1*412f47f9SXin Li /*
2*412f47f9SXin Li * Double-precision erf(x) function.
3*412f47f9SXin Li *
4*412f47f9SXin Li * Copyright (c) 2023, Arm Limited.
5*412f47f9SXin Li * SPDX-License-Identifier: MIT OR Apache-2.0 WITH LLVM-exception
6*412f47f9SXin Li */
7*412f47f9SXin Li
8*412f47f9SXin Li #include "math_config.h"
9*412f47f9SXin Li #include "pl_sig.h"
10*412f47f9SXin Li #include "pl_test.h"
11*412f47f9SXin Li
12*412f47f9SXin Li #define TwoOverSqrtPiMinusOne 0x1.06eba8214db69p-3
13*412f47f9SXin Li #define Shift 0x1p45
14*412f47f9SXin Li
15*412f47f9SXin Li /* Polynomial coefficients. */
16*412f47f9SXin Li #define OneThird 0x1.5555555555555p-2
17*412f47f9SXin Li #define TwoThird 0x1.5555555555555p-1
18*412f47f9SXin Li
19*412f47f9SXin Li #define TwoOverFifteen 0x1.1111111111111p-3
20*412f47f9SXin Li #define TwoOverFive 0x1.999999999999ap-2
21*412f47f9SXin Li #define Tenth 0x1.999999999999ap-4
22*412f47f9SXin Li
23*412f47f9SXin Li #define TwoOverNine 0x1.c71c71c71c71cp-3
24*412f47f9SXin Li #define TwoOverFortyFive 0x1.6c16c16c16c17p-5
25*412f47f9SXin Li #define Sixth 0x1.555555555555p-3
26*412f47f9SXin Li
27*412f47f9SXin Li /* Fast erf approximation based on series expansion near x rounded to
28*412f47f9SXin Li nearest multiple of 1/128.
29*412f47f9SXin Li Let d = x - r, and scale = 2 / sqrt(pi) * exp(-r^2). For x near r,
30*412f47f9SXin Li
31*412f47f9SXin Li erf(x) ~ erf(r)
32*412f47f9SXin Li + scale * d * [
33*412f47f9SXin Li + 1
34*412f47f9SXin Li - r d
35*412f47f9SXin Li + 1/3 (2 r^2 - 1) d^2
36*412f47f9SXin Li - 1/6 (r (2 r^2 - 3)) d^3
37*412f47f9SXin Li + 1/30 (4 r^4 - 12 r^2 + 3) d^4
38*412f47f9SXin Li - 1/90 (4 r^4 - 20 r^2 + 15) d^5
39*412f47f9SXin Li ]
40*412f47f9SXin Li
41*412f47f9SXin Li Maximum measure error: 2.29 ULP
42*412f47f9SXin Li erf(-0x1.00003c924e5d1p-8) got -0x1.20dd59132ebadp-8
43*412f47f9SXin Li want -0x1.20dd59132ebafp-8. */
44*412f47f9SXin Li double
erf(double x)45*412f47f9SXin Li erf (double x)
46*412f47f9SXin Li {
47*412f47f9SXin Li /* Get absolute value and sign. */
48*412f47f9SXin Li uint64_t ix = asuint64 (x);
49*412f47f9SXin Li uint64_t ia = ix & 0x7fffffffffffffff;
50*412f47f9SXin Li uint64_t sign = ix & ~0x7fffffffffffffff;
51*412f47f9SXin Li
52*412f47f9SXin Li /* |x| < 0x1p-508. Triggers exceptions. */
53*412f47f9SXin Li if (unlikely (ia < 0x2030000000000000))
54*412f47f9SXin Li return fma (TwoOverSqrtPiMinusOne, x, x);
55*412f47f9SXin Li
56*412f47f9SXin Li if (ia < 0x4017f80000000000) /* |x| < 6 - 1 / 128 = 5.9921875. */
57*412f47f9SXin Li {
58*412f47f9SXin Li /* Set r to multiple of 1/128 nearest to |x|. */
59*412f47f9SXin Li double a = asdouble (ia);
60*412f47f9SXin Li double z = a + Shift;
61*412f47f9SXin Li uint64_t i = asuint64 (z) - asuint64 (Shift);
62*412f47f9SXin Li double r = z - Shift;
63*412f47f9SXin Li /* Lookup erf(r) and scale(r) in table.
64*412f47f9SXin Li Set erf(r) to 0 and scale to 2/sqrt(pi) for |x| <= 0x1.cp-9. */
65*412f47f9SXin Li double erfr = __erf_data.tab[i].erf;
66*412f47f9SXin Li double scale = __erf_data.tab[i].scale;
67*412f47f9SXin Li
68*412f47f9SXin Li /* erf(x) ~ erf(r) + scale * d * poly (d, r). */
69*412f47f9SXin Li double d = a - r;
70*412f47f9SXin Li double r2 = r * r;
71*412f47f9SXin Li double d2 = d * d;
72*412f47f9SXin Li
73*412f47f9SXin Li /* poly (d, r) = 1 + p1(r) * d + p2(r) * d^2 + ... + p5(r) * d^5. */
74*412f47f9SXin Li double p1 = -r;
75*412f47f9SXin Li double p2 = fma (TwoThird, r2, -OneThird);
76*412f47f9SXin Li double p3 = -r * fma (OneThird, r2, -0.5);
77*412f47f9SXin Li double p4 = fma (fma (TwoOverFifteen, r2, -TwoOverFive), r2, Tenth);
78*412f47f9SXin Li double p5
79*412f47f9SXin Li = -r * fma (fma (TwoOverFortyFive, r2, -TwoOverNine), r2, Sixth);
80*412f47f9SXin Li
81*412f47f9SXin Li double p34 = fma (p4, d, p3);
82*412f47f9SXin Li double p12 = fma (p2, d, p1);
83*412f47f9SXin Li double y = fma (p5, d2, p34);
84*412f47f9SXin Li y = fma (y, d2, p12);
85*412f47f9SXin Li
86*412f47f9SXin Li y = fma (fma (y, d2, d), scale, erfr);
87*412f47f9SXin Li return asdouble (asuint64 (y) | sign);
88*412f47f9SXin Li }
89*412f47f9SXin Li
90*412f47f9SXin Li /* Special cases : erf(nan)=nan, erf(+inf)=+1 and erf(-inf)=-1. */
91*412f47f9SXin Li if (unlikely (ia >= 0x7ff0000000000000))
92*412f47f9SXin Li return (1.0 - (double) (sign >> 62)) + 1.0 / x;
93*412f47f9SXin Li
94*412f47f9SXin Li /* Boring domain (|x| >= 6.0). */
95*412f47f9SXin Li return asdouble (sign | asuint64 (1.0));
96*412f47f9SXin Li }
97*412f47f9SXin Li
98*412f47f9SXin Li PL_SIG (S, D, 1, erf, -6.0, 6.0)
99*412f47f9SXin Li PL_TEST_ULP (erf, 1.79)
100*412f47f9SXin Li PL_TEST_SYM_INTERVAL (erf, 0, 5.9921875, 40000)
101*412f47f9SXin Li PL_TEST_SYM_INTERVAL (erf, 5.9921875, inf, 40000)
102*412f47f9SXin Li PL_TEST_SYM_INTERVAL (erf, 0, inf, 40000)
103