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 94.16%. Comparing base (46a2874) to head (462a3cf).

Additional details and impacted files

@@            Coverage Diff             @@
##           master     #491      +/-   ##
==========================================
+ Coverage   94.02%   94.16%   +0.13%     
==========================================
  Files          14       14              
  Lines        2897     2965      +68     
==========================================
+ Hits         2724     2792      +68     
  Misses        173      173              

Flag	Coverage Δ
unittests	94.16% <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]

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.

[oscardssmith]

Other than missing tests for Inf, looks good to me. @devmotion any further sugestions?