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/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 namespace LIBC_NAMESPACE_DECL { 21 22 // Exceptional values 23 static constexpr int N_EXCEPTS = 6; 24 25 static constexpr uint32_t EXCEPT_INPUTS[N_EXCEPTS] = { 26 0x46199998, // x = 0x1.33333p13 x 27 0x55325019, // x = 0x1.64a032p43 x 28 0x5922aa80, // x = 0x1.4555p51 x 29 0x5f18b878, // x = 0x1.3170fp63 x 30 0x6115cb11, // x = 0x1.2b9622p67 x 31 0x7beef5ef, // x = 0x1.ddebdep120 x 32 }; 33 34 static constexpr uint32_t EXCEPT_OUTPUTS_SIN[N_EXCEPTS][4] = { 35 {0xbeb1fa5d, 0, 1, 0}, // x = 0x1.33333p13, sin(x) = -0x1.63f4bap-2 (RZ) 36 {0xbf171adf, 0, 1, 1}, // x = 0x1.64a032p43, sin(x) = -0x1.2e35bep-1 (RZ) 37 {0xbf587521, 0, 1, 1}, // x = 0x1.4555p51, sin(x) = -0x1.b0ea42p-1 (RZ) 38 {0x3dad60f6, 1, 0, 1}, // x = 0x1.3170fp63, sin(x) = 0x1.5ac1ecp-4 (RZ) 39 {0xbe7cc1e0, 0, 1, 1}, // x = 0x1.2b9622p67, sin(x) = -0x1.f983cp-3 (RZ) 40 {0xbf587d1b, 0, 1, 1}, // x = 0x1.ddebdep120, sin(x) = -0x1.b0fa36p-1 (RZ) 41 }; 42 43 static constexpr uint32_t EXCEPT_OUTPUTS_COS[N_EXCEPTS][4] = { 44 {0xbf70090b, 0, 1, 0}, // x = 0x1.33333p13, cos(x) = -0x1.e01216p-1 (RZ) 45 {0x3f4ea5d2, 1, 0, 0}, // x = 0x1.64a032p43, cos(x) = 0x1.9d4ba4p-1 (RZ) 46 {0x3f08aebe, 1, 0, 1}, // x = 0x1.4555p51, cos(x) = 0x1.115d7cp-1 (RZ) 47 {0x3f7f14bb, 1, 0, 0}, // x = 0x1.3170fp63, cos(x) = 0x1.fe2976p-1 (RZ) 48 {0x3f78142e, 1, 0, 1}, // x = 0x1.2b9622p67, cos(x) = 0x1.f0285cp-1 (RZ) 49 {0x3f08a21c, 1, 0, 0}, // x = 0x1.ddebdep120, cos(x) = 0x1.114438p-1 (RZ) 50 }; 51 52 LLVM_LIBC_FUNCTION(void, sincosf, (float x, float *sinp, float *cosp)) { 53 using FPBits = typename fputil::FPBits<float>; 54 FPBits xbits(x); 55 56 uint32_t x_abs = xbits.uintval() & 0x7fff'ffffU; 57 double xd = static_cast<double>(x); 58 59 // Range reduction: 60 // For |x| >= 2^-12, we perform range reduction as follows: 61 // Find k and y such that: 62 // x = (k + y) * pi/32 63 // k is an integer 64 // |y| < 0.5 65 // For small range (|x| < 2^45 when FMA instructions are available, 2^22 66 // otherwise), this is done by performing: 67 // k = round(x * 32/pi) 68 // y = x * 32/pi - k 69 // For large range, we will omit all the higher parts of 32/pi such that the 70 // least significant bits of their full products with x are larger than 63, 71 // since: 72 // sin((k + y + 64*i) * pi/32) = sin(x + i * 2pi) = sin(x), and 73 // cos((k + y + 64*i) * pi/32) = cos(x + i * 2pi) = cos(x). 74 // 75 // When FMA instructions are not available, we store the digits of 32/pi in 76 // chunks of 28-bit precision. This will make sure that the products: 77 // x * THIRTYTWO_OVER_PI_28[i] are all exact. 78 // When FMA instructions are available, we simply store the digits of326/pi in 79 // chunks of doubles (53-bit of precision). 80 // So when multiplying by the largest values of single precision, the 81 // resulting output should be correct up to 2^(-208 + 128) ~ 2^-80. By the 82 // worst-case analysis of range reduction, |y| >= 2^-38, so this should give 83 // us more than 40 bits of accuracy. For the worst-case estimation of range 84 // reduction, see for instances: 85 // Elementary Functions by J-M. Muller, Chapter 11, 86 // Handbook of Floating-Point Arithmetic by J-M. Muller et. al., 87 // Chapter 10.2. 88 // 89 // Once k and y are computed, we then deduce the answer by the sine and cosine 90 // of sum formulas: 91 // sin(x) = sin((k + y)*pi/32) 92 // = sin(y*pi/32) * cos(k*pi/32) + cos(y*pi/32) * sin(k*pi/32) 93 // cos(x) = cos((k + y)*pi/32) 94 // = cos(y*pi/32) * cos(k*pi/32) - sin(y*pi/32) * sin(k*pi/32) 95 // The values of sin(k*pi/32) and cos(k*pi/32) for k = 0..63 are precomputed 96 // and stored using a vector of 32 doubles. Sin(y*pi/32) and cos(y*pi/32) are 97 // computed using degree-7 and degree-6 minimax polynomials generated by 98 // Sollya respectively. 