1//===-- Implementation header for erff --------------------------*- C++ -*-===//
2//
3// Part of the LLVM Project, under the Apache License v2.0 with LLVM Exceptions.
4// See https://llvm.org/LICENSE.txt for license information.
5// SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception
6//
7//===----------------------------------------------------------------------===//
8
9#ifndef LLVM_LIBC_SRC___SUPPORT_MATH_ERFF_H
10#define LLVM_LIBC_SRC___SUPPORT_MATH_ERFF_H
11
12#include "src/__support/FPUtil/FPBits.h"
13#include "src/__support/FPUtil/except_value_utils.h"
14#include "src/__support/FPUtil/multiply_add.h"
15#include "src/__support/macros/config.h"
16#include "src/__support/macros/optimization.h" // LIBC_UNLIKELY
17
18namespace LIBC_NAMESPACE_DECL {
19
20namespace math {
21
22LIBC_INLINE float erff(float x) {
23
24 // Polynomials approximating erf(x)/x on ( k/8, (k + 1)/8 ) generated by
25 // Sollya with: > P = fpminimax(erf(x)/x, [|0, 2, 4, 6, 8, 10, 12, 14|],
26 // [|D...|],
27 // [k/8, (k + 1)/8]);
28 // for k = 0..31.
29 constexpr double COEFFS[32][8] = {
30 {0x1.20dd750429b6dp0, -0x1.812746b037753p-2, 0x1.ce2f219e8596ap-4,
31 -0x1.b82cdacb78fdap-6, 0x1.56479297dfda5p-8, -0x1.8b3ac5455ef02p-11,
32 -0x1.126fcac367e3bp-8, 0x1.2d0bdb3ba4984p-4},
33 {0x1.20dd750429b6dp0, -0x1.812746b0379a8p-2, 0x1.ce2f21a03cf2ap-4,
34 -0x1.b82ce30de083ep-6, 0x1.565bcad3eb60fp-8, -0x1.c02c66f659256p-11,
35 0x1.f92f673385229p-14, -0x1.def402648ae9p-17},
36 {0x1.20dd750429b34p0, -0x1.812746b032dcep-2, 0x1.ce2f219d84aaep-4,
37 -0x1.b82ce22dcf139p-6, 0x1.565b9efcd4af1p-8, -0x1.c021f1af414bcp-11,
38 0x1.f7c6d177eff82p-14, -0x1.c9e4410dcf865p-17},
39 {0x1.20dd750426eabp0, -0x1.812746ae592c7p-2, 0x1.ce2f211525f14p-4,
40 -0x1.b82ccc125e63fp-6, 0x1.56596f261cfd3p-8, -0x1.bfde1ff8eeecfp-11,
41 0x1.f31a9d15dc5d8p-14, -0x1.a5a4362844b3cp-17},
42 {0x1.20dd75039c705p0, -0x1.812746777e74dp-2, 0x1.ce2f17af98a1bp-4,
43 -0x1.b82be4b817cbep-6, 0x1.564bec2e2962ep-8, -0x1.bee86f9da3558p-11,
44 0x1.e9443689dc0ccp-14, -0x1.79c0f230805d8p-17},
45 {0x1.20dd74f811211p0, -0x1.81274371a3e8fp-2, 0x1.ce2ec038262e5p-4,
46 -0x1.b8265b82c5e1fp-6, 0x1.5615a2e239267p-8, -0x1.bc63ae023dcebp-11,
47 0x1.d87c2102f7e06p-14, -0x1.49584bea41d62p-17},
48 {0x1.20dd746d063e3p0, -0x1.812729a8a950fp-2, 0x1.ce2cb0a2df232p-4,
49 -0x1.b80eca1f51278p-6, 0x1.5572e26c46815p-8, -0x1.b715e5638b65ep-11,
50 0x1.bfbb195484968p-14, -0x1.177a565c15c52p-17},
