1 /*
2 * Double-precision inverse error function.
3 *
4 * Copyright (c) 2023-2024, Arm Limited.
5 * SPDX-License-Identifier: MIT OR Apache-2.0 WITH LLVM-exception
6 */
7 #include "math_config.h"
8 #include "poly_scalar_f64.h"
9 #include "test_sig.h"
10 #include "test_defs.h"
11
12 const static struct
13 {
14 /* We use P_N and Q_N to refer to arrays of coefficients, where P_N is the
15 coeffs of the numerator in table N of Blair et al, and Q_N is the coeffs
16 of the denominator. */
17 double P_17[7], Q_17[7], P_37[8], Q_37[8], P_57[9], Q_57[10];
18 } data = {
19 .P_17 = { 0x1.007ce8f01b2e8p+4, -0x1.6b23cc5c6c6d7p+6, 0x1.74e5f6ceb3548p+7,
20 -0x1.5200bb15cc6bbp+7, 0x1.05d193233a849p+6, -0x1.148c5474ee5e1p+3,
21 0x1.689181bbafd0cp-3 },
22 .Q_17 = { 0x1.d8fb0f913bd7bp+3, -0x1.6d7f25a3f1c24p+6, 0x1.a450d8e7f4cbbp+7,
23 -0x1.bc3480485857p+7, 0x1.ae6b0c504ee02p+6, -0x1.499dfec1a7f5fp+4,
24 0x1p+0 },
25 .P_37 = { -0x1.f3596123109edp-7, 0x1.60b8fe375999ep-2, -0x1.779bb9bef7c0fp+1,
26 0x1.786ea384470a2p+3, -0x1.6a7c1453c85d3p+4, 0x1.31f0fc5613142p+4,
27 -0x1.5ea6c007d4dbbp+2, 0x1.e66f265ce9e5p-3 },
28 .Q_37 = { -0x1.636b2dcf4edbep-7, 0x1.0b5411e2acf29p-2, -0x1.3413109467a0bp+1,
29 0x1.563e8136c554ap+3, -0x1.7b77aab1dcafbp+4, 0x1.8a3e174e05ddcp+4,
30 -0x1.4075c56404eecp+3, 0x1p+0 },
31 .P_57 = { 0x1.b874f9516f7f1p-14, 0x1.5921f2916c1c4p-7, 0x1.145ae7d5b8fa4p-2,
32 0x1.29d6dcc3b2fb7p+1, 0x1.cabe2209a7985p+2, 0x1.11859f0745c4p+3,
33 0x1.b7ec7bc6a2ce5p+2, 0x1.d0419e0bb42aep+1, 0x1.c5aa03eef7258p-1 },
34 .Q_57 = { 0x1.b8747e12691f1p-14, 0x1.59240d8ed1e0ap-7, 0x1.14aef2b181e2p-2,
35 0x1.2cd181bcea52p+1, 0x1.e6e63e0b7aa4cp+2, 0x1.65cf8da94aa3ap+3,
36 0x1.7e5c787b10a36p+3, 0x1.0626d68b6cea3p+3, 0x1.065c5f193abf6p+2,
37 0x1p+0 }
38 };
39
40 /* Inverse error function approximation, based on rational approximation as
41 described in
42 J. M. Blair, C. A. Edwards, and J. H. Johnson,
43 "Rational Chebyshev approximations for the inverse of the error function",
44 Math. Comp. 30, pp. 827--830 (1976).
45 https://doi.org/10.1090/S0025-5718-1976-0421040-7
46 Largest observed error is 24.46 ULP, in the extreme tail:
47 erfinv(0x1.fd9504351b757p-1) got 0x1.ff72c1092917p+0
48 want 0x1.ff72c10929158p+0. */
49 double
erfinv(double x)50 erfinv (double x)
51 {
52 double a = fabs (x);
53
54 if (a <= 0.75)
55 {
56 /* Largest observed error in this region is 6.06 ULP:
57 erfinv(0x1.1884650fd2d41p-2) got 0x1.fb65998cbd3fep-3
58 want 0x1.fb65998cbd404p-3. */
59 double t = x * x - 0.5625;
60 return x * horner_6_f64 (t, data.P_17) / horner_6_f64 (t, data.Q_17);
61 }
62
63 if (a <= 0.9375)
64 {
65 /* Largest observed error in this region is 6.95 ULP:
66 erfinv(0x1.a8d65b94d8c6p-1) got 0x1.f08325591b54p-1
67 want 0x1.f08325591b547p-1. */
68 double t = x * x - 0.87890625;
69 return x * horner_7_f64 (t, data.P_37) / horner_7_f64 (t, data.Q_37);
70 }
71
72 double t = 1.0 / (sqrt (-log (1 - a)));
73 return horner_8_f64 (t, data.P_57)
74 / (copysign (t, x) * horner_9_f64 (t, data.Q_57));
75 }
76
77 #if USE_MPFR
78 # warning Not generating tests for erfinv, as MPFR has no suitable reference
79 #else
80 TEST_DISABLE_FENV (erfinv)
81 TEST_SIG (S, D, 1, erfinv, -0.99, 0.99)
82 TEST_ULP (erfinv, 24.0)
83 TEST_INTERVAL (erfinv, 0, 1, 40000)
84 TEST_INTERVAL (erfinv, -0x1p-1022, -1, 40000)
85 #endif
86