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

source code of libc/src/math/generic/acosf.cpp