1//===-- Single-precision asin 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/asinf.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#include "src/__support/macros/properties/cpu_features.h" // LIBC_TARGET_CPU_HAS_FMA
19
20#include "inv_trigf_utils.h"
21
22namespace LIBC_NAMESPACE_DECL {
23
24#ifndef LIBC_MATH_HAS_SKIP_ACCURATE_PASS
25static constexpr size_t N_EXCEPTS = 2;
26
27// Exceptional values when |x| <= 0.5
28static constexpr fputil::ExceptValues<float, N_EXCEPTS> ASINF_EXCEPTS_LO = {{
29 // (inputs, RZ output, RU offset, RD offset, RN offset)
30 // x = 0x1.137f0cp-5, asinf(x) = 0x1.138c58p-5 (RZ)
31 {0x3d09bf86, 0x3d09c62c, 1, 0, 1},
32 // x = 0x1.cbf43cp-4, asinf(x) = 0x1.cced1cp-4 (RZ)
33 {0x3de5fa1e, 0x3de6768e, 1, 0, 0},
34}};
35
36// Exceptional values when 0.5 < |x| <= 1
37static constexpr fputil::ExceptValues<float, N_EXCEPTS> ASINF_EXCEPTS_HI = {{
38 // (inputs, RZ output, RU offset, RD offset, RN offset)
39 // x = 0x1.107434p-1, asinf(x) = 0x1.1f4b64p-1 (RZ)
40 {0x3f083a1a, 0x3f0fa5b2, 1, 0, 0},
41 // x = 0x1.ee836cp-1, asinf(x) = 0x1.4f0654p0 (RZ)
42 {0x3f7741b6, 0x3fa7832a, 1, 0, 0},
43}};
44#endif // !LIBC_MATH_HAS_SKIP_ACCURATE_PASS
45
46LLVM_LIBC_FUNCTION(float, asinf, (float x)) {
47 using FPBits = typename fputil::FPBits<float>;
48
49 FPBits xbits(x);
50 uint32_t x_uint = xbits.uintval();
51 uint32_t x_abs = xbits.uintval() & 0x7fff'ffffU;
52 constexpr double SIGN[2] = {1.0, -1.0};
53 uint32_t x_sign = x_uint >> 31;
54
55 // |x| <= 0.5-ish
56 if (x_abs < 0x3f04'471dU) {
57 // |x| < 0x1.d12edp-12
58 if (LIBC_UNLIKELY(x_abs < 0x39e8'9768U)) {
59 // When |x| < 2^-12, the relative error of the approximation asin(x) ~ x
60 // is:
61 // |asin(x) - x| / |asin(x)| < |x^3| / (6|x|)
62 // = x^2 / 6
63 // < 2^-25
64 // < epsilon(1)/2.
65 // So the correctly rounded values of asin(x) are:
66 // = x + sign(x)*eps(x) if rounding mode = FE_TOWARDZERO,
67 // or (rounding mode = FE_UPWARD and x is
68 // negative),
69 // = x otherwise.
70 // To simplify the rounding decision and make it more efficient, we use
71 // fma(x, 2^-25, x) instead.
72 // An exhaustive test shows that this formula work correctly for all
73 // rounding modes up to |x| < 0x1.d12edp-12.
74 // Note: to use the formula x + 2^-25*x to decide the correct rounding, we
75 // do need fma(x, 2^-25, x) to prevent underflow caused by 2^-25*x when
76 // |x| < 2^-125. For targets without FMA instructions, we simply use
77 // double for intermediate results as it is more efficient than using an
78 // emulated version of FMA.