51 {0x1.20dd701b44486p0, -0x1.812691145f237p-2, 0x1.ce23a06b8cfd9p-4,
52 -0x1.b7c1dc7245288p-6, 0x1.53e92f7f397ddp-8, -0x1.ad97cc4acf0b2p-11,
53 0x1.9f028b2b09b71p-14, -0x1.cdc4da08da8c1p-18},
54 {0x1.20dd5715ac332p0, -0x1.8123e680bd0ebp-2, 0x1.ce0457aded691p-4,
55 -0x1.b6f52d52bed4p-6, 0x1.50c291b84414cp-8, -0x1.9ea246b1ad4a9p-11,
56 0x1.77654674e0cap-14, -0x1.737c11a1bcebbp-18},
57 {0x1.20dce6593e114p0, -0x1.811a59c02eadcp-2, 0x1.cdab53c7cd7d5p-4,
58 -0x1.b526d2e321eedp-6, 0x1.4b1d32cd8b994p-8, -0x1.8963143ec0a1ep-11,
59 0x1.4ad5700e4db91p-14, -0x1.231e100e43ef2p-18},
60 {0x1.20db48bfd5a62p0, -0x1.80fdd84f9e308p-2, 0x1.ccd340d462983p-4,
61 -0x1.b196a2928768p-6, 0x1.4210c2c13a0f7p-8, -0x1.6dbdfb4ff71aep-11,
62 0x1.1bca2d17fbd71p-14, -0x1.bca36f90c7cf5p-19},
63 {0x1.20d64b2f8f508p0, -0x1.80b4d4f19fa8bp-2, 0x1.cb088197262e3p-4,
64 -0x1.ab51fd02e5b99p-6, 0x1.34e1e5e81a632p-8, -0x1.4c66377b502cep-11,
65 0x1.d9ad25066213cp-15, -0x1.4b0df7dd0cfa1p-19},
66 {0x1.20c8fc1243576p0, -0x1.8010cb2009e27p-2, 0x1.c7a47e9299315p-4,
67 -0x1.a155be5683654p-6, 0x1.233502694997bp-8, -0x1.26c94b7d813p-11,
68 0x1.8094f1de25fb9p-15, -0x1.e0e3d776c6eefp-20},
69 {0x1.20a9bd1611bc1p0, -0x1.7ec7fbce83f9p-2, 0x1.c1d757d7317b7p-4,
70 -0x1.92c160cd589fp-6, 0x1.0d307269cc5c2p-8, -0x1.fda5b0d2d1879p-12,
71 0x1.2fdd7b3b14a7fp-15, -0x1.54eed4a26af5ap-20},
72 {0x1.20682834f943dp0, -0x1.7c73f747bf5a9p-2, 0x1.b8c2db4a9ffd1p-4,
73 -0x1.7f0e4ffe989ecp-6, 0x1.e7061eae4166ep-9, -0x1.ad36e873fff2dp-12,
74 0x1.d39222396128ep-16, -0x1.d83dacec5ea6bp-21},
75 {0x1.1feb8d12676d7p0, -0x1.7898347284afep-2, 0x1.aba3466b34451p-4,
76 -0x1.663adc573e2f9p-6, 0x1.ae99fb17c3e08p-9, -0x1.602f950ad5535p-12,
77 0x1.5e9717490609dp-16, -0x1.3fca107bbc8d5p-21},
78 {0x1.1f12fe3c536fap0, -0x1.72b1d1f22e6d3p-2, 0x1.99fc0eed4a896p-4,
79 -0x1.48db0a87bd8c6p-6, 0x1.73e368895aa61p-9, -0x1.19b35d5301fc8p-12,
80 0x1.007987e4bb033p-16, -0x1.a7edcd4c2dc7p-22},
81 {0x1.1db7b0df84d5dp0, -0x1.6a4e4a41cde02p-2, 0x1.83bbded16455dp-4,
82 -0x1.2809b3b36977ep-6, 0x1.39c08bab44679p-9, -0x1.b7b45a70ed119p-13,
83 0x1.6e99b36410e7bp-17, -0x1.13619bb7ebc0cp-22},
84 {0x1.1bb1c85c4a527p0, -0x1.5f23b99a249a3p-2, 0x1.694c91fa0d12cp-4,
85 -0x1.053e1ce11c72dp-6, 0x1.02bf72c50ea78p-9, -0x1.4f478fb56cb02p-13,
86 0x1.005f80ecbe213p-17, -0x1.5f2446bde7f5bp-23},
87 {0x1.18dec3bd51f9dp0, -0x1.5123f58346186p-2, 0x1.4b8a1ca536ab4p-4,
88 -0x1.c4243015cc723p-7, 0x1.a1a8a01d351efp-10, -0x1.f466b34f1d86bp-14,
