Add routines for arithmetic operations with mixed real and complex types

[codecov-commenter]

Codecov Report

Merging #3 (371def4) into main (2cefd8a) will decrease coverage by 1.94%. The diff coverage is 81.57%.

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@@            Coverage Diff             @@
##             main       #3      +/-   ##
==========================================
- Coverage   87.16%   85.22%   -1.94%     
==========================================
  Files           5        5              
  Lines         148      176      +28     
==========================================
+ Hits          129      150      +21     
- Misses         19       26       +7     

Impacted Files	Coverage Δ
src/types.jl	100.00% <ø> (ø)
src/arithmetic.jl	45.65% <30.00%> (-4.35%)	:arrow_down:
src/complex.jl	100.00% <100.00%> (ø)
src/horner.jl	99.05% <100.00%> (+0.14%)	:arrow_up:

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[heltonmc]

Added support for complex SIMD evalpoly routines when argument is complex and coefficients are real. This is needed upstream in Bessels.jl

The regular evalpoly routine already SIMDs well for a single polynomial evaluation to the differnet complex evalpoly routine. So matching the single polynomial performance when evaluating two or four polynomials in parallel can't be achieved. But this quickly gains significant advantages compared to sequentially evaluating them.

julia> z
1.1 + 1.2im
julia> length(P1)
15

julia> const pp4 = pack_poly((P1, P2, P3, P4))

julia> @benchmark SIMDMath.horner_simd($z, pp4)
BenchmarkTools.Trial: 10000 samples with 1000 evaluations.
 Range (min … max):  6.458 ns … 9.000 ns  ┊ GC (min … max): 0.00% … 0.00%
 Time  (median):     6.583 ns             ┊ GC (median):    0.00%
 Time  (mean ± σ):   6.589 ns ± 0.057 ns  ┊ GC (mean ± σ):  0.00% ± 0.00%

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  6.46 ns        Histogram: frequency by time       6.71 ns <

 Memory estimate: 0 bytes, allocs estimate: 0.

julia> @benchmark evalpoly($z, $P1)
BenchmarkTools.Trial: 10000 samples with 1000 evaluations.
 Range (min … max):  4.000 ns … 20.291 ns  ┊ GC (min … max): 0.00% … 0.00%
 Time  (median):     4.084 ns              ┊ GC (median):    0.00%
 Time  (mean ± σ):   4.109 ns ±  0.311 ns  ┊ GC (mean ± σ):  0.00% ± 0.00%

                              █▁             ▃                
  ▂▁▁▁▁▁▁▁▁▁▁▁▁▁▃▁▁▁▁▁▁▁▁▁▁▁▁▁██▁▁▁▁▁▁▁▁▁▁▁▁▁█▁▁▁▁▁▁▁▁▁▁▁▁▁▃ ▂
  4 ns           Histogram: frequency by time        4.17 ns <

 Memory estimate: 0 bytes, allocs estimate: 0.

julia> @benchmark (evalpoly($z, $P1), evalpoly($z, $P2))
BenchmarkTools.Trial: 10000 samples with 1000 evaluations.
 Range (min … max):  7.417 ns … 31.208 ns  ┊ GC (min … max): 0.00% … 0.00%
 Time  (median):     7.542 ns              ┊ GC (median):    0.00%
 Time  (mean ± σ):   7.566 ns ±  0.451 ns  ┊ GC (mean ± σ):  0.00% ± 0.00%

                          █                                   
  ▂▁▁▁▁▁▁▁▂▁▁▁▁▁▁▁▅▁▁▁▁▁▁▁█▁▁▁▁▁▁▁▅▃▁▁▁▁▁▁▁▃▁▁▁▁▁▁▁▂▁▁▁▁▁▁▁▂ ▂
  7.42 ns        Histogram: frequency by time        7.71 ns <

 Memory estimate: 0 bytes, allocs estimate: 0.

julia> @benchmark (evalpoly($z, $P1), evalpoly($z, $P2), evalpoly($z, $P3))
BenchmarkTools.Trial: 10000 samples with 999 evaluations.
 Range (min … max):  11.177 ns … 16.266 ns  ┊ GC (min … max): 0.00% … 0.00%
 Time  (median):     11.261 ns              ┊ GC (median):    0.00%
 Time  (mean ± σ):   11.285 ns ±  0.074 ns  ┊ GC (mean ± σ):  0.00% ± 0.00%

                         █           ▅                         
  ▂▁▁▁▁▁▁▁▁▁▁▃▃▁▁▁▁▁▁▁▁▁▁█▁▁▁▁▁▁▁▁▁▁▁█▁▁▁▁▁▁▁▁▁▁▁▄▁▁▁▁▁▁▁▁▁▁▂ ▂
  11.2 ns         Histogram: frequency by time        11.4 ns <

 Memory estimate: 0 bytes, allocs estimate: 0.

julia> @benchmark (evalpoly($z, $P1), evalpoly($z, $P2), evalpoly($z, $P3), evalpoly($z, $P4))
BenchmarkTools.Trial: 10000 samples with 998 evaluations.
 Range (min … max):  14.695 ns … 41.458 ns  ┊ GC (min … max): 0.00% … 0.00%
 Time  (median):     14.822 ns              ┊ GC (median):    0.00%
 Time  (mean ± σ):   14.851 ns ±  0.634 ns  ┊ GC (mean ± σ):  0.00% ± 0.00%

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  14.7 ns      Histogram: log(frequency) by time      14.9 ns <

 Memory estimate: 0 bytes, allocs estimate: 0.

So the horner_simd routine is evaluating four different polynomials of length 15 with a complex argument in about 6.5 ns.

EDIT: I've switched the muladd function to match the usual instrinsics fmadd to match the other methods. I believe this is the best way to go as I frequently ran into issues where method errors using Vecs would throw errors in base. These are hard to track down when overloading that function.