4.998252492290094e-1 0 0 0 0 0 -2.889534690724711e-5,-1.0364141696186713e0 -2.889534690724711e-5,-1.0364141696186713e0 +e1_re I4F124 3000 4.996837251704399e-1 4.996837251704399e-1 0 0 0 0 0 3.141592653589793e0,2.9164238042875605e0 3.141592653589793e0,2.9164238042875605e0 +e1_re I3F125 2876 4.999290031736685e-1 4.999290031736685e-1 0 0 0 0 0 5e-1,-1.5707963267948966e0 5e-1,-1.5707963267948966e0 +e1_im I8F0 2973 4.9166377604968464e-1 4.9166377604968464e-1 0 0 0 0 0 -2e0,1.3e1 -2e0,1.3e1 +e1_im I7F1 2992 4.9931697247492307e-1 4.9931697247492307e-1 0 0 0 0 0 -2e0,-2.45e1 -2e0,-2.45e1 +e1_im I6F2 2997 4.997623694393793e-1 4.997623694393793e-1 0 0 0 0 0 -3e0,7.25e0 -3e0,7.25e0 +e1_im I5F3 2996 4.99866422520908e-1 4.99866422520908e-1 0 0 0 0 0 1.625e0,-6.25e-1 1.625e0,-6.25e-1 +e1_im I4F4 2996 4.995468966273117e-1 4.995468966273117e-1 0 0 0 0 0 -2.125e0,6.75e0 -2.125e0,6.75e0 +e1_im I3F5 2996 4.995939789583271e-1 4.995939789583271e-1 0 0 0 0 0 -1.09375e0,-1.75e0 -1.09375e0,-1.75e0 +e1_im I16F0 2993 4.99572842981395e-1 4.99572842981395e-1 0 0 0 0 0 -3e0,-3.2e1 -3e0,-3.2e1 +e1_im I15F1 2995 4.9938231094866187e-1 4.979774993286633e-1 0 0 0 0 0 -1e1,-2.919e3 -5e-1,6e0 +e1_im I14F2 2998 4.9899643730650467e-1 4.9899643730650467e-1 0 0 0 0 0 -1.5e0,1.25e0 -1.5e0,1.25e0 +e1_im I13F3 2999 4.992544972332515e-1 4.992544972332515e-1 0 0 0 0 0 0e0,4e0 0e0,4e0 +e1_im I12F4 3000 4.978873194454071e-1 4.978873194454071e-1 0 0 0 0 0 -1.3125e0,6.75e0 -1.3125e0,6.75e0 +e1_im I11F5 2997 4.990771774584222e-1 4.990771774584222e-1 0 0 0 0 0 1.5625e0,-7.53125e0 1.5625e0,-7.53125e0 +e1_im I10F6 2996 4.9956277239612723e-1 4.97888357709653e-1 0 0 0 0 0 -1.5625e0,-1.90625e0 -2e0,1.7809375e2 +e1_im I9F7 2999 4.999961320850615e-1 4.996600314432214e-1 0 0 0 0 0 -7.8125e-1,-5.078125e-1 1.359375e0,6.75e0 +e1_im I8F8 3000 4.998527113222565e-1 4.998527113222565e-1 0 0 0 0 0 -1.91796875e0,3.9296875e0 -1.91796875e0,3.9296875e0 +e1_im I7F9 3000 4.995657597686747e-1 4.995657597686747e-1 0 0 0 0 0 5e-1,3.15625e0 5e-1,3.15625e0 +e1_im I6F10 2999 4.9993447880274955e-1 4.9993447880274955e-1 0 0 0 0 0 1.9404296875e0,1.5474609375e1 1.9404296875e0,1.5474609375e1 +e1_im I5F11 2999 4.986164344792379e-1 4.9787770165107254e-1 0 0 0 0 0 -4.74658203125e0,2.080078125e-1 2.525390625e0,3.857421875e-2 +e1_im I4F12 3000 4.9997794722293515e-1 4.9997794722293515e-1 0 0 0 0 0 6.103515625e-3,-5.830810546875e0 6.103515625e-3,-5.830810546875e0 +e1_im I3F13 2998 4.999515915218645e-1 4.999515915218645e-1 0 0 0 0 0 7.054443359375e-1,1.897705078125e0 7.054443359375e-1,1.897705078125e0 +e1_im I32F0 2990 4.9251886298461806e-1 4.40091227651919e-1 0 0 0 0 0 -5e0,-2.2e1 -1e0,-2e0 +e1_im I31F1 2997 4.995136085441304e-1 4.9687435548600667e-1 0 0 0 0 0 -7e0,5e-1 -3e0,7.5e0 +e1_im I30F2 2999 4.9678204995786585e-1 4.9678204995786585e-1 0 0 0 0 0 1.25e0,-1e0 1.25e0,-1e0 +e1_im I29F3 2997 4.997091280084788e-1 4.997091280084788e-1 0 0 0 0 0 7.5e-1,-1.875e0 7.5e-1,-1.875e0 +e1_im I28F4 2998 4.99487736227938e-1 4.994476648437755e-1 0 0 0 0 0 -2.375e0,-1.8125e0 5e-1,-3.125e0 +e1_im I27F5 2999 4.987667968443263e-1 4.987667968443263e-1 0 0 0 0 0 -5.34375e0,8.321875e1 -5.34375e0,8.321875e1 +e1_im I26F6 3000 4.9881897040625844e-1 4.9881897040625844e-1 0 0 0 0 0 9.53125e-1,-1.890625e0 9.53125e-1,-1.890625e0 +e1_im I25F7 3000 4.992295025745932e-1 4.992295025745932e-1 0 0 0 0 0 1.2890625e0,-3.1328125e0 1.2890625e0,-3.1328125e0 +e1_im I24F8 3000 4.986606086982847e-1 4.986606086982847e-1 0 0 0 0 0 3.3359375e0,3.59765625e0 3.3359375e0,3.59765625e0 +e1_im I23F9 2998 4.990316407659428e-1 4.990316407659428e-1 0 0 0 0 0 2.091796875e0,-3.2421875e0 2.091796875e0,-3.2421875e0 +e1_im I22F10 3000 4.9989501515284823e-1 4.9989501515284823e-1 0 0 0 0 0 1.734375e0,-7.826171875e0 1.734375e0,-7.826171875e0 +e1_im I21F11 3000 4.993214708431173e-1 4.993214708431173e-1 0 0 0 0 0 5.97412109375e0,-7.28759765625e0 5.97412109375e0,-7.28759765625e0 +e1_im I20F12 3000 4.991182670005303e-1 4.991182670005303e-1 0 0 0 0 0 -1.76806640625e0,-6.22496826171875e2 -1.76806640625e0,-6.22496826171875e2 +e1_im I19F13 3000 4.993886639403105e-1 4.993886639403105e-1 0 0 0 0 0 4.998779296875e-1,1.000244140625e0 4.998779296875e-1,1.000244140625e0 +e1_im I18F14 3000 4.995174575005769e-1 4.995174575005769e-1 0 0 0 0 0 -2.44140625e-4,-1.5e0 -2.44140625e-4,-1.5e0 +e1_im I17F15 3000 4.9997604911286664e-1 4.9997604911286664e-1 0 0 0 0 0 -1.96319580078125e0,3.4665453186035156e3 -1.96319580078125e0,3.4665453186035156e3 +e1_im I16F16 2999 4.999664833929891e-1 4.996009418794253e-1 0 0 0 0 0 -3.001220703125e0,-5.4223175048828125e0 1.5702972412109375e0,1.587811279296875e1 +e1_im I15F17 3000 4.99880802325195e-1 4.99880802325195e-1 0 0 0 0 0 -1.0929107666015625e-1,-1.535289331817627e4 -1.0929107666015625e-1,-1.535289331817627e4 +e1_im I14F18 2998 4.998048206550034e-1 4.998048206550034e-1 0 0 0 0 0 2.93389892578125e0,-1.4665031433105469e0 2.93389892578125e0,-1.4665031433105469e0 +e1_im I13F19 3000 4.9976976676099283e-1 4.9963239916196855e-1 0 0 0 0 0 -1.8926620483398438e-2,-2.273845672607422e-1 3.880411148071289e0,2.1033571243286133e1 +e1_im I12F20 2999 4.9954165414696355e-1 4.9954165414696355e-1 0 0 0 0 0 3.141608238220215e0,-6.567191619873047e2 3.141608238220215e0,-6.567191619873047e2 +e1_im I11F21 3000 4.998838277138017e-1 4.997962026206014e-1 0 0 0 0 0 -4.572981834411621e0,2.31251859664917e0 -2.837080955505371e-1,-8.93349368095398e2 +e1_im I10F22 3000 4.9981254317371376e-1 4.9932553455192485e-1 0 0 0 0 0 -7.686280965805054e0,5.117383003234863e-1 -3.066904544830322e-1,-3.1415927410125732e0 +e1_im I9F23 2999 4.9980379042125e-1 4.9980379042125e-1 0 0 0 0 0 1.567149043083191e0,2.5599999976158142e2 1.567149043083191e0,2.5599999976158142e2 +e1_im I8F24 3000 4.9967396368942407e-1 4.9967396368942407e-1 0 0 0 0 0 -1.0000002980232239e0,-7.406179332733154e1 -1.0000002980232239e0,-7.406179332733154e1 +e1_im I7F25 2998 4.9957654016183517e-1 4.9957654016183517e-1 0 0 0 0 0 -3.1415926814079285e0,-4.362173667550087e1 -3.1415926814079285e0,-4.362173667550087e1 +e1_im I6F26 2999 4.996333228391795e-1 4.994482076195498e-1 0 0 0 0 0 -4.056398570537567e0,1.000030517578125e0 -4.9991077184677124e-1,-5.270884662866592e0 +e1_im I5F27 3000 4.999610599067613e-1 4.999610599067613e-1 0 0 0 0 0 -7.68987201154232e-1,1.570796325802803e0 -7.68987201154232e-1,1.570796325802803e0 +e1_im I4F28 2999 4.998998448035957e-1 4.9978077894713024e-1 0 0 0 0 0 -2.4350064247846603e0,-6.048465054482222e0 2.765269208699465e0,7.99999999627471e0 +e1_im I3F29 3000 4.99537933656938e-1 4.99537933656938e-1 0 0 0 0 0 1.086568297818303e0,-2.2351781129837036e0 1.086568297818303e0,-2.2351781129837036e0 +e1_im I64F0 2995 4.878060277309325e-1 4.40091227651919e-1 0 0 0 0 0 -3e0,-7e0 -1e0,2e0 +e1_im I63F1 2996 4.986920761420599e-1 4.979774993286633e-1 0 0 0 0 0 -7.5e0,-9.5e0 -5e-1,-6e0 +e1_im I62F2 2998 4.9885302571085444e-1 4.9885302571085444e-1 0 0 0 0 0 0e0,-1e0 0e0,-1e0 +e1_im I61F3 3000 4.996291888674418e-1 4.996291888674418e-1 0 0 0 0 0 1.75e0,-1e0 1.75e0,-1e0 +e1_im I60F4 2999 4.994476648437755e-1 4.994476648437755e-1 0 0 0 0 0 5e-1,3.125e0 5e-1,3.125e0 +e1_im I56F8 2999 4.990817316902816e-1 4.990817316902816e-1 0 0 0 0 0 -3.74609375e0,6.730546875e3 -3.74609375e0,6.730546875e3 +e1_im I52F12 3000 4.997915137210321e-1 4.997915137210321e-1 0 0 0 0 0 3.02734375e-2,3.37158203125e0 3.02734375e-2,3.37158203125e0 +e1_im I48F16 3000 4.999533820412425e-1 4.9882274423941214e-1 0 0 0 0 0 -6.4292144775390625e0,4.1337738037109375e0 9.998321533203125e-1,-4.3089752197265625e0 +e1_im I44F20 3000 4.979640347972462e-1 4.979640347972462e-1 0 0 0 0 0 4.999980926513672e-1,-4.4946956634521484e-1 4.999980926513672e-1,-4.4946956634521484e-1 +e1_im I40F24 3000 4.98967050669135e-1 4.98967050669135e-1 0 0 0 0 0 -4.470348358154297e-6,6.040468215942383e-1 -4.470348358154297e-6,6.040468215942383e-1 +e1_im I36F28 2999 4.999859261259441e-1 4.999859261259441e-1 0 0 0 0 0 -5.955694746226072e0,-1.638085840734467e4 -5.955694746226072e0,-1.638085840734467e4 +e1_im I33F31 2998 4.9968042914841043e-1 4.9968042914841043e-1 0 0 0 0 0 -2.594627120066434e0,3.469796339392536e7 -2.594627120066434e0,3.469796339392536e7 +e1_im I32F32 3000 4.9953981739713305e-1 4.9953981739713305e-1 0 0 0 0 0 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3.465592785857785e-3,4.218847493575595e-15 +e1_im I8F56 3000 4.996126984248699e-1 4.996126984248699e-1 0 0 0 0 0 -3.608262716392474e-10,-2.0000000596046448e0 -3.608262716392474e-10,-2.0000000596046448e0 +e1_im I6F58 3000 4.998790252884392e-1 4.9983228739667207e-1 0 0 0 0 0 -1.9249847473521589e0,-2.0403696742483275e0 -6.3647169164411654e-6,4.199070913892196e0 +e1_im I5F59 3000 4.998604037021146e-1 4.998604037021146e-1 0 0 0 0 0 1.5639962631298008e1,-7.646788885083917e-2 1.5639962631298008e1,-7.646788885083917e-2 +e1_im I4F60 3000 4.996440691719054e-1 4.996440691719054e-1 0 0 0 0 0 -1.5707963267948974e0,-3.9533361520316377e0 -1.5707963267948974e0,-3.9533361520316377e0 +e1_im I3F61 2999 4.9995337556859554e-1 4.9995337556859554e-1 0 0 0 0 0 1.9919608560596382e0,-1.7175540503733266e-13 1.9919608560596382e0,-1.7175540503733266e-13 +e1_im I128F0 2994 4.878060277309325e-1 4.835771993680733e-1 0 0 0 0 0 -3e0,-7e0 -3e0,2.9e1 +e1_im I127F1 3000 4.995136085441304e-1 4.9235439584145835e-1 0 0 0 0 0 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4.992580070687226e-1 0 0 0 0 0 -2.8920070030279383e0,1.4056643080310923e1 -2.8920070030279383e0,1.4056643080310923e1 +e1_im I72F56 3000 4.9974167145531206e-1 4.9974167145531206e-1 0 0 0 0 0 -4.8165745303209935e0,3.503351130517781e14 -4.8165745303209935e0,3.503351130517781e14 +e1_im I68F60 3000 4.9943367544385064e-1 4.9943367544385064e-1 0 0 0 0 0 3.1415926536897154e0,5.35007229490736e-11 3.1415926536897154e0,5.35007229490736e-11 +e1_im I67F61 3000 4.997901282240421e-1 4.991134552379191e-1 0 0 0 0 0 -1.5707963267948972e0,9.250663224448355e-2 1.681450048525118e-4,5.3992690315702765e0 +e1_im I66F62 3000 4.9963204209511985e-1 4.9963204209511985e-1 0 0 0 0 0 6.144275188460385e0,-2.217086154488485e-4 6.144275188460385e0,-2.217086154488485e-4 +e1_im I65F63 3000 4.999903506849672e-1 4.999903506849672e-1 0 0 0 0 0 -2.592235788909294e-4,-1.8446744073709552e19 -2.592235788909294e-4,-1.8446744073709552e19 +e1_im I64F64 3000 4.9982671265572776e-1 4.9982671265572776e-1 0 0 0 0 0 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-9.3125e0 -9.3125e0 +fresnel_c I11F5 3000 4.9794570317169445e-1 4.9794570317169445e-1 0 2954 0 0 0 1.971875e1 1.971875e1 +fresnel_c I10F6 3000 4.9900165977027505e-1 4.9900165977027505e-1 0 2960 0 0 0 -4.203125e0 -4.203125e0 +fresnel_c I9F7 3000 4.996460970506205e-1 4.996460970506205e-1 0 2974 0 0 0 -6.96875e0 -6.96875e0 +fresnel_c I8F8 3000 4.9964189347622606e-1 4.9964189347622606e-1 0 2948 0 0 0 -9.8046875e-1 -9.8046875e-1 +fresnel_c I7F9 3000 4.997458178224994e-1 4.997458178224994e-1 0 2958 0 0 0 -1.142578125e0 -1.142578125e0 +fresnel_c I6F10 3000 4.995300348267376e-1 4.995300348267376e-1 0 2950 0 0 0 -2.0927734375e0 -2.0927734375e0 +fresnel_c I5F11 3000 4.999319997870811e-1 4.999319997870811e-1 0 2960 0 0 0 -1.25361328125e1 -1.25361328125e1 +fresnel_c I4F12 3000 4.991962332259368e-1 4.991962332259368e-1 0 2951 0 0 0 2.103515625e0 2.103515625e0 +fresnel_c I3F13 3000 4.996470988784351e-1 4.996470988784351e-1 0 2954 0 0 0 -1.8994140625e-1 -1.8994140625e-1 +fresnel_c I32F0 3000 5.000000000000016e-1 5.000000000000016e-1 0 2948 0 0 0 4.0636e4 4.0636e4 +fresnel_c I31F1 3000 4.402131992463543e-1 4.402131992463543e-1 0 2938 0 0 0 -1e0 -1e0 +fresnel_c I30F2 3000 4.3950316021378144e-1 4.3950316021378144e-1 0 2956 0 0 0 -2.25e0 -2.25e0 +fresnel_c I29F3 3000 4.9095049036790217e-1 4.9095049036790217e-1 0 2949 0 0 0 -5e0 -5e0 +fresnel_c I28F4 3000 4.9193204212784064e-1 4.9193204212784064e-1 0 2953 0 0 0 -5.625e0 -5.625e0 +fresnel_c I27F5 3000 4.980755959620598e-1 4.980755959620598e-1 0 2948 0 0 0 1.4125e1 1.4125e1 +fresnel_c I26F6 3000 4.9900165977027505e-1 4.9900165977027505e-1 0 2949 0 0 0 4.203125e0 4.203125e0 +fresnel_c I25F7 3000 4.9962841626123256e-1 4.9962841626123256e-1 0 2952 0 0 0 -1.984375e0 -1.984375e0 +fresnel_c I24F8 3000 4.9982203742491366e-1 4.9982203742491366e-1 