Leaves-first order for large power-of-two transforms - #5
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W butterflies per iteration on SIMD.jl vectors (2 for ComplexF64, 4 for ComplexF32); the twiddle table keeps its compact layout and the triplets are rearranged in registers. Falls back to the scalar loop for strided or non-contiguous output and for other element types.
Above 2^18 elements the depth-first recursion reads each leaf at a stride that uses one element per cache line. Compute the 4096-point sub-transforms first, in input order, gathering one cache line of pencils at a time into a plan-owned buffer, then run the remaining butterfly passes level by level. Same operations, identical output; 2^20 43 -> 21 ms, 2^22 188 -> 101 ms (ComplexF64, Neoverse-N1, with the SIMD pass).
…of two), test for 2^18-2^21 N/B must be a power of 4; with a fixed B = 4096 that failed for odd log2 N (e.g. the 2^19 complex transform inside a 2^20 rfft) and the digit reversal wrote out of bounds.
This was referenced Aug 30, 2026
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Stacked on #3 (the diff includes the SIMD pass). Above 2^18 elements the depth-first recursion reads each 64-point leaf at a stride that uses one element per cache line (2.2–3.8× penalty at 2^18–2^22,
benchmark/KERNEL_STAGES.md).src/leaffirst.jl: compute the 4096/2048-point sub-transforms first in input order, gathering one cache line of pencils at a time into a plan-owned buffer, then the remaining butterfly passes level by level; same operations, identical output.compare3 vs
integration/all, aarch64 Neoverse-N1, 1 thread, 505 cases (with #3): 1.21× geomean; ComplexF64 fft 2^18 6.4→4.1 ms, 2^20 44.5→21.1 ms, 2^22 193→103 ms = FFTW MEASURE's 20.8 / 102.2 ms (ESTIMATE 52.7 / 238 ms); pow2 class 1.30× (ComplexF64 now 0.94× of FFTW ESTIMATE), ComplexF32 pow2 1.54×, primes/awkward 1.2–1.5× (Bluestein's padded transforms). Table:benchmark/compare3/aarch64/large/COMPARE3.mdonbench/compare3.Tested: full suite green on 4 threads; new
test/leaffirst.jl(2^18–2^21, both parities of log2 n, complex/real/backward, zero allocation) — added after an rfft of 2^20 exposed a wrong digit reversal for odd log2 n.x86-64 (companion session,
benchmark/x86-64/compare3/leaves/): 1.25× geomean; pow2 1.25× ComplexF64 / 1.68× ComplexF32; primes 1.19× / 1.64× (Float32 primes reach parity with FFTW ESTIMATE, 1.01×).