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/optimization.h" // LIBC_UNLIKELY 17 #include "src/__support/macros/properties/cpu_features.h" // LIBC_TARGET_CPU_HAS_FMA 18 19 #include <errno.h> 20 21 #include "inv_trigf_utils.h" 22 23 namespace LIBC_NAMESPACE { 24 25 static constexpr size_t N_EXCEPTS = 2; 26 27 // Exceptional values when |x| <= 0.5 28 static 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 37 static 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 45 LLVM_LIBC_FUNCTION(float, asinf, (float x)) { 46 using FPBits = typename fputil::FPBits<float>; 47 48 FPBits xbits(x); 49 uint32_t x_uint = xbits.uintval(); 50 uint32_t x_abs = xbits.uintval() & 0x7fff'ffffU; 51 constexpr double SIGN[2] = {1.0, -1.0}; 52 uint32_t x_sign = x_uint >> 31; 53 54 // |x| <= 0.5-ish 55 if (x_abs < 0x3f04'471dU) { 56 // |x| < 0x1.d12edp-12 57 if (LIBC_UNLIKELY(x_abs < 0x39e8'9768U)) { 58 // When |x| < 2^-12, the relative error of the approximation asin(x) ~ x 59 // is: 60 // |asin(x) - x| / |asin(x)| < |x^3| / (6|x|) 61 // = x^2 / 6 62 // < 2^-25 63 // < epsilon(1)/2. 64 // So the correctly rounded values of asin(x) are: 65 // = x + sign(x)*eps(x) if rounding mode = FE_TOWARDZERO, 66 // or (rounding mode = FE_UPWARD and x is 67 // negative), 68 // = x otherwise. 69 // To simplify the rounding decision and make it more efficient, we use 70 // fma(x, 2^-25, x) instead. 71 // An exhaustive test shows that this formula work correctly for all 72 // rounding modes up to |x| < 0x1.d12edp-12. 73 // Note: to use the formula x + 2^-25*x to decide the correct rounding, we 74 // do need fma(x, 2^-25, x) to prevent underflow caused by 2^-25*x when 75 // |x| < 2^-125. For targets without FMA instructions, we simply use 76 // double for intermediate results as it is more efficient than using an 77 // emulated version of FMA. 78 #if defined(LIBC_TARGET_CPU_HAS_FMA) 79 return fputil::multiply_add(x, 0x1.0p-25f, x); 80 #else 81 double xd = static_cast<double>(x); 82 return static_cast<float>(fputil::multiply_add(xd, 0x1.0p-25, xd)); 83 #endif // LIBC_TARGET_CPU_HAS_FMA 84 } 85 86 // Check for exceptional values 87 if (auto r = ASINF_EXCEPTS_LO.lookup_odd(x_abs, x_sign); 88 LIBC_UNLIKELY(r.has_value())) 89 return r.value(); 90 91 // For |x| <= 0.5, we approximate asinf(x) by: 92 // asin(x) = x * P(x^2) 93 // Where P(X^2) = Q(X) is a degree-20 minimax even polynomial approximating 94 // asin(x)/x on [0, 0.5] generated by Sollya with: 95 // > Q = fpminimax(asin(x)/x, [|0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20|], 96 // [|1, D...|], [0, 0.5]); 97 // An exhaustive test shows that this approximation works well up to a 98 // little more than 0.5. 99 double xd = static_cast<double>(x); 100 double xsq = xd * xd; 101 double x3 = xd * xsq; 102 double r = asin_eval(xsq); 103 return static_cast<float>(fputil::multiply_add(x3, r, xd)); 104 } 105 106 // |x| > 1, return NaNs. 107 if (LIBC_UNLIKELY(x_abs > 0x3f80'0000U)) { 108 if (x_abs <= 0x7f80'0000U) { 109 fputil::set_errno_if_required(EDOM); 110 fputil::raise_except_if_required(FE_INVALID); 111 } 112 return FPBits::quiet_nan().get_val(); 113 } 114 115 // Check for exceptional values 116 if (auto r = ASINF_EXCEPTS_HI.lookup_odd(x_abs, x_sign); 117 LIBC_UNLIKELY(r.has_value())) 118 return r.value(); 119 120 // When |x| > 0.5, we perform range reduction as follow: 121 // 122 // Assume further that 0.5 < x <= 1, and let: 123 // y = asin(x) 124 // We will use the double angle formula: 125 // cos(2y) = 1 - 2 sin^2(y) 126 // and the complement angle identity: 127 // x = sin(y) = cos(pi/2 - y) 128 // = 1 - 2 sin^2 (pi/4 - y/2) 129 // So: 130 // sin(pi/4 - y/2) = sqrt( (1 - x)/2 ) 131 // And hence: 132 // pi/4 - y/2 = asin( sqrt( (1 - x)/2 ) ) 133 // Equivalently: 134 // asin(x) = y = pi/2 - 2 * asin( sqrt( (1 - x)/2 ) ) 135 // Let u = (1 - x)/2, then: 136 // asin(x) = pi/2 - 2 * asin( sqrt(u) ) 137 // Moreover, since 0.5 < x <= 1: 138 // 0 <= u < 1/4, and 0 <= sqrt(u) < 0.5, 139 // And hence we can reuse the same polynomial approximation of asin(x) when 140 // |x| <= 0.5: 141 // asin(x) ~ pi/2 - 2 * sqrt(u) * P(u), 142 143 xbits.set_sign(Sign::POS); 144 double sign = SIGN[x_sign]; 145 double xd = static_cast<double>(xbits.get_val()); 146 double u = fputil::multiply_add(-0.5, xd, 0.5); 147 double c1 = sign * (-2 * fputil::sqrt(u)); 148 double c2 = fputil::multiply_add(sign, M_MATH_PI_2, c1); 149 double c3 = c1 * u; 150 151 double r = asin_eval(u); 152 return static_cast<float>(fputil::multiply_add(c3, r, c2)); 153 } 154 155 } // namespace LIBC_NAMESPACE 156