0 2945 0 0 0 5.19921875e1 5.19921875e1 +fresnel_c I23F9 3000 4.991756412084946e-1 4.991756412084946e-1 0 2938 0 0 0 1.779296875e0 1.779296875e0 +fresnel_c I22F10 3000 4.999750260292993e-1 4.999750260292993e-1 0 2945 0 0 0 7.5869140625e0 7.5869140625e0 +fresnel_c I21F11 3000 4.9998330224412957e-1 4.9998330224412957e-1 0 2954 0 0 0 -2.8134765625e0 -2.8134765625e0 +fresnel_c I20F12 3000 4.997032169622454e-1 4.997032169622454e-1 0 2948 0 0 0 -2.7451171875e0 -2.7451171875e0 +fresnel_c I19F13 3000 4.9859871307371906e-1 4.9859871307371906e-1 0 2952 0 0 0 -1.149169921875e0 -1.149169921875e0 +fresnel_c I18F14 3000 4.999551298822459e-1 4.999551298822459e-1 0 2954 0 0 0 1.6580810546875e0 1.6580810546875e0 +fresnel_c I17F15 3000 4.9981705767295465e-1 4.9981705767295465e-1 0 2951 0 0 0 -1.0743368530273438e2 -1.0743368530273438e2 +fresnel_c I16F16 3000 4.995068417459395e-1 4.995068417459395e-1 0 2952 0 0 0 4.999542236328125e-1 4.999542236328125e-1 +fresnel_c I15F17 3000 4.998924301885697e-1 4.998924301885697e-1 0 2937 0 0 0 2.620208740234375e0 2.620208740234375e0 +fresnel_c I14F18 3000 4.998606184807153e-1 4.998606184807153e-1 0 2939 0 0 0 -5.73011701965332e3 -5.73011701965332e3 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4.9979072069925096e-1 0 2952 0 0 0 5.994011200964451e0 5.994011200964451e0 +fresnel_c I4F28 3000 4.9975578856852604e-1 4.9975578856852604e-1 0 2956 0 0 0 5.590128414332867e-1 5.590128414332867e-1 +fresnel_c I3F29 3000 4.9935196798633075e-1 4.9935196798633075e-1 0 2952 0 0 0 1.5216165836900473e0 1.5216165836900473e0 +fresnel_c I64F0 3000 5e-1 5e-1 0 2932 0 0 0 -8.036781066238358e18 -8.036781066238358e18 +fresnel_c I63F1 3000 4.402131992463543e-1 4.402131992463543e-1 0 2941 0 0 0 1e0 1e0 +fresnel_c I62F2 3000 4.3950316021378144e-1 4.3950316021378144e-1 0 2947 0 0 0 -2.25e0 -2.25e0 +fresnel_c I61F3 3000 4.9095049036790217e-1 4.9095049036790217e-1 0 2939 0 0 0 5e0 5e0 +fresnel_c I60F4 3000 4.9193204212784064e-1 4.9193204212784064e-1 0 2941 0 0 0 5.625e0 5.625e0 +fresnel_c I56F8 3000 4.9954252926073434e-1 4.9954252926073434e-1 0 2942 0 0 0 -8.21875e0 -8.21875e0 +fresnel_c I52F12 3000 4.9977519361537526e-1 4.9977519361537526e-1 0 2948 0 0 0 -3.7372705078125e2 -3.7372705078125e2 +fresnel_c I48F16 3000 4.997710865896781e-1 4.997710865896781e-1 0 2950 0 0 0 -9.367218017578125e0 -9.367218017578125e0 +fresnel_c I44F20 3000 4.996404723698251e-1 4.996404723698251e-1 0 2948 0 0 0 -1.1707645416259766e1 -1.1707645416259766e1 +fresnel_c I40F24 3000 4.999184939397601e-1 4.999184939397601e-1 0 2937 0 0 0 4.999999403953552e-1 4.999999403953552e-1 +fresnel_c I36F28 3000 4.998094462918782e-1 4.998094462918782e-1 0 2951 0 0 0 -9.999879375100136e-1 -9.999879375100136e-1 +fresnel_c I33F31 3000 4.996663775496641e-1 4.996663775496641e-1 0 2956 0 0 0 -6.5083921654426716e7 -6.5083921654426716e7 +fresnel_c I32F32 3000 4.993621859707469e-1 4.993621859707469e-1 0 2953 0 0 0 -1.023e3 -1.023e3 +fresnel_c I31F33 3000 4.9990954426741446e-1 4.9990954426741446e-1 0 2942 0 0 0 -5.2327216231497005e0 -5.2327216231497005e0 +fresnel_c I28F36 3000 4.9991768487945065e-1 4.9991768487945065e-1 0 2955 0 0 0 9.950101020597766e7 9.950101020597766e7 +fresnel_c I24F40 3000 4.9975571752971987e-1 4.9975571752971987e-1 0 2956 0 0 0 -5.226903174871166e6 -5.226903174871166e6 +fresnel_c I20F44 3000 4.9998241575540486e-1 4.9998241575540486e-1 0 2946 0 0 0 -2.28922359418939e5 -2.28922359418939e5 +fresnel_c I16F48 3000 4.9958043700746213e-1 4.9958043700746213e-1 0 2951 0 0 0 -3.757007472960403e0 -3.757007472960403e0 +fresnel_c I12F52 3000 4.9994441578921534e-1 4.9994441578921534e-1 0 2949 0 0 0 1.5707963267953513e0 1.5707963267953513e0 +fresnel_c I8F56 3000 4.996418444477809e-1 4.996418444477809e-1 0 2936 0 0 0 4.508281407786539e0 4.508281407786539e0 +fresnel_c I6F58 3000 4.999980722177399e-1 4.999980722177399e-1 0 2944 0 0 0 2.937977701644603e1 2.937977701644603e1 +fresnel_c I5F59 3000 4.9991380929793383e-1 4.9991380929793383e-1 0 2947 0 0 0 6.061367972917776e0 6.061367972917776e0 +fresnel_c I4F60 3000 