[Paullo9]

Requested by Paul · project thread

Fixes #4

Summary

• MixedRadixPlan gains a generic odd-radix pass for any prime factor other than 2, 3, 4, 5 and 7. It uses the pocketfft passg butterfly, works on one output group at a time, and runs four j at a time on fused multiply-adds. Before this, lengths off vDSP with factors such as 11, 13 or 23 always went through Bluestein.

• FFT2D chooses between the plan and Bluestein with the shape of pocketfft's make_cfft_plan:

  • lengths below 50, or whose largest prime factor is at most √n, always use the plan;
  • otherwise Bluestein is used when 2.5 · m · log2(m) < cost_guess(n).

  pocketfft's own Bluestein weight is 6 · m · log2(m). vDSP's power-of-two DFT is cheaper than pocketfft's, so the weight here is measured with FFTBenchmarkTests instead.

• Issue perf: mixed-radix FFT for lengths off vDSP that factor into 2, 3, 5 and 7 #42 already brought in the 2, 3, 5, 7 plan. With this PR, every length now takes the cheaper of the two paths.

Timings

Local forward fft2, median of 21, one thread:

n	before	after
506 = 2 · 11 · 23	3.77 ms (Bluestein)	3.06 ms (mixed radix)
1001 = 7 · 11 · 13	about 19 ms (Bluestein, as at 1009)	11.5 ms (mixed radix)
97, 106, 127, 509	Bluestein	Bluestein, unchanged (the forced generic pass is 1.3x to 10x slower there)

NumPy 2.5.4 takes 2.34 ms at 506², so a gap of about 1.3x remains there. Lengths in 2, 3, 5 and 7 are unchanged from #42.

Tests

• lengthsPickTheirFFTPath: red before the change, because 506, 11, 13 and 1001 took Bluestein. It now lists the boundary lengths on both sides.
• mixedRadixCostRuleIsPocketffts: the factorisation, cost_guess and the rule at worked values.
• The direct-DFT comparison now covers every element at generic lengths: (97, 11), (13, 1001), (496, 127) and (47, 198).
• A new test compares the inverse with a direct inverse DFT at 121, 169 and 198. These include two generic passes in a row.
• All tests use numpy assert_allclose rtol 1e-7. In the reviewer's scratch run, the generic lengths agreed with a direct DFT to between 1e-16 and 8e-15 relative, in both directions.
• Fast gate green: swift test -c release --parallel --skip SimulatoKitParityTests. The parity target was not run in this session.
• Reviewed before the commit: no must-fix. The inverse test was added from the review.

Left for later

docs/open-items.md and the Grid row in docs/parity.md still describe the old FFT routing. Both files are touched by open docs PRs, so they get a line once those PRs land. README, docs/status.md and the DocC article are updated here.

🤖 Generated with Claude Code