1 //===-- Single-precision sin 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/sinf.h" 10 #include "sincosf_utils.h" 11 #include "src/__support/FPUtil/BasicOperations.h" 12 #include "src/__support/FPUtil/FEnvImpl.h" 13 #include "src/__support/FPUtil/FPBits.h" 14 #include "src/__support/FPUtil/PolyEval.h" 15 #include "src/__support/FPUtil/multiply_add.h" 16 #include "src/__support/FPUtil/rounding_mode.h" 17 #include "src/__support/common.h" 18 #include "src/__support/macros/optimization.h" // LIBC_UNLIKELY 19 #include "src/__support/macros/properties/cpu_features.h" // LIBC_TARGET_CPU_HAS_FMA 20 21 #include <errno.h> 22 23 #if defined(LIBC_TARGET_CPU_HAS_FMA) 24 #include "range_reduction_fma.h" 25 #else 26 #include "range_reduction.h" 27 #endif 28 29 namespace LIBC_NAMESPACE { 30 31 LLVM_LIBC_FUNCTION(float, sinf, (float x)) { 32 using FPBits = typename fputil::FPBits<float>; 33 FPBits xbits(x); 34 35 uint32_t x_u = xbits.uintval(); 36 uint32_t x_abs = x_u & 0x7fff'ffffU; 37 double xd = static_cast<double>(x); 38 39 // Range reduction: 40 // For |x| > pi/32, we perform range reduction as follows: 41 // Find k and y such that: 42 // x = (k + y) * pi/32 43 // k is an integer 44 // |y| < 0.5 45 // For small range (|x| < 2^45 when FMA instructions are available, 2^22 46 // otherwise), this is done by performing: 47 // k = round(x * 32/pi) 48 // y = x * 32/pi - k 49 // For large range, we will omit all the higher parts of 32/pi such that the 50 // least significant bits of their full products with x are larger than 63, 51 // since sin((k + y + 64*i) * pi/32) = sin(x + i * 2pi) = sin(x). 52 // 53 // When FMA instructions are not available, we store the digits of 32/pi in 54 // chunks of 28-bit precision. This will make sure that the products: 55 // x * THIRTYTWO_OVER_PI_28[i] are all exact. 56 // When FMA instructions are available, we simply store the digits of 32/pi in 57 // chunks of doubles (53-bit of precision). 58 // So when multiplying by the largest values of single precision, the 59 // resulting output should be correct up to 2^(-208 + 128) ~ 2^-80. By the 60 // worst-case analysis of range reduction, |y| >= 2^-38, so this should give 61 // us more than 40 bits of accuracy. For the worst-case estimation of range 62 // reduction, see for instances: 63 // Elementary Functions by J-M. Muller, Chapter 11, 64 // Handbook of Floating-Point Arithmetic by J-M. Muller et. al., 65 // Chapter 10.2. 66 // 67 // Once k and y are computed, we then deduce the answer by the sine of sum 68 // formula: 69 // sin(x) = sin((k + y)*pi/32) 70 // = sin(y*pi/32) * cos(k*pi/32) + cos(y*pi/32) * sin(k*pi/32) 71 // The values of sin(k*pi/32) and cos(k*pi/32) for k = 0..31 are precomputed 72 // and stored using a vector of 32 doubles. Sin(y*pi/32) and cos(y*pi/32) are 73 // computed using degree-7 and degree-6 minimax polynomials generated by 74 // Sollya respectively. 