Optimize 1D direct NFFT by carrying the phase by recurrence.

[jenskeiner]

This PR adds an optimization to the 1D direct NFFT algorithm that should result in performance improvements.

It's an example of a result obtained by using a coding agent with a harness to guide the code optimization process and ensure correctness. I'll leave it here for a bit before merging.

Unfortunately, the benchmarks suffer from noise from different GitHub runners and thus are not really comparable with develop; see https://codspeed.io/blog/why-glibc-faster-github-actions.

To fix this, we should switch to CodSpeeds Macro Runners. The drawback may be that we can easily hit the 500 minutes per month quota on the free tier. However, for open-source projects, they may be willing to increase the quota as per their website.

It would be best to merge this PR only when we have removed the noise from our CI.

[codspeed-hq]

Merging this PR will improve performance by ×4.4

⚠️ Different runtime environments detected

Some benchmarks with significant performance changes were compared across different runtime environments,
which may affect the accuracy of the results.

Open the report in CodSpeed to investigate

⚡ 30 improved benchmarks
✅ 102 untouched benchmarks

Performance Changes

	Mode	Benchmark	BASE	HEAD	Efficiency
⚡	WallTime	codspeed-macro_gcc_kaiserbessel_double/nfft_forward_direct_1d[512/1600]	41.3 ms	4 ms	×10
⚡	WallTime	codspeed-macro_gcc_kaiserbessel_double/nfft_forward_direct_1d_omp[512/1600]	3,003.1 µs	291.1 µs	×10
⚡	WallTime	codspeed-macro_gcc_kaiserbessel_double/nfft_forward_direct_1d[256/800]	10.4 ms	1 ms	×10
⚡	WallTime	codspeed-macro_gcc_kaiserbessel_double/nfft_forward_direct_1d[128/400]	2,633.1 µs	281.7 µs	×9.3
⚡	WallTime	codspeed-macro_gcc_kaiserbessel_double/nfft_forward_direct_1d_omp[256/800]	769.8 µs	84.6 µs	×9.1
⚡	WallTime	codspeed-macro_gcc_kaiserbessel_double/nfft_forward_direct_1d[64/200]	652 µs	80.9 µs	×8.1
⚡	WallTime	codspeed-macro_gcc_kaiserbessel_double/nfft_forward_direct_1d_omp[128/400]	206.7 µs	28.8 µs	×7.2
⚡	WallTime	codspeed-macro_gcc_kaiserbessel_double/nfft_forward_direct_1d[32/100]	159.8 µs	24.9 µs	×6.4
⚡	WallTime	codspeed-macro_gcc_kaiserbessel_float/nfft_forward_direct_1d_omp[512/1600]	1,567.5 µs	285.2 µs	×5.5
⚡	WallTime	codspeed-macro_gcc_kaiserbessel_float/nfft_forward_direct_1d[512/1600]	21.2 ms	3.9 ms	×5.4
⚡	WallTime	codspeed-macro_gcc_kaiserbessel_float/nfft_forward_direct_1d[256/800]	4,875.7 µs	993.5 µs	×4.9
⚡	WallTime	codspeed-macro_gcc_kaiserbessel_float/nfft_forward_direct_1d_omp[256/800]	373.7 µs	80.9 µs	×4.6
⚡	WallTime	codspeed-macro_gcc_kaiserbessel_double/nfft_forward_direct_1d_omp[64/200]	57.7 µs	13.5 µs	×4.3
⚡	WallTime	codspeed-macro_gcc_kaiserbessel_float/nfft_forward_direct_1d[128/400]	1,100.1 µs	259 µs	×4.2
⚡	WallTime	codspeed-macro_gcc_kaiserbessel_float/nfft_forward_direct_1d[64/200]	262.5 µs	68.6 µs	×3.8
⚡	WallTime	codspeed-macro_gcc_kaiserbessel_long-double/nfft_forward_direct_1d[512/1600]	2,019.3 ms	574.2 ms	×3.5
⚡	WallTime	codspeed-macro_gcc_kaiserbessel_long-double/nfft_forward_direct_1d_omp[512/1600]	144.8 ms	41.4 ms	×3.5
⚡	WallTime	codspeed-macro_gcc_kaiserbessel_float/nfft_forward_direct_1d_omp[128/400]	93.1 µs	26.9 µs	×3.5
⚡	WallTime	codspeed-macro_gcc_kaiserbessel_long-double/nfft_forward_direct_1d[256/800]	503.1 ms	145.8 ms	×3.5
⚡	WallTime	codspeed-macro_gcc_kaiserbessel_long-double/nfft_forward_direct_1d_omp[256/800]	36.5 ms	10.6 ms	×3.4
...	...	...	...	...	...

ℹ️ Only the first 20 benchmarks are displayed. Go to the app to view all benchmarks.

Tip

Curious why this is faster? Comment @codspeedbot explain why this is faster on this PR, or directly use the CodSpeed MCP with your agent.

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_{Comparing feature/nfft-trafo-direct-recurrence (56f7684) with develop (783cb56)}