89 0x1.5f835eea0bf6ap-18, -0x1.b83165b939234p-24},
90 {0x1.152804c3369f4p0, -0x1.4084cd4afd4bcp-2, 0x1.2ba2e836e47aap-4,
91 -0x1.800f2dfc6904bp-7, 0x1.4a6daf0669c59p-10, -0x1.6e326ab872317p-14,
92 0x1.d9761a6a755a5p-19, -0x1.0fca33f9dd4b5p-24},
93 {0x1.1087ad68356aap0, -0x1.2dbb044707459p-2, 0x1.0aea8ceaa0384p-4,
94 -0x1.40b516d52b3d2p-7, 0x1.00c9e05f01d22p-10, -0x1.076afb0dc0ff7p-14,
95 0x1.39fadec400657p-19, -0x1.4b5761352e7e3p-25},
96 {0x1.0b0a7a8ba4a22p0, -0x1.196990d22d4a1p-2, 0x1.d5551e6ac0c4dp-5,
97 -0x1.07cce1770bd1ap-7, 0x1.890347b8848bfp-11, -0x1.757ec96750b6ap-15,
98 0x1.9b258a1e06bcep-20, -0x1.8fc6d22da7572p-26},
99 {0x1.04ce2be70fb47p0, -0x1.0449e4b0b9cacp-2, 0x1.97f7424f4b0e7p-5,
100 -0x1.ac825439c42f4p-8, 0x1.28f5f65426dfbp-11, -0x1.05b699a90f90fp-15,
101 0x1.0a888eecf4593p-20, -0x1.deace2b32bb31p-27},
102 {0x1.fbf9fb0e11cc8p-1, -0x1.de2640856545ap-3, 0x1.5f5b1f47f851p-5,
103 -0x1.588bc71eb41b9p-8, 0x1.bc6a0a772f56dp-12, -0x1.6b9fad1f1657ap-16,
104 0x1.573204ba66504p-21, -0x1.1d38065c94e44p-27},
105 {0x1.ed8f18c99e031p-1, -0x1.b4cb6acd903b4p-3, 0x1.2c7f3dddd6fc1p-5,
106 -0x1.13052067df4ep-8, 0x1.4a5027444082fp-12, -0x1.f672bab0e2554p-17,
107 0x1.b83c756348cc9p-22, -0x1.534f1a1079499p-28},
108 {0x1.debd33044166dp-1, -0x1.8d7cd9053f7d8p-3, 0x1.ff9957fb3d6e7p-6,
109 -0x1.b50be55de0f36p-9, 0x1.e92c8ec53a628p-13, -0x1.5a4b88d508007p-17,
110 0x1.1a27737559e26p-22, -0x1.942ae62cb2c14p-29},
111 {0x1.cfdbf0386f3bdp-1, -0x1.68e33d93b0dc4p-3, 0x1.b2683d58f53dep-6,
112 -0x1.5a9174e70d26fp-9, 0x1.69ddd326d49cdp-13, -0x1.dd8f397a8219cp-18,
113 0x1.6a755016ad4ddp-23, -0x1.e366e0139187dp-30},
114 {0x1.c132adb8d7464p-1, -0x1.475a899f61b46p-3, 0x1.70a431397a77cp-6,
115 -0x1.12e3d35beeee2p-9, 0x1.0c16b05738333p-13, -0x1.4a47f873e144ep-18,
116 0x1.d3d494c698c02p-24, -0x1.2302c59547fe5p-30},
117 {0x1.b2f5fd05555e7p-1, -0x1.28feefbe03ec7p-3, 0x1.3923acbb3a676p-6,
118 -0x1.b4ff793cd6358p-10, 0x1.8ea0eb8c913bcp-14, -0x1.cb31ec2baceb1p-19,
119 0x1.30011e7e80c04p-24, -0x1.617710635cb1dp-31},
120 {0x1.a54853cd9593ep-1, -0x1.0dbdbaea4dc8ep-3, 0x1.0a93e2c20a0fdp-6,
121 -0x1.5c969ff401ea8p-10, 0x1.29e0cc64fe627p-14, -0x1.4160d8e9d3c2ap-19,
122 0x1.8e7b67594624ap-25, -0x1.b1cf2c975b09bp-32},
123 {0x1.983ceece09ff8p-1, -0x1.eacc78f7a2dp-4, 0x1.c74418410655fp-7,
124 -0x1.1756a050e441ep-10, 0x1.bff3650f7f548p-15, -0x1.c56c0217d3adap-20,
125 0x1.07b4918d0b489p-25, -0x1.0d4be8c1c50f8p-32},
126 };
127
