1 //===-- Single-precision sincos 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/sincosf.h" 10 #include "sincosf_utils.h" 11 #include "src/__support/FPUtil/FEnvImpl.h" 12 #include "src/__support/FPUtil/FPBits.h" 13 #include "src/__support/FPUtil/multiply_add.h" 14 #include "src/__support/FPUtil/rounding_mode.h" 15 #include "src/__support/common.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 namespace LIBC_NAMESPACE { 22 23 // Exceptional values 24 static constexpr int N_EXCEPTS = 6; 25 26 static constexpr uint32_t EXCEPT_INPUTS[N_EXCEPTS] = { 27 0x46199998, // x = 0x1.33333p13 x 28 0x55325019, // x = 0x1.64a032p43 x 29 0x5922aa80, // x = 0x1.4555p51 x 30 0x5f18b878, // x = 0x1.3170fp63 x 31 0x6115cb11, // x = 0x1.2b9622p67 x 32 0x7beef5ef, // x = 0x1.ddebdep120 x 33 }; 34 35 static constexpr uint32_t EXCEPT_OUTPUTS_SIN[N_EXCEPTS][4] = { 36 {0xbeb1fa5d, 0, 1, 0}, // x = 0x1.33333p13, sin(x) = -0x1.63f4bap-2 (RZ) 37 {0xbf171adf, 0, 1, 1}, // x = 0x1.64a032p43, sin(x) = -0x1.2e35bep-1 (RZ) 38 {0xbf587521, 0, 1, 1}, // x = 0x1.4555p51, sin(x) = -0x1.b0ea42p-1 (RZ) 39 {0x3dad60f6, 1, 0, 1}, // x = 0x1.3170fp63, sin(x) = 0x1.5ac1ecp-4 (RZ) 40 {0xbe7cc1e0, 0, 1, 1}, // x = 0x1.2b9622p67, sin(x) = -0x1.f983cp-3 (RZ) 41 {0xbf587d1b, 0, 1, 1}, // x = 0x1.ddebdep120, sin(x) = -0x1.b0fa36p-1 (RZ) 42 }; 43 44 static constexpr uint32_t EXCEPT_OUTPUTS_COS[N_EXCEPTS][4] = { 45 {0xbf70090b, 0, 1, 0}, // x = 0x1.33333p13, cos(x) = -0x1.e01216p-1 (RZ) 46 {0x3f4ea5d2, 1, 0, 0}, // x = 0x1.64a032p43, cos(x) = 0x1.9d4ba4p-1 (RZ) 47 {0x3f08aebe, 1, 0, 1}, // x = 0x1.4555p51, cos(x) = 0x1.115d7cp-1 (RZ) 48 {0x3f7f14bb, 1, 0, 0}, // x = 0x1.3170fp63, cos(x) = 0x1.fe2976p-1 (RZ) 49 {0x3f78142e, 1, 0, 1}, // x = 0x1.2b9622p67, cos(x) = 0x1.f0285cp-1 (RZ) 50 {0x3f08a21c, 1, 0, 0}, // x = 0x1.ddebdep120, cos(x) = 0x1.114438p-1 (RZ) 51 }; 52 53 LLVM_LIBC_FUNCTION(void, sincosf, (float x, float *sinp, float *cosp)) { 54 using FPBits = typename fputil::FPBits<float>; 55 FPBits xbits(x); 56 57 uint32_t x_abs = xbits.uintval() & 0x7fff'ffffU; 58 double xd = static_cast<double>(x); 59 60 // Range reduction: 61 // For |x| >= 2^-12, we perform range reduction as follows: 62 // Find k and y such that: 63 // x = (k + y) * pi/32 64 // k is an integer 65 // |y| < 0.5 66 // For small range (|x| < 2^45 when FMA instructions are available, 2^22 67 // otherwise), this is done by performing: 68 // k = round(x * 32/pi) 69 // y = x * 32/pi - k 70 // For large range, we will omit all the higher parts of 32/pi such that the 71 // least significant bits of their full products with x are larger than 63, 72 // since: 73 // sin((k + y + 64*i) * pi/32) = sin(x + i * 2pi) = sin(x), and 74 // cos((k + y + 64*i) * pi/32) = cos(x + i * 2pi) = cos(x). 75 // 76 // When FMA instructions are not available, we store the digits of 32/pi in 77 // chunks of 28-bit precision. This will make sure that the products: 78 // x * THIRTYTWO_OVER_PI_28[i] are all exact. 79 // When FMA instructions are available, we simply store the digits of326/pi in 80 // chunks of doubles (53-bit of precision). 81 // So when multiplying by the largest values of single precision, the 82 // resulting output should be correct up to 2^(-208 + 128) ~ 2^-80. By the 83 // worst-case analysis of range reduction, |y| >= 2^-38, so this should give 84 // us more than 40 bits of accuracy. For the worst-case estimation of range 85 // reduction, see for instances: 86 // Elementary Functions by J-M. Muller, Chapter 11, 87 // Handbook of Floating-Point Arithmetic by J-M. Muller et. al., 88 // Chapter 10.2. 89 // 90 // Once k and y are computed, we then deduce the answer by the sine and cosine 91 // of sum formulas: 92 // sin(x) = sin((k + y)*pi/32) 93 // = sin(y*pi/32) * cos(k*pi/32) + cos(y*pi/32) * sin(k*pi/32) 94 // cos(x) = cos((k + y)*pi/32) 95 // = cos(y*pi/32) * cos(k*pi/32) - sin(y*pi/32) * sin(k*pi/32) 96 // The values of sin(k*pi/32) and cos(k*pi/32) for k = 0..63 are precomputed 97 // and stored using a vector of 32 doubles. Sin(y*pi/32) and cos(y*pi/32) are 98 // computed using degree-7 and degree-6 minimax polynomials generated by 99 // Sollya respectively. 