As I continue to work my way through the report on the accuracy of math library functions by Innocente and Zimmermann, log10f seemed an obvious target for improvements. I managed to reduce the maximum error from 2.08518 ulps in CUDA 11 to 0.95920 ulps for my implementation below. On my sm_75 GPU, the performance of my_log10f() equals that of the CUDA 11’s built-in log10f() function.
/*
Copyright (c) 2022, Norbert Juffa
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*/
/* logarithm base 10; maxulp=0.95920 ulp1=12844035 */
__inline__ __device__ float my_log10f (float a)
{
const float TWO_TO_23 = 8388608.0f; // 0x1.0p+23
const float TWO_TO_M23 = 1.19209290e-7f; // 0x1.0p-23
const float TWO_TO_M126 = 1.175494351e-38f; // 0x1.0p-126
const float LOG10_TWO_HI = 3.01025391e-1f; // 0x1.344000p-2
const float LOG10_TWO_LO = 4.60503907e-6f; // 0x1.3509f8p-18
const float INF_F = __int_as_float (0x7f800000); // infinity
float f, m, r, i, t;
int e;
i = 0.0f;
if (a < TWO_TO_M126) {
a = a * TWO_TO_23;
i = -23.0f;
}
e = (__float_as_int (a) - 0x3f3504f3) & 0xff800000;
m = __int_as_float (__float_as_int (a) - e);
i = fmaf ((float)e, TWO_TO_M23, i);
f = m - 1.0f;
/* Compute log10_1p(m) for m in [sqrt(0.5)-1, sqrt(2.0)-1] */
r = -3.30963135e-2f; // -0x1.0f2000p-5
r = fmaf (r, f, 5.60370684e-2f); // 0x1.cb0e40p-5
r = fmaf (r, f, -5.75154796e-2f); // -0x1.d72ab4p-5
r = fmaf (r, f, 6.15895353e-2f); // 0x1.f88a9ep-5
r = fmaf (r, f, -7.21328035e-2f); // -0x1.2774bap-4
r = fmaf (r, f, 8.68686363e-2f); // 0x1.63d05ep-4
r = fmaf (r, f, -1.08580485e-1f); // -0x1.bcbee4p-4
r = fmaf (r, f, 1.44764677e-1f); // 0x1.287a62p-3
r = fmaf (r, f, -2.17147186e-1f); // -0x1.bcb7aap-3
r = fmaf (r, f, -6.57055154e-2f); // -0x1.0d213ap-4
/* log10(a) = i*log10(2) + log10(m) */
t = fmaf (LOG10_TWO_LO, i, 0.5f * f);
r = fmaf (r, f, t);
r = fmaf (LOG10_TWO_HI, i, r);
/* handle special cases: INFs, NANs, zeros, negative arguments */
if (!((a > 0.0f) && (a < INF_F))) {
asm ("lg2.approx.ftz.f32 %0,%1;" : "=f"(r) : "f"(a));
}
return r;
}