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Improved std::complex sqrt and rsqrt.

Antonio Sánchez requested to merge cantonios/eigen:complex_sqrt into master

Replaces std::sqrt with complex_sqrt for all platforms (previously complex_sqrt was only used for CUDA and MSVC), and implements custom complex_rsqrt.

Also introduces numext::rsqrt to simplify implementation, and modified numext::hypot to adhere to IEEE IEC 6059 for special cases.

The complex_sqrt and complex_rsqrt implementations were found to be significantly faster than std::sqrt<std::complex<T>> and 1/numext::sqrt<std::complex<T>>.

Benchmark file attached. complex_sqrt_benchmarks.cpp

GCC 10, Intel Xeon, x86_64:
---------------------------------------------------------------------------
Benchmark                                 Time             CPU   Iterations
---------------------------------------------------------------------------
BM_Sqrt<std::complex<float>>           9.21 ns         9.21 ns     73225448
BM_StdSqrt<std::complex<float>>        17.1 ns         17.1 ns     40966545
BM_Sqrt<std::complex<double>>          8.53 ns         8.53 ns     81111062
BM_StdSqrt<std::complex<double>>       21.5 ns         21.5 ns     32757248
BM_Rsqrt<std::complex<float>>          10.3 ns         10.3 ns     68047474
BM_DivSqrt<std::complex<float>>        16.3 ns         16.3 ns     42770127
BM_Rsqrt<std::complex<double>>         11.3 ns         11.3 ns     61322028
BM_DivSqrt<std::complex<double>>       16.5 ns         16.5 ns     42200711

Clang 11, Intel Xeon, x86_64:
---------------------------------------------------------------------------
Benchmark                                 Time             CPU   Iterations
---------------------------------------------------------------------------
BM_Sqrt<std::complex<float>>           7.46 ns         7.45 ns     90742042
BM_StdSqrt<std::complex<float>>        16.6 ns         16.6 ns     42369878
BM_Sqrt<std::complex<double>>          8.49 ns         8.49 ns     81629030
BM_StdSqrt<std::complex<double>>       21.8 ns         21.7 ns     31809588
BM_Rsqrt<std::complex<float>>          8.39 ns         8.39 ns     82933666
BM_DivSqrt<std::complex<float>>        14.4 ns         14.4 ns     48638676
BM_Rsqrt<std::complex<double>>         9.83 ns         9.82 ns     70068956
BM_DivSqrt<std::complex<double>>       15.7 ns         15.7 ns     44487798

Clang 9, Pixel 2, aarch64:
---------------------------------------------------------------------------
Benchmark                                 Time             CPU   Iterations
---------------------------------------------------------------------------
BM_Sqrt<std::complex<float>>           24.2 ns         24.1 ns     28616031
BM_StdSqrt<std::complex<float>>         104 ns          103 ns      6826926
BM_Sqrt<std::complex<double>>          31.8 ns         31.8 ns     22157591
BM_StdSqrt<std::complex<double>>        128 ns          128 ns      5437375
BM_Rsqrt<std::complex<float>>          31.9 ns         31.8 ns     22384383
BM_DivSqrt<std::complex<float>>        99.2 ns         98.9 ns      7250438
BM_Rsqrt<std::complex<double>>         46.0 ns         45.8 ns     15338689
BM_DivSqrt<std::complex<double>>        119 ns          119 ns      5898944
Edited by Antonio Sánchez

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