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 22 namespace LIBC_NAMESPACE { 23 24 static constexpr size_t N_EXCEPTS = 4; 25 26 // Exceptional values when |x| <= 0.5 27 static 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 39 LLVM_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_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 -0x1.5555555555555p-3 * xd, xd * xd, 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(-x3, r, 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 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(-0.5, xd, 0.5); 116 double cv = 2 * fputil::sqrt(u); 117 118 double r3 = asin_eval(u); 119 double r = fputil::multiply_add(cv * u, r3, cv); 120 return static_cast<float>(x_sign ? M_MATH_PI - r : r); 121 } 122 123 } // namespace LIBC_NAMESPACE 124