A very fast low-precision implementation of erff() for compute capability >= 7.5

Accelerated Computing CUDA CUDA Programming and Performance
1

With compute capability 7.5 NVIDIA added hardware support for the computation of tanhf() to the GPU’s multifunction unit (MUFU). Various publications and multiple questions on Mathematics Stackexchange discuss how to use the hyperbolic tangent tanh to compute the error function erf, motivated by similarity of the graphs of the two functions. The simplest such scheme I am aware of (due to Vedder) uses a simple cubic polynomial to translate the argument of erf to the argument of tanh and achieves an accuracy of 0.039%.

For the implementation of fast_erff() below I tuned the coefficients of Vedder’s proposed cubic polynomial specifically for best results with the MUFU.TANH instruction. This implementation requires just five machine instructions.

/*
  Copyright 2023, Norbert Juffa

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__forceinline__ __device__ float raw_tanhf (float a)
{
#if defined (__CUDA_ARCH__) && (__CUDA_ARCH__ < 750)
#error unsupported __CUDA_ARCH__
#else // __CUDA_ARCH__
    asm ("tanh.approx.f32 %0,%1; \n\t" : "+f"(a) : "f"(a));
#endif // __CUDA_ARCH__
    return a;
}

/*
 Based on John D. Vedder, "Simple approximations for the error function and its
 inverse." American Journal of Physics, Vol. 55, No. 8, Aug. 1987, pp. 762-763.

 maximum ulp error: 5735.81, maximum relative error: 3.8735e-4
*/
__forceinline__ __device__ float fast_erff (float x)
{
    float x2 = x * x;
    x = fmaf (fmaf (0.100646973f, x2, 0.128759325f), x, x); // 0x1.9c4000p-4, 0x1.07b2f8p-3
    return raw_tanhf (x);
}