100 101 // |x| < 0x1.0p-12f 102 if (LIBC_UNLIKELY(x_abs < 0x3980'0000U)) { 103 if (LIBC_UNLIKELY(x_abs == 0U)) { 104 // For signed zeros. 105 *sinp = x; 106 *cosp = 1.0f; 107 return; 108 } 109 // When |x| < 2^-12, the relative errors of the approximations 110 // sin(x) ~ x, cos(x) ~ 1 111 // are: 112 // |sin(x) - x| / |sin(x)| < |x^3| / (6|x|) 113 // = x^2 / 6 114 // < 2^-25 115 // < epsilon(1)/2. 116 // |cos(x) - 1| < |x^2 / 2| = 2^-25 < epsilon(1)/2. 117 // So the correctly rounded values of sin(x) and cos(x) are: 118 // sin(x) = x - sign(x)*eps(x) if rounding mode = FE_TOWARDZERO, 119 // or (rounding mode = FE_UPWARD and x is 120 // negative), 121 // = x otherwise. 122 // cos(x) = 1 - eps(x) if rounding mode = FE_TOWARDZERO or FE_DOWWARD, 123 // = 1 otherwise. 124 // To simplify the rounding decision and make it more efficient and to 125 // prevent compiler to perform constant folding, we use 126 // sin(x) = fma(x, -2^-25, x), 127 // cos(x) = fma(x*0.5f, -x, 1) 128 // instead. 129 // Note: to use the formula x - 2^-25*x to decide the correct rounding, we 130 // do need fma(x, -2^-25, x) to prevent underflow caused by -2^-25*x when 131 // |x| < 2^-125. For targets without FMA instructions, we simply use 132 // double for intermediate results as it is more efficient than using an 133 // emulated version of FMA. 134 #if defined(LIBC_TARGET_CPU_HAS_FMA) 135 *sinp = fputil::multiply_add(x, -0x1.0p-25f, x); 136 *cosp = fputil::multiply_add(FPBits(x_abs).get_val(), -0x1.0p-25f, 1.0f); 137 #else 138 *sinp = static_cast<float>(fputil::multiply_add(xd, -0x1.0p-25, xd)); 139 *cosp = static_cast<float>(fputil::multiply_add( 140 static_cast<double>(FPBits(x_abs).get_val()), -0x1.0p-25, 1.0)); 141 #endif // LIBC_TARGET_CPU_HAS_FMA 142 return; 143 } 144 145 // x is inf or nan. 146 if (LIBC_UNLIKELY(x_abs >= 0x7f80'0000U)) { 147 if (x_abs == 0x7f80'0000U) { 148 fputil::set_errno_if_required(EDOM); 149 fputil::raise_except_if_required(FE_INVALID); 150 } 151 *sinp = FPBits::quiet_nan().get_val(); 152 *cosp = *sinp; 153 return; 154 } 155 156 // Check exceptional values. 157 for (int i = 0; i < N_EXCEPTS; ++i) { 158 if (LIBC_UNLIKELY(x_abs == EXCEPT_INPUTS[i])) { 159 uint32_t s = EXCEPT_OUTPUTS_SIN[i][0]; // FE_TOWARDZERO 160 uint32_t c = EXCEPT_OUTPUTS_COS[i][0]; // FE_TOWARDZERO 161 bool x_sign = x < 0; 162 switch (fputil::quick_get_round()) { 163 case FE_UPWARD: 164 s += x_sign ? EXCEPT_OUTPUTS_SIN[i][2] : EXCEPT_OUTPUTS_SIN[i][1]; 165 c += EXCEPT_OUTPUTS_COS[i][1]; 166 break; 167 case FE_DOWNWARD: 168 s += x_sign ? EXCEPT_OUTPUTS_SIN[i][1] : EXCEPT_OUTPUTS_SIN[i][2]; 169 c += EXCEPT_OUTPUTS_COS[i][2]; 170 break; 171 case FE_TONEAREST: 172 s += EXCEPT_OUTPUTS_SIN[i][3]; 173 c += EXCEPT_OUTPUTS_COS[i][3]; 174 break; 175 } 176 *sinp = x_sign ? -FPBits(s).get_val() : FPBits(s).get_val(); 177 *cosp = FPBits(c).get_val(); 178 179 return; 180 } 181 } 182 183 // Combine the results with the sine and cosine of sum formulas: 184 // sin(x) = sin((k + y)*pi/32) 185 // = sin(y*pi/32) * cos(k*pi/32) + cos(y*pi/32) * sin(k*pi/32) 186 // = sin_y * cos_k + (1 + cosm1_y) * sin_k 187 // = sin_y * cos_k + (cosm1_y * sin_k + sin_k) 188 // cos(x) = cos((k + y)*pi/32) 189 // = cos(y*pi/32) * cos(k*pi/32) - sin(y*pi/32) * sin(k*pi/32) 190 // = cosm1_y * cos_k + sin_y * sin_k 191 // = (cosm1_y * cos_k + cos_k) + sin_y * sin_k 192 double sin_k, cos_k, sin_y, cosm1_y; 193 194 sincosf_eval(xd, x_abs, sin_k, cos_k, sin_y, cosm1_y); 195 196 *sinp = static_cast<float>(fputil::multiply_add( 197 sin_y, cos_k, fputil::multiply_add(cosm1_y, sin_k, sin_k))); 198 *cosp = static_cast<float>(fputil::multiply_add( 199 sin_y, -sin_k, fputil::multiply_add(cosm1_y, cos_k, cos_k))); 200 } 201 202 } // namespace LIBC_NAMESPACE 203