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.  */
44double
45erf (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
98PL_SIG (S, D, 1, erf, -6.0, 6.0)
99PL_TEST_ULP (erf, 1.79)
100PL_TEST_SYM_INTERVAL (erf, 0, 5.9921875, 40000)
101PL_TEST_SYM_INTERVAL (erf, 5.9921875, inf, 40000)
102PL_TEST_SYM_INTERVAL (erf, 0, inf, 40000)
103