Added erf(x) Float64 and Float32 Julia implementations

[AhmedYKadah]

Faster than current wrapper function call (including Float32 function call).
Uses algorithm based on https://github.com/ARM-software/optimized-routines/blob/master/math/erf.c

[codecov]

Codecov Report

✅ All modified and coverable lines are covered by tests.
✅ Project coverage is 94.24%. Comparing base (1f0527c) to head (1d45644).

Additional details and impacted files

@@            Coverage Diff             @@
##           master     #491      +/-   ##
==========================================
+ Coverage   94.11%   94.24%   +0.13%     
==========================================
  Files          14       14              
  Lines        2905     2973      +68     
==========================================
+ Hits         2734     2802      +68     
  Misses        171      171              

Flag	Coverage Δ
unittests	94.24% <100.00%> (+0.13%)	⬆️

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[AhmedYKadah]

Old:
Float 64
@benchmark SpecialFunctions.erf(data) setup=(data=6*rand(Float64)-3) samples=1000000

BenchmarkTools.Trial: 217729 samples with 1000 evaluations per sample.
Range (min … max): 6.300 ns … 283.700 ns ┊ GC (min … max): 0.00% … 0.00%
Time (median): 29.500 ns ┊ GC (median): 0.00%
Time (mean ± σ): 21.993 ns ± 13.209 ns ┊ GC (mean ± σ): 0.00% ± 0.00%

Float32
@benchmark SpecialFunctions.erf(data) setup=(data=6*rand(Float32)-3) samples=1000000

BenchmarkTools.Trial: 312732 samples with 1000 evaluations per sample.
Range (min … max): 4.300 ns … 125.100 ns ┊ GC (min … max): 0.00% … 0.00%
Time (median): 19.900 ns ┊ GC (median): 0.00%
Time (mean ± σ): 15.035 ns ± 7.951 ns ┊ GC (mean ± σ): 0.00% ± 0.00%

New:
Float64
@benchmark erf(data) setup=(data=6*rand(Float64)-3) samples=1000000

BenchmarkTools.Trial: 507504 samples with 1000 evaluations per sample.
Range (min … max): 5.400 ns … 4.890 μs ┊ GC (min … max): 0.00% … 0.00%
Time (median): 8.700 ns ┊ GC (median): 0.00%
Time (mean ± σ): 8.775 ns ± 9.855 ns ┊ GC (mean ± σ): 0.00% ± 0.00%

Float32
@benchmark Float32(erf(data)) setup=(data=6*rand(Float64)-3) samples=1000000

BenchmarkTools.Trial: 526797 samples with 1000 evaluations per sample.
Range (min … max): 5.400 ns … 195.500 ns ┊ GC (min … max): 0.00% … 0.00%
Time (median): 8.800 ns ┊ GC (median): 0.00%
Time (mean ± σ): 8.521 ns ± 2.236 ns ┊ GC (mean ± σ): 0.00% ± 0.00%

[AhmedYKadah]

Float32 implementation available, but not faster than Float64 version due to a exp() call.
Float64 version still faster than old Float32.

[AhmedYKadah]

need to clean up polynomial evaluations.
code also could use more organization

[AhmedYKadah]

Remaining: erfc Float64 and Float32 implementations, and the erf Float32 implementation

[oscardssmith]

any chance you can regenerate the polies to make this be 0x40180000? If you can, you would be able to use UInt16 literals for all of these by only taking the top 16 rather than top 32 bits.

[AhmedYKadah]

would that have a meaningful impact?

[oscardssmith]

probably not much. It might just save a cycle or 2.

[oscardssmith]

if you wanted to test the speed, you could try it without regenerating the polys and it should give you a good idea.

[oscardssmith]

if you wanted to do a Float16 impl, it should be easier than the others. Specifically, the domain is only to 2, and the accuracy required is much reduced.

[oscardssmith]

100% could wait for a followup PR.

[AhmedYKadah]

I'm thinking that too to be honest.
this and the poli regen.

[oscardssmith]

Given that this is faster and accurate, seems good to merge to me!

[mschauer]

Are there any tests for edge cases/ULP in the c version we do not do ourselves?

[devmotion]

The implementation does not handle NaN32 and NaN16 correctly:

julia> erf(NaN32)
1.0f0

julia> erf(NaN16)
Float16(1.0)

[mschauer]

Then we should also add a test for these

[AhmedYKadah]

There aren't any tests for erfc. Is that expected?

[AhmedYKadah]

Any other changes needed?

[oscardssmith]

we should probably should test erfc.