[luanluciano93]

fftRec calls math.Cos/math.Sin once per (k,j) pair in the naive DFT leg and once per (k,q) pair in the mixed-radix leg, even though a transform of length n only has n distinct roots of unity. It also allocates a fresh make([]cpx, n) at every node of the recursion tree.

I found this while profiling a WhatsApp call bridge built on meowcaller. In a CPU profile of one 60 ms frame through MlowEncoder:

	% of total CPU
math.cos	27.4%
math.sin	11.4%
mlow.fftRec (cumulative)	46.7%

So roughly 39% of the entire media path was spent recomputing values that never change.

The change

• cache the roots of unity per (size, sign) in a sync.Map, indexed by (k*j) mod n;
• carve the recursion scratch out of one pooled buffer instead of allocating per node.

Indexing by the remainder is also slightly more accurate than the previous sign*2*pi*k*j/n float32 product, whose error grew with k*j.

Numbers

AMD EPYC 9354P, go1.26, -benchtime 200x -count=5, medians:

BenchmarkMlowEncodeFrame   7.18ms -> 2.76ms   (2.6x)
BenchmarkPercFFT            267us ->   53us   (5.0x)
allocs/frame (encoder)      3855  ->  1966
allocs/op (rfft)             129  ->     3

After the change math.cos drops to 2.4% and fftRec to 21.9% cumulative; what is left at the top is real codec work (addPulse, celpDotProd, SmplPitch).

Correctness

The more accurate twiddles flip a few quantiser decisions, so the bitstream changes very slightly — 200 encoded speech frames came to 34,161 bytes instead of 34,181. Everything I could check says the output is still valid:

• the existing mlow tests pass unmodified;
• an unpatched MlowDecoder reads back all 200 frames produced by the patched encoder: no empty frames, no NaNs, every frame carrying energy;
• three real WhatsApp calls placed with this patch carried audio in both directions, and the far end heard the transmitted tones cleanly — indistinguishable from the same test on unpatched code.

If you would rather keep the bitstream bit-identical, this cannot be done with a table at all, since the old code's angle is a float32 product rather than a value with period n. I would be happy to put the table behind a build tag or an option instead, if you prefer.

Benchmarks

I added mlow/fft_bench_test.go at both levels so the change can be re-measured. The encoder benchmark synthesises voice-like audio on purpose: on silence the VAD takes a much shorter path and the numbers flatter the encoder.

[luanluciano93]

Closing as a duplicate of #30, which predates this by a month and is the better change — it shares one twiddle table across all recursion levels, also removes the rfft allocations in perc.go, and quantifies the accuracy improvement against a float64 reference DFT.

I have moved my reproduction and the real-call validation of the bitstream change into a comment on #30.