4.9991306355366266e-1 4.9991306355366266e-1 0 2949 0 0 0 4.629890476159426e-1 4.629890476159426e-1 +fresnel_c I3F61 3000 4.999472500947364e-1 4.999472500947364e-1 0 2957 0 0 0 -2.0000032092580105e0 -2.0000032092580105e0 +fresnel_c I128F0 3000 5e-1 5e-1 0 2942 0 0 0 1.7014118346046923e38 1.7014118346046923e38 +fresnel_c I127F1 3000 4.402131992463543e-1 4.402131992463543e-1 0 2934 0 0 0 -1e0 -1e0 +fresnel_c I126F2 3000 4.3950316021378144e-1 4.3950316021378144e-1 0 2940 0 0 0 -2.25e0 -2.25e0 +fresnel_c I124F4 3000 4.9193204212784064e-1 4.9193204212784064e-1 0 2942 0 0 0 5.625e0 5.625e0 +fresnel_c I120F8 3000 4.9923224534043453e-1 4.9923224534043453e-1 0 2944 0 0 0 3.48828125e0 3.48828125e0 +fresnel_c I112F16 3000 4.9997440947031574e-1 4.9997440947031574e-1 0 2925 0 0 0 8.500518798828125e-1 8.500518798828125e-1 +fresnel_c I104F24 3000 4.999184939397601e-1 4.999184939397601e-1 0 2947 0 0 0 4.999999403953552e-1 4.999999403953552e-1 +fresnel_c I96F32 3000 4.993621859707469e-1 4.993621859707469e-1 0 2949 0 0 0 1.023e3 1.023e3 +fresnel_c I88F40 3000 4.9937041483670713e-1 4.9937041483670713e-1 0 2926 0 0 0 4.78719952785741e0 4.78719952785741e0 +fresnel_c I80F48 3000 4.999363065862922e-1 4.999363065862922e-1 0 2950 0 0 0 -4.466990763786935e0 -4.466990763786935e0 +fresnel_c I72F56 3000 4.9936000362591e-1 4.9936000362591e-1 0 2942 0 0 0 3.21195762385549e0 3.21195762385549e0 +fresnel_c I68F60 3000 4.9999276387073016e-1 4.9999276387073016e-1 0 2931 0 0 0 8.158756749421425e-1 8.158756749421425e-1 +fresnel_c I67F61 3000 4.9983533304619004e-1 4.9983533304619004e-1 0 2938 0 0 0 -5e-1 -5e-1 +fresnel_c I66F62 3000 4.995841036321998e-1 4.995841036321998e-1 0 2936 0 0 0 1.801439850948198e16 1.801439850948198e16 +fresnel_c I65F63 3000 4.998633853097088e-1 4.998633853097088e-1 0 2939 0 0 0 2.5684413750211754e17 2.5684413750211754e17 +fresnel_c I64F64 3000 4.998661487423339e-1 4.998661487423339e-1 0 2931 0 0 0 -7.778031238423134e18 -7.778031238423134e18 +fresnel_c I63F65 3000 4.997232479810813e-1 4.997232479810813e-1 0 2951 0 0 0 -3.3155651222464384e18 -3.3155651222464384e18 +fresnel_c I62F66 3000 4.9998684536172316e-1 4.9998684536172316e-1 0 2950 0 0 0 3.9500208985833236e-2 3.9500208985833236e-2 +fresnel_c I56F72 3000 4.996436889096111e-1 4.996436889096111e-1 0 2947 0 0 0 -2.553223758940244e16 -2.553223758940244e16 +fresnel_c I48F80 3000 4.992417222536063e-1 4.992417222536063e-1 0 2939 0 0 0 7.038797519061161e0 7.038797519061161e0 +fresnel_c I40F88 3000 4.999697887108413e-1 4.999697887108413e-1 0 2942 0 0 0 -1.4242603704424655e8 -1.4242603704424655e8 +fresnel_c I32F96 3000 4.995988951580011e-1 4.995988951580011e-1 0 2938 0 0 0 3.1415964682870587e0 3.1415964682870587e0 +fresnel_c I24F104 3000 4.996614389981741e-1 4.996614389981741e-1 0 2940 0 0 0 1.6452331724325087e0 1.6452331724325087e0 +fresnel_c I16F112 3000 4.998066999937705e-1 4.998066999937705e-1 0 2951 0 0 0 4.496586271812084e0 4.496586271812084e0 +fresnel_c I10F118 3000 4.9974110352778667e-1 4.9974110352778667e-1 0 2929 0 0 0 -1.2619343542346633e2 -1.2619343542346633e2 +fresnel_c I8F120 3000 4.994425901838095e-1 4.994425901838095e-1 0 2927 0 0 0 9.427978638866348e1 9.427978638866348e1 +fresnel_c I6F122 3000 4.9978259736108904e-1 4.9978259736108904e-1 0 2937 0 0 0 2.6119375121479326e1 2.6119375121479326e1 +fresnel_c I5F123 3000 4.999042990302647e-1 4.999042990302647e-1 0 2945 0 0 0 1.2627400988875373e-1 1.2627400988875373e-1 +fresnel_c I4F124 3000 4.9991415209473694e-1 4.9991415209473694e-1 0 2373 0 0 0 4.591936506839664e0 4.591936506839664e0 +fresnel_c I3F125 3000 4.997507292364346e-1 4.997507292364346e-1 0 2943 0 0 0 -3.2290594098750036e0 -3.2290594098750036e0 +fresnel_s I8F0 3000 4.999999505360164e-1 4.999999505360164e-1 0 2950 0 0 0 -1.27e2 -1.27e2 +fresnel_s I7F1 3000 3.95009920164186e-1 3.95009920164186e-1 0 2953 0 0 0 -1.5e0 -1.5e0 +fresnel_s I6F2 3000 4.7672702327837174e-1 4.7672702327837174e-1 0 2958 0 0 0 2.5e0 2.5e0 +fresnel_s I5F3 3000 4.939268208771619e-1 4.939268208771619e-1 0 2944 0 0 0 -1e0 -1e0 +fresnel_s I4F4 3000 