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

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