75 76 // |x| <= pi/16 77 if (LIBC_UNLIKELY(x_abs <= 0x3e49'0fdbU)) { 78 79 // |x| < 0x1.d12ed2p-12f 80 if (LIBC_UNLIKELY(x_abs < 0x39e8'9769U)) { 81 if (LIBC_UNLIKELY(x_abs == 0U)) { 82 // For signed zeros. 83 return x; 84 } 85 // When |x| < 2^-12, the relative error of the approximation sin(x) ~ x 86 // is: 87 // |sin(x) - x| / |sin(x)| < |x^3| / (6|x|) 88 // = x^2 / 6 89 // < 2^-25 90 // < epsilon(1)/2. 91 // So the correctly rounded values of sin(x) are: 92 // = x - sign(x)*eps(x) if rounding mode = FE_TOWARDZERO, 93 // or (rounding mode = FE_UPWARD and x is 94 // negative), 95 // = x otherwise. 96 // To simplify the rounding decision and make it more efficient, we use 97 // fma(x, -2^-25, x) instead. 98 // An exhaustive test shows that this formula work correctly for all 99 // rounding modes up to |x| < 0x1.c555dep-11f. 100 // Note: to use the formula x - 2^-25*x to decide the correct rounding, we 101 // do need fma(x, -2^-25, x) to prevent underflow caused by -2^-25*x when 102 // |x| < 2^-125. For targets without FMA instructions, we simply use 103 // double for intermediate results as it is more efficient than using an 104 // emulated version of FMA. 105 #if defined(LIBC_TARGET_CPU_HAS_FMA) 106 return fputil::multiply_add(x, -0x1.0p-25f, x); 107 #else 108 return static_cast<float>(fputil::multiply_add(xd, -0x1.0p-25, xd)); 109 #endif // LIBC_TARGET_CPU_HAS_FMA 110 } 111 112 // |x| < pi/16. 113 double xsq = xd * xd; 114 115 // Degree-9 polynomial approximation: 116 // sin(x) ~ x + a_3 x^3 + a_5 x^5 + a_7 x^7 + a_9 x^9 117 // = x (1 + a_3 x^2 + ... + a_9 x^8) 118 // = x * P(x^2) 119 // generated by Sollya with the following commands: 120 // > display = hexadecimal; 121 // > Q = fpminimax(sin(x)/x, [|0, 2, 4, 6, 8|], [|1, D...|], [0, pi/16]); 122 double result = 123 fputil::polyeval(xsq, 1.0, -0x1.55555555554c6p-3, 0x1.1111111085e65p-7, 124 -0x1.a019f70fb4d4fp-13, 0x1.718d179815e74p-19); 125 return static_cast<float>(xd * result); 126 } 127 128 if (LIBC_UNLIKELY(x_abs == 0x4619'9998U)) { // x = 0x1.33333p13 129 float r = -0x1.63f4bap-2f; 130 int rounding = fputil::quick_get_round(); 131 if ((rounding == FE_DOWNWARD && xbits.is_pos()) || 132 (rounding == FE_UPWARD && xbits.is_neg())) 133 r = -0x1.63f4bcp-2f; 134 return xbits.is_neg() ? -r : r; 135 } 136 137 if (LIBC_UNLIKELY(x_abs >= 0x7f80'0000U)) { 138 if (x_abs == 0x7f80'0000U) { 139 fputil::set_errno_if_required(EDOM); 140 fputil::raise_except_if_required(FE_INVALID); 141 } 142 return x + FPBits::quiet_nan().get_val(); 143 } 144 145 // Combine the results with the sine of sum formula: 146 // sin(x) = sin((k + y)*pi/32) 147 // = sin(y*pi/32) * cos(k*pi/32) + cos(y*pi/32) * sin(k*pi/32) 148 // = sin_y * cos_k + (1 + cosm1_y) * sin_k 149 // = sin_y * cos_k + (cosm1_y * sin_k + sin_k) 150 double sin_k, cos_k, sin_y, cosm1_y; 151 152 sincosf_eval(xd, x_abs, sin_k, cos_k, sin_y, cosm1_y); 153 154 return static_cast<float>(fputil::multiply_add( 155 sin_y, cos_k, fputil::multiply_add(cosm1_y, sin_k, sin_k))); 156 } 157 158 } // namespace LIBC_NAMESPACE 159