The exponentially scaled complementary error function computes ex*x and is useful in preventing premature underflow when the complementary error function is combined with fast-growing terms, for example when computing the Mills ratio: M(x) = exp(x*x) * sqrt(π/2) * erfc (x/sqrt(2)). The single-precision implementation of it, erfcxf(), in CUDA 11 has a maximum error of 3.90275 ulps.
In my implementation below, my_erfcxf(), the maximum error is reduced to 2.4504 ulps. On my Quadro RTX 4000 (compute capability 7.5) it performs identically to CUDA’s built-in function within a measurement noise level of 2%. I would expect this to hold across all GPUs currently supported by CUDA.
/*
Copyright (c) 2022, Norbert Juffa
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*/
/* Compute exponential function with maximum error 0.86565 ulps */
__forceinline__ __device__ float my_expf (float a)
{
float f, r, j, s, t;
int i, ia;
// exp(a) = 2**i * exp(f); i = rintf (a / log(2))
j = fmaf (1.442695f, a, 12582912.f) - 12582912.f; // 0x1.715476p0, 0x1.8p23
f = fmaf (j, -6.93145752e-1f, a); // -0x1.62e400p-1 // log_2_hi
f = fmaf (j, -1.42860677e-6f, f); // -0x1.7f7d1cp-20 // log_2_lo
i = (int)j;
// approximate r = exp(f) on interval [-log(2)/2, +log(2)/2]
r = 1.37805939e-3f; // 0x1.694000p-10
r = fmaf (r, f, 8.37312452e-3f); // 0x1.125edcp-7
r = fmaf (r, f, 4.16695364e-2f); // 0x1.555b5ap-5
r = fmaf (r, f, 1.66664720e-1f); // 0x1.555450p-3
r = fmaf (r, f, 4.99999851e-1f); // 0x1.fffff6p-2
r = fmaf (r, f, 1.00000000e+0f); // 0x1.000000p+0
r = fmaf (r, f, 1.00000000e+0f); // 0x1.000000p+0
// exp(a) = 2**i * r
ia = (i > 0) ? 0 : 0x83000000;
s = __int_as_float (0x7f000000 + ia);
t = __int_as_float ((i << 23) - ia);
r = r * s;
r = r * t;
// handle special cases: severe overflow / underflow
if (fabsf (a) >= 104.0f) r = s * s;
return r;
}
/*
Compute the exponentially scaled complementary error function exp(x*x)*erfc(x)
maximum error positive half-plane: 1.97521 ulps
maximum error negative half-plane: 2.45040 ulps
*/
__inline__ __device__ float my_erfcxf (float x)
{
float MY_INF = __int_as_float (0x7f800000);
float a, d, e, p, q, r, s, t;
a = fabsf (x);
/* Compute q = (a-2)/(a+2) accurately. [0,INF) -> [-1,1] */
p = a + 2.0f;
asm ("rcp.approx.ftz.f32 %0,%1;" : "=f"(r) : "f"(p)); // r = 1.0f / p
q = fmaf (-4.0f, r, 1.0f);
t = fmaf (q + 1.0f, -2.0f, a);
e = fmaf (-a, q, t);
q = fmaf (r, e, q);
/* Approximate (1+2*a)*exp(a*a)*erfc(a) as p(q)+1 for q in [-1,1] */
p = 5.92619181e-5f; // 0x1.f12000p-15
p = fmaf (p, q, 1.61231728e-4f); // 0x1.5220a0p-13
p = fmaf (p, q, -3.46499495e-4f); // -0x1.6b54c0p-12
p = fmaf (p, q, -1.39683776e-3f); // -0x1.6e2c32p-10
p = fmaf (p, q, 1.20587996e-3f); // 0x1.3c1d3cp-10
p = fmaf (p, q, 8.69013276e-3f); // 0x1.1cc21ep-7
p = fmaf (p, q, -8.01389851e-3f); // -0x1.069974p-7
p = fmaf (p, q, -5.42122647e-2f); // -0x1.bc1b5cp-5
p = fmaf (p, q, 1.64048553e-1f); // 0x1.4ff8b0p-3
p = fmaf (p, q, -1.66031078e-1f); // -0x1.54081ap-3
p = fmaf (p, q, -9.27637145e-2f); // -0x1.7bf5cep-4
p = fmaf (p, q, 2.76978403e-1f); // 0x1.1ba03ap-2
/* Divide (1+p) by (1+2*a) ==> exp(a*a)*erfc(a) */
d = fmaf (0.25f, a, 0.125f);
asm ("rcp.approx.ftz.f32 %0,%1;" : "=f"(r) : "f"(d)); // r = 1.0f / d
q = fmaf (p, r, r);
e = fmaf (fmaf (-a, q, 4.0f), 0.25f, fmaf (-0.125f, q, p)); // residual
r = 0.125f * fmaf (e, r, q);
if (a == MY_INF) r = 0.0f;
/* Handle negative arguments: erfcx(x) = 2*exp(x*x) - erfcx(|x|) */
if (x < 0.0f) {
s = x * x;
d = fmaf (x, x, -s);
e = my_expf (s);
r = e - r;
r = fmaf (e, d + d, r);
r = r + e;
if (e == MY_INF) r = e; // avoid creating NaN
}
return r;
}