4.991224466658228e-1 4.991224466658228e-1 0 2948 0 0 0 6.4375e0 6.4375e0 +fresnel_s I3F5 3000 4.982025576266881e-1 4.982025576266881e-1 0 2934 0 0 0 -3.09375e0 -3.09375e0 +fresnel_s I16F0 3000 4.999999999999971e-1 4.999999999999971e-1 0 2959 0 0 0 3.2767e4 3.2767e4 +fresnel_s I15F1 3000 3.95009920164186e-1 3.95009920164186e-1 0 2949 0 0 0 -1.5e0 -1.5e0 +fresnel_s I14F2 3000 4.7672702327837174e-1 4.7672702327837174e-1 0 2957 0 0 0 2.5e0 2.5e0 +fresnel_s I13F3 3000 4.939268208771619e-1 4.939268208771619e-1 0 2947 0 0 0 -1e0 -1e0 +fresnel_s I12F4 3000 4.991224466658228e-1 4.991224466658228e-1 0 2952 0 0 0 -6.4375e0 -6.4375e0 +fresnel_s I11F5 3000 4.9878153991267576e-1 4.9878153991267576e-1 0 2961 0 0 0 -6.78125e0 -6.78125e0 +fresnel_s I10F6 3000 4.9814679254512184e-1 4.9814679254512184e-1 0 2954 0 0 0 2.90625e0 2.90625e0 +fresnel_s I9F7 3000 4.9921465398142717e-1 4.9921465398142717e-1 0 2947 0 0 0 1.953125e-1 1.953125e-1 +fresnel_s I8F8 3000 4.9987500105559074e-1 4.9987500105559074e-1 0 2952 0 0 0 -1.076171875e2 -1.076171875e2 +fresnel_s I7F9 3000 4.996144781231063e-1 4.996144781231063e-1 0 2953 0 0 0 4.0078125e0 4.0078125e0 +fresnel_s I6F10 3000 4.998100643406575e-1 4.998100643406575e-1 0 2953 0 0 0 -7.17578125e0 -7.17578125e0 +fresnel_s I5F11 3000 4.9995822735797135e-1 4.9995822735797135e-1 0 2947 0 0 0 1.544921875e0 1.544921875e0 +fresnel_s I4F12 3000 4.99960037233665e-1 4.99960037233665e-1 0 2950 0 0 0 -1.7509765625e0 -1.7509765625e0 +fresnel_s I3F13 3000 4.998075262158528e-1 4.998075262158528e-1 0 2947 0 0 0 -3.34716796875e0 -3.34716796875e0 +fresnel_s I32F0 3000 5.000000000000017e-1 5.000000000000017e-1 0 2962 0 0 0 3.9693e4 3.9693e4 +fresnel_s I31F1 3000 3.95009920164186e-1 3.95009920164186e-1 0 2950 0 0 0 -1.5e0 -1.5e0 +fresnel_s I30F2 3000 4.7672702327837174e-1 4.7672702327837174e-1 0 2957 0 0 0 2.5e0 2.5e0 +fresnel_s I29F3 3000 4.939268208771619e-1 4.939268208771619e-1 0 2950 0 0 0 1e0 1e0 +fresnel_s I28F4 3000 4.991224466658228e-1 4.991224466658228e-1 0 2956 0 0 0 -6.4375e0 -6.4375e0 +fresnel_s I27F5 3000 4.9878153991267576e-1 4.9878153991267576e-1 0 2960 0 0 0 6.78125e0 6.78125e0 +fresnel_s I26F6 3000 4.9814679254512184e-1 4.9814679254512184e-1 0 2944 0 0 0 -2.90625e0 -2.90625e0 +fresnel_s I25F7 3000 4.9921465398142717e-1 4.9921465398142717e-1 0 2961 0 0 0 -1.953125e-1 -1.953125e-1 +fresnel_s I24F8 3000 4.9915345589911186e-1 4.9915345589911186e-1 0 2949 0 0 0 7.6953125e0 7.6953125e0 +fresnel_s I23F9 3000 4.996144781231063e-1 4.996144781231063e-1 0 2957 0 0 0 4.0078125e0 4.0078125e0 +fresnel_s I22F10 3000 4.9933470038211836e-1 4.9933470038211836e-1 0 2949 0 0 0 9.765625e-2 9.765625e-2 +fresnel_s I21F11 3000 4.9984051135836416e-1 4.9984051135836416e-1 0 2953 0 0 0 5.6630859375e0 5.6630859375e0 +fresnel_s I20F12 3000 4.99386375496209e-1 4.99386375496209e-1 0 2964 0 0 0 -1.569580078125e0 -1.569580078125e0 +fresnel_s I19F13 3000 4.998475866128252e-1 4.998475866128252e-1 0 2962 0 0 0 3.16619970703125e3 3.16619970703125e3 +fresnel_s I18F14 3000 4.992679344305356e-1 4.992679344305356e-1 0 2959 0 0 0 -5.372314453125e0 -5.372314453125e0 +fresnel_s I17F15 3000 4.993684623535274e-1 4.993684623535274e-1 0 2955 0 0 0 -5.224334716796875e0 -5.224334716796875e0 +fresnel_s I16F16 3000 4.9969405684431867e-1 4.9969405684431867e-1 0 2944 0 0 0 8.268953979492188e3 8.268953979492188e3 +fresnel_s I15F17 3000 4.9959704953670714e-1 4.9959704953670714e-1 0 2944 0 0 0 -1.0022811889648438e0 -1.0022811889648438e0 +fresnel_s I14F18 3000 4.998717096828945e-1 4.998717096828945e-1 0 2958 0 0 0 -2.0707328300476074e3 -2.0707328300476074e3 +fresnel_s I13F19 3000 4.997495634977062e-1 4.997495634977062e-1 0 2947 0 0 0 2.0574909992218018e3 2.0574909992218018e3 +fresnel_s I12F20 3000 4.995126463914939e-1 4.995126463914939e-1 0 2954 0 0 0 -1.1305859670639038e3 -1.1305859670639038e3 +fresnel_s I11F21 3000 4.9996938103591365e-1 4.9996938103591365e-1 0 2948 0 0 0 -1.2378454208374023e0 -1.2378454208374023e0 +fresnel_s I10F22 3000 4.994823590707503e-1 4.994823590707503e-1 0 2944 0 0 0 -6.281189680099487e0 -6.281189680099487e0 +fresnel_s I9F23 3000 4.9923363746037724e-1 4.9923363746037724e-1 0 2960 0 0 0 2.6415926218032837e0 2.6415926218032837e0 +fresnel_s I8F24 3000 4.999910877924636e-1 4.999910877924636e-1 0 2951 0 0 0 -8.120208382606506e-1 -8.120208382606506e-1 +fresnel_s I7F25 3000 4.999048758284963e-1 4.999048758284963e-1 0 2954 0 0 0 -3.6607249677181244e0 -3.6607249677181244e0 +fresnel_s I6F26 3000 4.9987922190361983e-1 4.9987922190361983e-1 0 2947 0 0 0 5.000011473894119e-1 5.000011473894119e-1 +fresnel_s I5F27 3000 4.996713958233998e-1 4.996713958233998e-1 0 2953 0 0 0 1.9414039701223373e0 1.9414039701223373e0 +fresnel_s I4F28 3000 4.9990762317658394e-1 4.9990762317658394e-1 0 2943 0 0 0 -7.999999303370714e0 -7.999999303370714e0 +fresnel_s I3F29 3000 4.9984027883332255e-1 4.9984027883332255e-1 0 2960 0 0 0 -4.999999962747097e-1 -4.999999962747097e-1 +fresnel_s I64F0 3000 5e-1 5e-1 0 2956 0 0 0 -8.760056552503842e18 -8.760056552503842e18 +fresnel_s I63F1 3000 3.95009920164186e-1 3.95009920164186e-1 0 2949 0 0 0 -1.5e0 -1.5e0 +fresnel_s I62F2 3000 4.7672702327837174e-1 4.7672702327837174e-1 0 2943 0 0 0 2.5e0 2.5e0 +fresnel_s I61F3 3000 4.939268208771619e-1 4.939268208771619e-1 0 2952 0 0 0 -1e0 -1e0 +fresnel_s I60F4 3000 4.991224466658228e-1 4.991224466658228e-1 0 2953 0 0 0 6.4375e0 6.4375e0 +fresnel_s I56F8 3000 4.9922290505374606e-1 4.9922290505374606e-1 0 2931 0 0 0 3.546875e0 3.546875e0 +fresnel_s I52F12 3000 4.9943531970543553e-1 4.9943531970543553e-1 0 2946 0 0 0 6.15234375e-2 6.15234375e-2 +fresnel_s I48F16 3000 4.986592072606126e-1 4.986592072606126e-1 0 2952 0 0 0 6.196746826171875e-1 6.196746826171875e-1 +fresnel_s I44F20 3000 4.9993322392607237e-1 4.9993322392607237e-1 0 2938 0 0 0 1.3858528137207031e0 1.3858528137207031e0 +fresnel_s I40F24 3000 4.9965180551855753e-1 4.9965180551855753e-1 0 2946 0 0 0 1.7893640398979187e0 1.7893640398979187e0 +fresnel_s I36F28 3000 4.998307872498343e-1 4.998307872498343e-1 0 2951 0 0 0 1.9560569860041142e0 1.9560569860041142e0 +fresnel_s I33F31 3000 4.9998609828063073e-1 4.9998609828063073e-1 0 2955 0 0 0 -4.864741861820221e0 -4.864741861820221e0 +fresnel_s I32F32 3000 4.99685009675477e-1 4.99685009675477e-1 0 2946 0 0 0 9.100066028249716e8 9.100066028249716e8 +fresnel_s I31F33 3000 4.997870859842761e-1 4.997870859842761e-1 0 2941 0 0 0 -9.755406073982245e3 -9.755406073982245e3 +fresnel_s I28F36 3000 4.9993841537929784e-1 4.9993841537929784e-1 0 2952 0 0 0 6.630698311520973e0 6.630698311520973e0 +fresnel_s I24F40 3000 4.9957037462070975e-1 4.9957037462070975e-1 0 2943 0 0 0 4.61649283034317e6 4.61649283034317e6 +fresnel_s I20F44 3000 4.997289303428136e-1 4.997289303428136e-1 0 2952 0 0 0 -4.455101475286938e-4 -4.455101475286938e-4 +fresnel_s I16F48 3000 4.997568809515502e-1 4.997568809515502e-1 0 2958 0 0 0 -3.0429203673205105e1 -3.0429203673205105e1 +fresnel_s I12F52 3000 4.9979142666905957e-1 4.9979142666905957e-1 0 2951 0 0 0 -3.141592653124132e0 -3.141592653124132e0 +fresnel_s I8F56 3000 4.9996386517852287e-1 4.9996386517852287e-1 0 2938 0 0 0 2.197652949656627e0 2.197652949656627e0 +fresnel_s I6F58 3000 4.9997329689029807e-1 4.9997329689029807e-1 0 2943 0 0 0 -9.998026257926383e-1 -9.998026257926383e-1 +fresnel_s I5F59 3000 4.989659141251394e-1 4.989659141251394e-1 0 2942 0 0 0 3.125e-2 3.125e-2 +fresnel_s I4F60 3000 4.998042999527519e-1 4.998042999527519e-1 0 2961 0 0 0 7.337125906523586e0 7.337125906523586e0 +fresnel_s I3F61 3000 4.9999713315778865e-1 4.9999713315778865e-1 0 2955 0 0 0 2.9843850617283807e0 2.9843850617283807e0 +fresnel_s I128F0 3000 5e-1 5e-1 0 2953 0 0 0 -2.5627803411410166e31 -2.5627803411410166e31 +fresnel_s I127F1 3000 3.95009920164186e-1 3.95009920164186e-1 0 2949 0 0 0 -1.5e0 -1.5e0 +fresnel_s I126F2 3000 4.7672702327837174e-1 4.7672702327837174e-1 0 2937 0 0 0 2.5e0 2.5e0 +fresnel_s I124F4 3000 4.991224466658228e-1 4.991224466658228e-1 0 2945 0 0 0 6.4375e0 6.4375e0 +fresnel_s I120F8 3000 4.990765809769937e-1 4.990765809769937e-1 0 2953 0 0 0 3.58203125e0 3.58203125e0 +fresnel_s I112F16 3000 