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