1*f3087befSAndrew Turner /*
2*f3087befSAndrew Turner * Single-precision erf(x) function.
3*f3087befSAndrew Turner *
4*f3087befSAndrew Turner * Copyright (c) 2023-2024, Arm Limited.
5*f3087befSAndrew Turner * SPDX-License-Identifier: MIT OR Apache-2.0 WITH LLVM-exception
6*f3087befSAndrew Turner */
7*f3087befSAndrew Turner
8*f3087befSAndrew Turner #include "math_config.h"
9*f3087befSAndrew Turner #include "test_sig.h"
10*f3087befSAndrew Turner #include "test_defs.h"
11*f3087befSAndrew Turner
12*f3087befSAndrew Turner #define TwoOverSqrtPiMinusOne 0x1.06eba8p-3f
13*f3087befSAndrew Turner #define Shift 0x1p16f
14*f3087befSAndrew Turner #define OneThird 0x1.555556p-2f
15*f3087befSAndrew Turner
16*f3087befSAndrew Turner /* Fast erff approximation based on series expansion near x rounded to
17*f3087befSAndrew Turner nearest multiple of 1/128.
18*f3087befSAndrew Turner Let d = x - r, and scale = 2 / sqrt(pi) * exp(-r^2). For x near r,
19*f3087befSAndrew Turner
20*f3087befSAndrew Turner erf(x) ~ erf(r)
21*f3087befSAndrew Turner + scale * d * [
22*f3087befSAndrew Turner + 1
23*f3087befSAndrew Turner - r d
24*f3087befSAndrew Turner + 1/3 (2 r^2 - 1) d^2
25*f3087befSAndrew Turner - 1/6 (r (2 r^2 - 3) ) d^3
26*f3087befSAndrew Turner + 1/30 (4 r^4 - 12 r^2 + 3) d^4
27*f3087befSAndrew Turner ]
28*f3087befSAndrew Turner
29*f3087befSAndrew Turner This single precision implementation uses only the following terms:
30*f3087befSAndrew Turner
31*f3087befSAndrew Turner erf(x) ~ erf(r) + scale * d * [1 - r * d - 1/3 * d^2]
32*f3087befSAndrew Turner
33*f3087befSAndrew Turner Values of erf(r) and scale are read from lookup tables.
34*f3087befSAndrew Turner For |x| > 3.9375, erf(|x|) rounds to 1.0f.
35*f3087befSAndrew Turner
36*f3087befSAndrew Turner Maximum error: 1.93 ULP
37*f3087befSAndrew Turner erff(0x1.c373e6p-9) got 0x1.fd686cp-9
38*f3087befSAndrew Turner want 0x1.fd6868p-9. */
39*f3087befSAndrew Turner float
arm_math_erff(float x)40*f3087befSAndrew Turner arm_math_erff (float x)
41*f3087befSAndrew Turner {
42*f3087befSAndrew Turner /* Get absolute value and sign. */
43*f3087befSAndrew Turner uint32_t ix = asuint (x);
44*f3087befSAndrew Turner uint32_t ia = ix & 0x7fffffff;
45*f3087befSAndrew Turner uint32_t sign = ix & ~0x7fffffff;
46*f3087befSAndrew Turner
47*f3087befSAndrew Turner /* |x| < 0x1p-62. Triggers exceptions. */
48*f3087befSAndrew Turner if (unlikely (ia < 0x20800000))
49*f3087befSAndrew Turner return fmaf (TwoOverSqrtPiMinusOne, x, x);
50*f3087befSAndrew Turner
51*f3087befSAndrew Turner if (ia < 0x407b8000) /* |x| < 4 - 8 / 128 = 3.9375. */
52*f3087befSAndrew Turner {
53*f3087befSAndrew Turner /* Lookup erf(r) and scale(r) in tables, e.g. set erf(r) to 0 and scale
54*f3087befSAndrew Turner to 2/sqrt(pi), when x reduced to r = 0. */
55*f3087befSAndrew Turner float a = asfloat (ia);
56*f3087befSAndrew Turner float z = a + Shift;
57*f3087befSAndrew Turner uint32_t i = asuint (z) - asuint (Shift);
58*f3087befSAndrew Turner float r = z - Shift;
59*f3087befSAndrew Turner float erfr = __v_erff_data.tab[i].erf;
60*f3087befSAndrew Turner float scale = __v_erff_data.tab[i].scale;
61*f3087befSAndrew Turner
62*f3087befSAndrew Turner /* erf(x) ~ erf(r) + scale * d * (1 - r * d - 1/3 * d^2). */
63*f3087befSAndrew Turner float d = a - r;
64*f3087befSAndrew Turner float d2 = d * d;
65*f3087befSAndrew Turner float y = -fmaf (OneThird, d, r);
66*f3087befSAndrew Turner y = fmaf (fmaf (y, d2, d), scale, erfr);
67*f3087befSAndrew Turner return asfloat (asuint (y) | sign);
68*f3087befSAndrew Turner }
69*f3087befSAndrew Turner
70*f3087befSAndrew Turner /* Special cases : erff(nan)=nan, erff(+inf)=+1 and erff(-inf)=-1. */
71*f3087befSAndrew Turner if (unlikely (ia >= 0x7f800000))
72*f3087befSAndrew Turner return (1.0f - (float) (sign >> 30)) + 1.0f / x;
73*f3087befSAndrew Turner
74*f3087befSAndrew Turner /* Boring domain (|x| >= 4.0). */
75*f3087befSAndrew Turner return asfloat (sign | asuint (1.0f));
76*f3087befSAndrew Turner }
77*f3087befSAndrew Turner
78*f3087befSAndrew Turner TEST_ULP (arm_math_erff, 1.43)
79*f3087befSAndrew Turner TEST_SYM_INTERVAL (arm_math_erff, 0, 3.9375, 40000)
80*f3087befSAndrew Turner TEST_SYM_INTERVAL (arm_math_erff, 3.9375, inf, 40000)
81*f3087befSAndrew Turner TEST_SYM_INTERVAL (arm_math_erff, 0, inf, 40000)
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