4.9948009428227813e-1 4.9948009428227813e-1 0 2943 0 0 0 -1.3546046020507813e4 -1.3546046020507813e4 +fresnel_s I104F24 3000 4.997854710083194e-1 4.997854710083194e-1 0 2938 0 0 0 -5.000038146972656e-1 -5.000038146972656e-1 +fresnel_s I96F32 3000 4.9978542353541444e-1 4.9978542353541444e-1 0 2938 0 0 0 -3.723839344456792e-1 -3.723839344456792e-1 +fresnel_s I88F40 3000 4.992924761126698e-1 4.992924761126698e-1 0 2937 0 0 0 -4.373647301343772e1 -4.373647301343772e1 +fresnel_s I80F48 3000 4.9974154919984354e-1 4.9974154919984354e-1 0 2949 0 0 0 -1.638085840734641e4 -1.638085840734641e4 +fresnel_s I72F56 3000 4.9988039285979263e-1 4.9988039285979263e-1 0 2945 0 0 0 1.42281743424525e9 1.42281743424525e9 +fresnel_s I68F60 3000 4.9971386616370345e-1 4.9971386616370345e-1 0 2936 0 0 0 -2.6082249066393078e10 -2.6082249066393078e10 +fresnel_s I67F61 3000 4.9991239646835184e-1 4.9991239646835184e-1 0 2953 0 0 0 5.4841955745495326e1 5.4841955745495326e1 +fresnel_s I66F62 3000 4.9999883722768657e-1 4.9999883722768657e-1 0 2942 0 0 0 -3.1415926554524383e0 -3.1415926554524383e0 +fresnel_s I65F63 3000 4.999292689375404e-1 4.999292689375404e-1 0 2922 0 0 0 -3.7292786464773466e0 -3.7292786464773466e0 +fresnel_s I64F64 3000 4.9968514425436594e-1 4.9968514425436594e-1 0 2955 0 0 0 -3.8566488749672975e18 -3.8566488749672975e18 +fresnel_s I63F65 3000 4.996974240388933e-1 4.996974240388933e-1 0 2942 0 0 0 -3.152519713880292e0 -3.152519713880292e0 +fresnel_s I62F66 3000 4.9987354399247863e-1 4.9987354399247863e-1 0 2945 0 0 0 -3.141592713194438e0 -3.141592713194438e0 +fresnel_s I56F72 3000 4.9992787054392346e-1 4.9992787054392346e-1 0 2942 0 0 0 1.5438376348667344e0 1.5438376348667344e0 +fresnel_s I48F80 3000 4.99929571866387e-1 4.99929571866387e-1 0 2947 0 0 0 -1.8042539547958095e-3 -1.8042539547958095e-3 +fresnel_s I40F88 3000 4.9944617552579407e-1 4.9944617552579407e-1 0 2953 0 0 0 1.200332979433338e0 1.200332979433338e0 +fresnel_s I32F96 3000 4.998712683644509e-1 4.998712683644509e-1 0 2937 0 0 0 -3.4992813510301954e0 -3.4992813510301954e0 +fresnel_s I24F104 3000 4.9980994381786503e-1 4.9980994381786503e-1 0 2953 0 0 0 6.21073141566883e0 6.21073141566883e0 +fresnel_s I16F112 3000 4.99842695757963e-1 4.99842695757963e-1 0 2925 0 0 0 1.6648672664535132e-7 1.6648672664535132e-7 +fresnel_s I10F118 3000 4.9969105640274053e-1 4.9969105640274053e-1 0 2940 0 0 0 4.999999999999992e-1 4.999999999999992e-1 +fresnel_s I8F120 3000 4.999516229399862e-1 4.999516229399862e-1 0 2957 0 0 0 -1.000000000007276e0 -1.000000000007276e0 +fresnel_s I6F122 3000 4.994566692267189e-1 4.994566692267189e-1 0 2937 0 0 0 1.5707963267948681e0 1.5707963267948681e0 +fresnel_s I5F123 3000 4.993756775083898e-1 4.993756775083898e-1 0 2920 0 0 0 8.934327218154609e0 8.934327218154609e0 +fresnel_s I4F124 3000 4.993816694522103e-1 4.993816694522103e-1 0 2366 0 0 0 7.918062026681481e0 7.918062026681481e0 +fresnel_s I3F125 3000 4.9982004972139543e-1 4.9982004972139543e-1 0 2928 0 0 0 9.788705634847252e-1 9.788705634847252e-1 +gamma I8F0 127 0e0 0e0 0 0 0 0 0 +gamma I7F1 127 4.5509229818896796e-1 2.567583341910251e-1 0 0 0 0 0 5e-1 5e-1 +gamma I6F2 127 4.9756036711236673e-1 4.5509229818896796e-1 0 0 0 0 0 2.5e-1 1.5e0 +gamma I5F3 127 4.761507847244542e-1 4.660584012023881e-1 0 0 0 0 0 6.25e-1 1.125e0 +gamma I4F4 127 4.9872835458351156e-1 4.9872835458351156e-1 0 0 0 0 0 1.6875e0 1.6875e0 +gamma I3F5 127 4.943331943795519e-1 4.9068084288537694e-1 0 0 0 0 0 2.53125e0 1.1875e0 +gamma I16F0 32767 0e0 0e0 0 0 0 0 0 +gamma I15F1 32767 4.9138840442330706e-1 2.567583341910251e-1 0 0 0 0 0 7.5e0 5e-1 +gamma I14F2 32767 4.9756036711236673e-1 4.5509229818896796e-1 0 0 0 0 0 2.5e-1 1.5e0 +gamma I13F3 32767 4.761507847244542e-1 4.660584012023881e-1 0 0 0 0 0 6.25e-1 1.125e0 +gamma I12F4 32767 4.9872835458351156e-1 4.9872835458351156e-1 0 0 0 0 0 1.6875e0 1.6875e0 +gamma I11F5 32767 4.9769157258653934e-1 4.9068084288537694e-1 0 0 0 0 0 6.125e0 1.1875e0 +gamma I10F6 32767 4.992604449621926e-1 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