Fix double precision for exp, sin, cos, erf, erfinv on CPU

[Anri-Lombard]

Proposed changes

Fixes #3047 - These functions were using float32 polynomial approximations for all inputs, causing precision loss for float64. Added proper double precision paths using element-wise std:: calls for exp/sin/cos/erf, and Julia's SpecialFunctions.jl rational approximation for erfinv.

Questions for @awni:

1. For erfinv, I used coefficients from Julia's SpecialFunctions.jl (Blair et al. 1976, 3-region rational approximation, ~1e-13 relative error). Is this acceptable?
2. I added float64 variants and a --cpu flag to the benchmark. Is it useful to include this or rather remove these changes?

Checklist

• I have read the CONTRIBUTING document
• I have run pre-commit run --all-files to format my code / installed pre-commit prior to committing changes
• I have added tests that prove my fix is effective or that my feature works
• I have updated the necessary documentation (if needed)

[awni]

Is the check for N=1 here necessary?

[Anri-Lombard]

Right now Simd<T, 1> (in base_simd.h) doesn't have operator[] - only the general Simd<T, N> in accelerate_simd.h does. So we need the N=1 check to use .value instead of [i]... I think adding operator there could remove this check so that's probably cleaner 🙏

[awni]

The purpose of the issue (#3047), perhaps implicitly, was to add actual vectorized implementations rather than unrolled element-wise implementations. Are you interested in making that update? So basically we'd need to find good approximations for exp, sin/cos, erf and use simd functions to implement them. And similarly for erfinv it should be translated into a simd version.

[Anri-Lombard]

Hey @awni, so my original approach was using Cephes polynomials, but I ran the benchmark and it was slower for everything except exp (where it was only ~4% faster). For sin and cos it was about 50% slower. But I'm happy to change the implementation if a vectorized implementation is preferred 👍 Perhaps there are reasons I'm not thinking of as well?

[awni]

but I ran the benchmark and it was slower for everything except exp (where it was only ~4% faster). For sin and cos it was about 50% slower.

That's interesting and unexpected. What machine were you running benchmarks on?

[Anri-Lombard]

but I ran the benchmark and it was slower for everything except exp (where it was only ~4% faster). For sin and cos it was about 50% slower.

That's interesting and unexpected. What machine were you running benchmarks on?

I'm using an M1 Macbook Pro 🙏