79#if defined(LIBC_TARGET_CPU_HAS_FMA_FLOAT)
80 return fputil::multiply_add(x, 0x1.0p-25f, x);
81#else
82 double xd = static_cast<double>(x);
83 return static_cast<float>(fputil::multiply_add(xd, 0x1.0p-25, xd));
84#endif // LIBC_TARGET_CPU_HAS_FMA_FLOAT
85 }
86
87#ifndef LIBC_MATH_HAS_SKIP_ACCURATE_PASS
88 // Check for exceptional values
89 if (auto r = ASINF_EXCEPTS_LO.lookup_odd(x_abs, x_sign);
90 LIBC_UNLIKELY(r.has_value()))
91 return r.value();
92#endif // !LIBC_MATH_HAS_SKIP_ACCURATE_PASS
93
94 // For |x| <= 0.5, we approximate asinf(x) by:
95 // asin(x) = x * P(x^2)
96 // Where P(X^2) = Q(X) is a degree-20 minimax even polynomial approximating
97 // asin(x)/x on [0, 0.5] generated by Sollya with:
98 // > Q = fpminimax(asin(x)/x, [|0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20|],
99 // [|1, D...|], [0, 0.5]);
100 // An exhaustive test shows that this approximation works well up to a
101 // little more than 0.5.
102 double xd = static_cast<double>(x);
103 double xsq = xd * xd;
104 double x3 = xd * xsq;
105 double r = asin_eval(xsq);
106 return static_cast<float>(fputil::multiply_add(x3, r, xd));
107 }
108
109 // |x| > 1, return NaNs.
110 if (LIBC_UNLIKELY(x_abs > 0x3f80'0000U)) {
111 if (xbits.is_signaling_nan()) {
112 fputil::raise_except_if_required(FE_INVALID);
113 return FPBits::quiet_nan().get_val();
114 }
115
116 if (x_abs <= 0x7f80'0000U) {
117 fputil::set_errno_if_required(EDOM);
118 fputil::raise_except_if_required(FE_INVALID);
119 }
120
121 return FPBits::quiet_nan().get_val();
122 }
123
124#ifndef LIBC_MATH_HAS_SKIP_ACCURATE_PASS
125 // Check for exceptional values
126 if (auto r = ASINF_EXCEPTS_HI.lookup_odd(x_abs, x_sign);
127 LIBC_UNLIKELY(r.has_value()))
128 return r.value();
129#endif // !LIBC_MATH_HAS_SKIP_ACCURATE_PASS
130
131 // When |x| > 0.5, we perform range reduction as follow:
132 //
133 // Assume further that 0.5 < x <= 1, and let:
134 // y = asin(x)
135 // We will use the double angle formula:
136 // cos(2y) = 1 - 2 sin^2(y)
137 // and the complement angle identity:
138 // x = sin(y) = cos(pi/2 - y)
139 // = 1 - 2 sin^2 (pi/4 - y/2)
140 // So:
141 // sin(pi/4 - y/2) = sqrt( (1 - x)/2 )
142 // And hence:
143 // pi/4 - y/2 = asin( sqrt( (1 - x)/2 ) )
144 // Equivalently:
145 // asin(x) = y = pi/2 - 2 * asin( sqrt( (1 - x)/2 ) )
146 // Let u = (1 - x)/2, then:
147 // asin(x) = pi/2 - 2 * asin( sqrt(u) )
148 // Moreover, since 0.5 < x <= 1:
149 // 0 <= u < 1/4, and 0 <= sqrt(u) < 0.5,
150 // And hence we can reuse the same polynomial approximation of asin(x) when
151 // |x| <= 0.5:
152 // asin(x) ~ pi/2 - 2 * sqrt(u) * P(u),
153
154 xbits.set_sign(Sign::POS);
155 double sign = SIGN[x_sign];
156 double xd = static_cast<double>(xbits.get_val());
157 double u = fputil::multiply_add(-0.5, xd, 0.5);
158 double c1 = sign * (-2 * fputil::sqrt<double>(u));
159 double c2 = fputil::multiply_add(sign, M_MATH_PI_2, c1);
160 double c3 = c1 * u;
161
162 double r = asin_eval(u);
163 return static_cast<float>(fputil::multiply_add(c3, r, c2));
164}
165
166} // namespace LIBC_NAMESPACE_DECL
167

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