1 //===-- Single-precision cos 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/cosf.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/except_value_utils.h" 15 #include "src/__support/FPUtil/multiply_add.h" 16 #include "src/__support/common.h" 17 #include "src/__support/macros/config.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 namespace LIBC_NAMESPACE_DECL { 22 23 // Exceptional cases for cosf. 24 static constexpr size_t N_EXCEPTS = 6; 25 26 static constexpr fputil::ExceptValues<float, N_EXCEPTS> COSF_EXCEPTS{{ 27 // (inputs, RZ output, RU offset, RD offset, RN offset) 28 // x = 0x1.64a032p43, cos(x) = 0x1.9d4ba4p-1 (RZ) 29 {0x55325019, 0x3f4ea5d2, 1, 0, 0}, 30 // x = 0x1.4555p51, cos(x) = 0x1.115d7cp-1 (RZ) 31 {0x5922aa80, 0x3f08aebe, 1, 0, 1}, 32 // x = 0x1.48a858p54, cos(x) = 0x1.f48148p-2 (RZ) 33 {0x5aa4542c, 0x3efa40a4, 1, 0, 0}, 34 // x = 0x1.3170fp63, cos(x) = 0x1.fe2976p-1 (RZ) 35 {0x5f18b878, 0x3f7f14bb, 1, 0, 0}, 36 // x = 0x1.2b9622p67, cos(x) = 0x1.f0285cp-1 (RZ) 37 {0x6115cb11, 0x3f78142e, 1, 0, 1}, 38 // x = 0x1.ddebdep120, cos(x) = 0x1.114438p-1 (RZ) 39 {0x7beef5ef, 0x3f08a21c, 1, 0, 0}, 40 }}; 41 42 LLVM_LIBC_FUNCTION(float, cosf, (float x)) { 43 using FPBits = typename fputil::FPBits<float>; 44 45 FPBits xbits(x); 46 xbits.set_sign(Sign::POS); 47 48 uint32_t x_abs = xbits.uintval(); 49 double xd = static_cast<double>(xbits.get_val()); 50 51 // Range reduction: 52 // For |x| > pi/16, we perform range reduction as follows: 53 // Find k and y such that: 54 // x = (k + y) * pi/32 55 // k is an integer 56 // |y| < 0.5 57 // For small range (|x| < 2^45 when FMA instructions are available, 2^22 58 // otherwise), this is done by performing: 59 // k = round(x * 32/pi) 60 // y = x * 32/pi - k 61 // For large range, we will omit all the higher parts of 16/pi such that the 62 // least significant bits of their full products with x are larger than 63, 63 // since cos((k + y + 64*i) * pi/32) = cos(x + i * 2pi) = cos(x). 64 // 65 // When FMA instructions are not available, we store the digits of 32/pi in 66 // chunks of 28-bit precision. This will make sure that the products: 67 // x * THIRTYTWO_OVER_PI_28[i] are all exact. 68 // When FMA instructions are available, we simply store the digits of 32/pi in 69 // chunks of doubles (53-bit of precision). 70 // So when multiplying by the largest values of single precision, the 71 // resulting output should be correct up to 2^(-208 + 128) ~ 2^-80. By the 72 // worst-case analysis of range reduction, |y| >= 2^-38, so this should give 73 // us more than 40 bits of accuracy. For the worst-case estimation of range 74 // reduction, see for instances: 75 // Elementary Functions by J-M. Muller, Chapter 11, 76 // Handbook of Floating-Point Arithmetic by J-M. Muller et. al., 77 // Chapter 10.2. 78 // 79 // Once k and y are computed, we then deduce the answer by the cosine of sum 80 // formula: 81 // cos(x) = cos((k + y)*pi/32) 82 // = cos(y*pi/32) * cos(k*pi/32) - sin(y*pi/32) * sin(k*pi/32) 83 // The values of sin(k*pi/32) and cos(k*pi/32) for k = 0..63 are precomputed 84 // and stored using a vector of 32 doubles. Sin(y*pi/32) and cos(y*pi/32) are 85 // computed using degree-7 and degree-6 minimax polynomials generated by 86 // Sollya respectively. 87 88 // |x| < 0x1.0p-12f 89 if (LIBC_UNLIKELY(x_abs < 0x3980'0000U)) { 90 // When |x| < 2^-12, the relative error of the approximation cos(x) ~ 1 91 // is: 92 // |cos(x) - 1| < |x^2 / 2| = 2^-25 < epsilon(1)/2. 93 // So the correctly rounded values of cos(x) are: 94 // = 1 - eps(x) if rounding mode = FE_TOWARDZERO or FE_DOWWARD, 95 // = 1 otherwise. 96 // To simplify the rounding decision and make it more efficient and to 97 // prevent compiler to perform constant folding, we use 98 // fma(x, -2^-25, 1) instead. 99 // Note: to use the formula 1 - 2^-25*x to decide the correct rounding, we 100 // do need fma(x, -2^-25, 1) to prevent underflow caused by -2^-25*x when 101 // |x| < 2^-125. For targets without FMA instructions, we simply use 102 // double for intermediate results as it is more efficient than using an 103 // emulated version of FMA. 104 #if defined(LIBC_TARGET_CPU_HAS_FMA) 105 return fputil::multiply_add(xbits.get_val(), -0x1.0p-25f, 1.0f); 106 #else 107 return static_cast<float>(fputil::multiply_add(xd, -0x1.0p-25, 1.0)); 108 #endif // LIBC_TARGET_CPU_HAS_FMA 109 } 110 111 if (auto r = COSF_EXCEPTS.lookup(x_abs); LIBC_UNLIKELY(r.has_value())) 112 return r.value(); 113 114 // x is inf or nan. 115 if (LIBC_UNLIKELY(x_abs >= 0x7f80'0000U)) { 116 if (x_abs == 0x7f80'0000U) { 117 fputil::set_errno_if_required(EDOM); 118 fputil::raise_except_if_required(FE_INVALID); 119 } 120 return x + FPBits::quiet_nan().get_val(); 121 } 122 123 // Combine the results with the sine of sum formula: 124 // cos(x) = cos((k + y)*pi/32) 125 // = cos(y*pi/32) * cos(k*pi/32) - sin(y*pi/32) * sin(k*pi/32) 126 // = cosm1_y * cos_k + sin_y * sin_k 127 // = (cosm1_y * cos_k + cos_k) + sin_y * sin_k 128 double sin_k, cos_k, sin_y, cosm1_y; 129 130 sincosf_eval(xd, x_abs, sin_k, cos_k, sin_y, cosm1_y); 131 132 return static_cast<float>(fputil::multiply_add( 133 sin_y, -sin_k, fputil::multiply_add(cosm1_y, cos_k, cos_k))); 134 } 135 136 } // namespace LIBC_NAMESPACE_DECL 137