1//===-- Implementation header for acosf -------------------------*- 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_ACOSF_H
10#define LLVM_LIBC_SRC___SUPPORT_MATH_ACOSF_H
11
12#include "inv_trigf_utils.h"
13#include "src/__support/FPUtil/FEnvImpl.h"
14#include "src/__support/FPUtil/FPBits.h"
15#include "src/__support/FPUtil/except_value_utils.h"
16#include "src/__support/FPUtil/multiply_add.h"
17#include "src/__support/FPUtil/sqrt.h"
18#include "src/__support/macros/config.h"
19#include "src/__support/macros/optimization.h" // LIBC_UNLIKELY
20
21namespace LIBC_NAMESPACE_DECL {
22
23namespace math {
24
25namespace acosf_internal {
26
27#ifndef LIBC_MATH_HAS_SKIP_ACCURATE_PASS
28
29LIBC_INLINE_VAR constexpr size_t N_EXCEPTS = 4;
30
31// Exceptional values when |x| <= 0.5
32LIBC_INLINE_VAR constexpr fputil::ExceptValues<float, N_EXCEPTS> ACOSF_EXCEPTS =
33 {.values: {
34 // (inputs, RZ output, RU offset, RD offset, RN offset)
35 // x = 0x1.110b46p-26, acosf(x) = 0x1.921fb4p0 (RZ)
36 {.input: 0x328885a3, .rnd_towardzero_result: 0x3fc90fda, .rnd_upward_offset: 1, .rnd_downward_offset: 0, .rnd_tonearest_offset: 1},
37 // x = -0x1.110b46p-26, acosf(x) = 0x1.921fb4p0 (RZ)
38 {.input: 0xb28885a3, .rnd_towardzero_result: 0x3fc90fda, .rnd_upward_offset: 1, .rnd_downward_offset: 0, .rnd_tonearest_offset: 1},
39 // x = 0x1.04c444p-12, acosf(x) = 0x1.920f68p0 (RZ)
40 {.input: 0x39826222, .rnd_towardzero_result: 0x3fc907b4, .rnd_upward_offset: 1, .rnd_downward_offset: 0, .rnd_tonearest_offset: 1},
41 // x = -0x1.04c444p-12, acosf(x) = 0x1.923p0 (RZ)
42 {.input: 0xb9826222, .rnd_towardzero_result: 0x3fc91800, .rnd_upward_offset: 1, .rnd_downward_offset: 0, .rnd_tonearest_offset: 1},
43 }};
44
45#endif // !LIBC_MATH_HAS_SKIP_ACCURATE_PASS
46
47} // namespace acosf_internal
48
49LIBC_INLINE float acosf(float x) {
50 using namespace acosf_internal;
51 using namespace inv_trigf_utils_internal;
52 using FPBits = typename fputil::FPBits<float>;
53
54 FPBits xbits(x);
55 uint32_t x_uint = xbits.uintval();
56 uint32_t x_abs = xbits.uintval() & 0x7fff'ffffU;
57 uint32_t x_sign = x_uint >> 31;
58
59 // |x| <= 0.5
60 if (LIBC_UNLIKELY(x_abs <= 0x3f00'0000U)) {
61 // |x| < 0x1p-10
62 if (LIBC_UNLIKELY(x_abs < 0x3a80'0000U)) {
63 // When |x| < 2^-10, we use the following approximation:
64 // acos(x) = pi/2 - asin(x)
65 // ~ pi/2 - x - x^3 / 6
66
67#ifndef LIBC_MATH_HAS_SKIP_ACCURATE_PASS
68 // Check for exceptional values
69 if (auto r = ACOSF_EXCEPTS.lookup(x_bits: x_uint); LIBC_UNLIKELY(r.has_value()))
70 return r.value();
71#endif // !LIBC_MATH_HAS_SKIP_ACCURATE_PASS
72
73 double xd = static_cast<double>(x);
74 return static_cast<float>(fputil::multiply_add(
75 x: -0x1.5555555555555p-3 * xd, y: xd * xd, z: M_MATH_PI_2 - xd));
76 }
77
78 // For |x| <= 0.5, we approximate acosf(x) by:
79 // acos(x) = pi/2 - asin(x) = pi/2 - x * P(x^2)
80 // Where P(X^2) = Q(X) is a degree-24 minimax even polynomial approximating
81 // asin(x)/x on [0, 0.5] generated by Sollya with:
82 // > Q = fpminimax(asin(x)/x, [|0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20,
83 // 22, 24|], [|1, D...|], [0, 0.5]);
84 double xd = static_cast<double>(x);
85 double xsq = xd * xd;
86 double x3 = xd * xsq;
87 double r = asin_eval(xsq);
88 return static_cast<float>(fputil::multiply_add(x: -x3, y: r, z: M_MATH_PI_2 - xd));
89 }
90
91 // |x| >= 1, return 0, 2pi, or NaNs.
92 if (LIBC_UNLIKELY(x_abs >= 0x3f80'0000U)) {
93 if (x_abs == 0x3f80'0000U)
94 return x_sign ? /* x == -1.0f */ fputil::round_result_slightly_down(
95 value_rn: 0x1.921fb6p+1f)
96 : /* x == 1.0f */ 0.0f;
97
98 if (xbits.is_signaling_nan()) {
99 fputil::raise_except_if_required(FE_INVALID);
100 return FPBits::quiet_nan().get_val();
101 }
102
103 // |x| <= +/-inf
104 if (x_abs <= 0x7f80'0000U) {
105 fputil::set_errno_if_required(EDOM);
106 fputil::raise_except_if_required(FE_INVALID);
107 }
108
109 return x + FPBits::quiet_nan().get_val();
110 }
111
112 // When 0.5 < |x| < 1, we perform range reduction as follow:
113 //
114 // Assume further that 0.5 < x <= 1, and let:
115 // y = acos(x)
116 // We use the double angle formula:
117 // x = cos(y) = 1 - 2 sin^2(y/2)
118 // So:
119 // sin(y/2) = sqrt( (1 - x)/2 )
120 // And hence:
121 // y = 2 * asin( sqrt( (1 - x)/2 ) )
122 // Let u = (1 - x)/2, then
123 // acos(x) = 2 * asin( sqrt(u) )
124 // Moreover, since 0.5 < x <= 1,
125 // 0 <= u < 1/4, and 0 <= sqrt(u) < 0.5,
126 // And hence we can reuse the same polynomial approximation of asin(x) when
127 // |x| <= 0.5:
128 // acos(x) ~ 2 * sqrt(u) * P(u).
129 //
130 // When -1 < x <= -0.5, we use the identity:
131 // acos(x) = pi - acos(-x)
132 // which is reduced to the postive case.
133
134 xbits.set_sign(Sign::POS);
135 double xd = static_cast<double>(xbits.get_val());
136 double u = fputil::multiply_add(x: -0.5, y: xd, z: 0.5);
137 double cv = 2 * fputil::sqrt<double>(x: u);
138
139 double r3 = asin_eval(xsq: u);
140 double r = fputil::multiply_add(x: cv * u, y: r3, z: cv);
141 return static_cast<float>(x_sign ? M_MATH_PI - r : r);
142}
143
144} // namespace math
145
146} // namespace LIBC_NAMESPACE_DECL
147
148#endif // LLVM_LIBC_SRC___SUPPORT_MATH_ACOSF_H
149