128 using FPBits = typename fputil::FPBits<float>;
129 FPBits xbits(x);
130
131 uint32_t x_u = xbits.uintval();
132 uint32_t x_abs = x_u & 0x7fff'ffffU;
133
134 if (LIBC_UNLIKELY(x_abs >= 0x4080'0000U)) {
135 constexpr float ONE[2] = {1.0f, -1.0f};
136 constexpr float SMALL[2] = {-0x1.0p-25f, 0x1.0p-25f};
137
138 int sign = xbits.is_neg() ? 1 : 0;
139
140 if (LIBC_UNLIKELY(x_abs >= 0x7f80'0000U)) {
141 if (xbits.is_signaling_nan()) {
142 fputil::raise_except_if_required(FE_INVALID);
143 return FPBits::quiet_nan().get_val();
144 }
145 return (x_abs > 0x7f80'0000) ? x : ONE[sign];
146 }
147
148 return ONE[sign] + SMALL[sign];
149 }
150
151#ifndef LIBC_MATH_HAS_SKIP_ACCURATE_PASS
152 // Exceptional mask = common 0 bits of 2 exceptional values.
153 constexpr uint32_t EXCEPT_MASK = 0x809a'6184U;
154
155 if (LIBC_UNLIKELY((x_abs & EXCEPT_MASK) == 0)) {
156 // Exceptional values
157 if (LIBC_UNLIKELY(x_abs == 0x3f65'9229U)) // |x| = 0x1.cb2452p-1f
158 return x < 0.0f ? fputil::round_result_slightly_down(value_rn: -0x1.972ea8p-1f)
159 : fputil::round_result_slightly_up(value_rn: 0x1.972ea8p-1f);
160 if (LIBC_UNLIKELY(x_abs == 0x4004'1e6aU)) // |x| = 0x1.083cd4p+1f
161 return x < 0.0f ? fputil::round_result_slightly_down(value_rn: -0x1.fe3462p-1f)
162 : fputil::round_result_slightly_up(value_rn: 0x1.fe3462p-1f);
163 if (x_abs == 0U)
164 return x;
165 }
166#endif // !LIBC_MATH_HAS_SKIP_ACCURATE_PASS
167
168 // Polynomial approximation:
169 // erf(x) ~ x * (c0 + c1 * x^2 + c2 * x^4 + ... + c7 * x^14)
170 double xd = static_cast<double>(x);
171 double xsq = xd * xd;
172
173 constexpr uint32_t EIGHT = 3 << FPBits::FRACTION_LEN;
174 int idx = static_cast<int>(FPBits(x_abs + EIGHT).get_val());
175
176 double x4 = xsq * xsq;
177 double c0 = fputil::multiply_add(x: xsq, y: COEFFS[idx][1], z: COEFFS[idx][0]);
178 double c1 = fputil::multiply_add(x: xsq, y: COEFFS[idx][3], z: COEFFS[idx][2]);
179 double c2 = fputil::multiply_add(x: xsq, y: COEFFS[idx][5], z: COEFFS[idx][4]);
180 double c3 = fputil::multiply_add(x: xsq, y: COEFFS[idx][7], z: COEFFS[idx][6]);
181
182 double x8 = x4 * x4;
183 double p0 = fputil::multiply_add(x: x4, y: c1, z: c0);
184 double p1 = fputil::multiply_add(x: x4, y: c3, z: c2);
185
186 return static_cast<float>(xd * fputil::multiply_add(x: x8, y: p1, z: p0));
187}
188
189} // namespace math
190
191} // namespace LIBC_NAMESPACE_DECL
192
193#endif // LLVM_LIBC_SRC___SUPPORT_MATH_ERFF_H
194