99 100 // |x| < 0x1.0p-12f 101 if (LIBC_UNLIKELY(x_abs < 0x3980'0000U)) { 102 if (LIBC_UNLIKELY(x_abs == 0U)) { 103 // For signed zeros. 104 *sinp = x; 105 *cosp = 1.0f; 106 return; 107 } 108 // When |x| < 2^-12, the relative errors of the approximations 109 // sin(x) ~ x, cos(x) ~ 1 110 // are: 111 // |sin(x) - x| / |sin(x)| < |x^3| / (6|x|) 112 // = x^2 / 6 113 // < 2^-25 114 // < epsilon(1)/2. 115 // |cos(x) - 1| < |x^2 / 2| = 2^-25 < epsilon(1)/2. 116 // So the correctly rounded values of sin(x) and cos(x) are: 117 // sin(x) = x - sign(x)*eps(x) if rounding mode = FE_TOWARDZERO, 118 // or (rounding mode = FE_UPWARD and x is 119 // negative), 120 // = x otherwise. 121 // cos(x) = 1 - eps(x) if rounding mode = FE_TOWARDZERO or FE_DOWWARD, 122 // = 1 otherwise. 123 // To simplify the rounding decision and make it more efficient and to 124 // prevent compiler to perform constant folding, we use 125 // sin(x) = fma(x, -2^-25, x), 126 // cos(x) = fma(x*0.5f, -x, 1) 127 // instead. 128 // Note: to use the formula x - 2^-25*x to decide the correct rounding, we 129 // do need fma(x, -2^-25, x) to prevent underflow caused by -2^-25*x when 130 // |x| < 2^-125. For targets without FMA instructions, we simply use 131 // double for intermediate results as it is more efficient than using an 132 // emulated version of FMA. 133 #if defined(LIBC_TARGET_CPU_HAS_FMA) 134 *sinp = fputil::multiply_add(x, -0x1.0p-25f, x); 135 *cosp = fputil::multiply_add(FPBits(x_abs).get_val(), -0x1.0p-25f, 1.0f); 136 #else 137 *sinp = static_cast<float>(fputil::multiply_add(xd, -0x1.0p-25, xd)); 138 *cosp = static_cast<float>(fputil::multiply_add( 139 static_cast<double>(FPBits(x_abs).get_val()), -0x1.0p-25, 1.0)); 140 #endif // LIBC_TARGET_CPU_HAS_FMA 141 return; 142 } 143 144 // x is inf or nan. 145 if (LIBC_UNLIKELY(x_abs >= 0x7f80'0000U)) { 146 if (x_abs == 0x7f80'0000U) { 147 fputil::set_errno_if_required(EDOM); 148 fputil::raise_except_if_required(FE_INVALID); 149 } 150 *sinp = FPBits::quiet_nan().get_val(); 151 *cosp = *sinp; 152 return; 153 } 154 155 // Check exceptional values. 156 for (int i = 0; i < N_EXCEPTS; ++i) { 157 if (LIBC_UNLIKELY(x_abs == EXCEPT_INPUTS[i])) { 158 uint32_t s = EXCEPT_OUTPUTS_SIN[i][0]; // FE_TOWARDZERO 159 uint32_t c = EXCEPT_OUTPUTS_COS[i][0]; // FE_TOWARDZERO 160 bool x_sign = x < 0; 161 switch (fputil::quick_get_round()) { 162 case FE_UPWARD: 163 s += x_sign ? EXCEPT_OUTPUTS_SIN[i][2] : EXCEPT_OUTPUTS_SIN[i][1]; 164 c += EXCEPT_OUTPUTS_COS[i][1]; 165 break; 166 case FE_DOWNWARD: 167 s += x_sign ? EXCEPT_OUTPUTS_SIN[i][1] : EXCEPT_OUTPUTS_SIN[i][2]; 168 c += EXCEPT_OUTPUTS_COS[i][2]; 169 break; 170 case FE_TONEAREST: 171 s += EXCEPT_OUTPUTS_SIN[i][3]; 172 c += EXCEPT_OUTPUTS_COS[i][3]; 173 break; 174 } 175 *sinp = x_sign ? -FPBits(s).get_val() : FPBits(s).get_val(); 176 *cosp = FPBits(c).get_val(); 177 178 return; 179 } 180 } 181 182 // Combine the results with the sine and cosine of sum formulas: 183 // sin(x) = sin((k + y)*pi/32) 184 // = sin(y*pi/32) * cos(k*pi/32) + cos(y*pi/32) * sin(k*pi/32) 185 // = sin_y * cos_k + (1 + cosm1_y) * sin_k 186 // = sin_y * cos_k + (cosm1_y * sin_k + sin_k) 187 // cos(x) = cos((k + y)*pi/32) 188 // = cos(y*pi/32) * cos(k*pi/32) - sin(y*pi/32) * sin(k*pi/32) 189 // = cosm1_y * cos_k + sin_y * sin_k 190 // = (cosm1_y * cos_k + cos_k) + sin_y * sin_k 191 double sin_k, cos_k, sin_y, cosm1_y; 192 193 sincosf_eval(xd, x_abs, sin_k, cos_k, sin_y, cosm1_y); 194 195 *sinp = static_cast<float>(fputil::multiply_add( 196 sin_y, cos_k, fputil::multiply_add(cosm1_y, sin_k, sin_k))); 197 *cosp = static_cast<float>(fputil::multiply_add( 198 sin_y, -sin_k, fputil::multiply_add(cosm1_y, cos_k, cos_k))); 199 } 200 201 } // namespace LIBC_NAMESPACE_DECL 202