Cody-Waite argument reduction for trigonometric functions

[pleroy]

Benchmarks:

2024-10-11T15:14:59+02:00
Running C:\Users\phl\Projects\GitHub\Principia\Principia\Release\x64\benchmarks.exe
Run on (48 X 3793 MHz CPU s)
CPU Caches:
  L1 Data 32 KiB (x24)
  L1 Instruction 32 KiB (x24)
  L2 Unified 512 KiB (x24)
  L3 Unified 32768 KiB (x4)
------------------------------------------------------------------------------------------------------
Benchmark                                                            Time             CPU   Iterations
------------------------------------------------------------------------------------------------------
BM_EvaluateElementaryFunction<Metric::Latency, std::sin>          13.4 ns         13.4 ns     56000000 cycles: 50.6264
BM_EvaluateElementaryFunction<Metric::Throughput, std::sin>       3.88 ns         3.84 ns    179200000 cycles: 14.6875
BM_EvaluateElementaryFunction<Metric::Latency, cr_sin>            32.6 ns         32.2 ns     21334000 cycles: 123.51
BM_EvaluateElementaryFunction<Metric::Throughput, cr_sin>         21.4 ns         21.8 ns     34462000 cycles: 81.165
BM_EvaluateElementaryFunction<Metric::Latency, Sin>               16.1 ns         16.0 ns     44800000 cycles: 60.9541
BM_EvaluateElementaryFunction<Metric::Throughput, Sin>            7.18 ns         7.25 ns    112000000 cycles: 27.202
BM_EvaluateElementaryFunction<Metric::Latency, std::cos>          13.8 ns         13.8 ns     49778000 cycles: 52.2563
BM_EvaluateElementaryFunction<Metric::Throughput, std::cos>       3.97 ns         3.90 ns    172308000 cycles: 15.0264
BM_EvaluateElementaryFunction<Metric::Latency, cr_cos>            35.9 ns         36.1 ns     19479000 cycles: 136.108
BM_EvaluateElementaryFunction<Metric::Throughput, cr_cos>         22.3 ns         22.0 ns     29867000 cycles: 84.3854
BM_EvaluateElementaryFunction<Metric::Latency, Cos>               14.7 ns         14.6 ns     44800000 cycles: 55.8376
BM_EvaluateElementaryFunction<Metric::Throughput, Cos>            7.12 ns         7.15 ns     89600000 cycles: 26.9696

#1760.

[eggrobin]

Add a comment noting that it so happens that 2 / π is correctly rounded.

[pleroy]

Added a comment explaining why we don't care too much about the rounding here.

[eggrobin]

Is this faster than computing it using _mm_floor_sd?

[pleroy]

It is actually significantly faster, shaving 7 cycles on Sin and 5 on Cos. The code is rather ugly, though. Updated the benchmark numbers.