1 //===-- Single-precision e^x - 1 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/expm1f.h" 10 #include "common_constants.h" // Lookup tables EXP_M1 and EXP_M2. 11 #include "src/__support/FPUtil/BasicOperations.h" 12 #include "src/__support/FPUtil/FEnvImpl.h" 13 #include "src/__support/FPUtil/FMA.h" 14 #include "src/__support/FPUtil/FPBits.h" 15 #include "src/__support/FPUtil/PolyEval.h" 16 #include "src/__support/FPUtil/multiply_add.h" 17 #include "src/__support/FPUtil/nearest_integer.h" 18 #include "src/__support/FPUtil/rounding_mode.h" 19 #include "src/__support/common.h" 20 #include "src/__support/macros/optimization.h" // LIBC_UNLIKELY 21 #include "src/__support/macros/properties/cpu_features.h" // LIBC_TARGET_CPU_HAS_FMA 22 23 #include <errno.h> 24 25 namespace LIBC_NAMESPACE { 26 27 LLVM_LIBC_FUNCTION(float, expm1f, (float x)) { 28 using FPBits = typename fputil::FPBits<float>; 29 FPBits xbits(x); 30 31 uint32_t x_u = xbits.uintval(); 32 uint32_t x_abs = x_u & 0x7fff'ffffU; 33 34 // Exceptional value 35 if (LIBC_UNLIKELY(x_u == 0x3e35'bec5U)) { // x = 0x1.6b7d8ap-3f 36 int round_mode = fputil::quick_get_round(); 37 if (round_mode == FE_TONEAREST || round_mode == FE_UPWARD) 38 return 0x1.8dbe64p-3f; 39 return 0x1.8dbe62p-3f; 40 } 41 42 #if !defined(LIBC_TARGET_CPU_HAS_FMA) 43 if (LIBC_UNLIKELY(x_u == 0xbdc1'c6cbU)) { // x = -0x1.838d96p-4f 44 int round_mode = fputil::quick_get_round(); 45 if (round_mode == FE_TONEAREST || round_mode == FE_DOWNWARD) 46 return -0x1.71c884p-4f; 47 return -0x1.71c882p-4f; 48 } 49 #endif // LIBC_TARGET_CPU_HAS_FMA 50 51 // When |x| > 25*log(2), or nan 52 if (LIBC_UNLIKELY(x_abs >= 0x418a'a123U)) { 53 // x < log(2^-25) 54 if (xbits.is_neg()) { 55 // exp(-Inf) = 0 56 if (xbits.is_inf()) 57 return -1.0f; 58 // exp(nan) = nan 59 if (xbits.is_nan()) 60 return x; 61 int round_mode = fputil::quick_get_round(); 62 if (round_mode == FE_UPWARD || round_mode == FE_TOWARDZERO) 63 return -0x1.ffff'fep-1f; // -1.0f + 0x1.0p-24f 64 return -1.0f; 65 } else { 66 // x >= 89 or nan 67 if (xbits.uintval() >= 0x42b2'0000) { 68 if (xbits.uintval() < 0x7f80'0000U) { 69 int rounding = fputil::quick_get_round(); 70 if (rounding == FE_DOWNWARD || rounding == FE_TOWARDZERO) 71 return FPBits::max_normal().get_val(); 72 73 fputil::set_errno_if_required(ERANGE); 74 fputil::raise_except_if_required(FE_OVERFLOW); 75 } 76 return x + FPBits::inf().get_val(); 77 } 78 } 79 } 80 81 // |x| < 2^-4 82 if (x_abs < 0x3d80'0000U) { 83 // |x| < 2^-25 84 if (x_abs < 0x3300'0000U) { 85 // x = -0.0f 86 if (LIBC_UNLIKELY(xbits.uintval() == 0x8000'0000U)) 87 return x; 88 // When |x| < 2^-25, the relative error of the approximation e^x - 1 ~ x 89 // is: 90 // |(e^x - 1) - x| / |e^x - 1| < |x^2| / |x| 91 // = |x| 92 // < 2^-25 93 // < epsilon(1)/2. 94 // So the correctly rounded values of expm1(x) are: 95 // = x + eps(x) if rounding mode = FE_UPWARD, 96 // or (rounding mode = FE_TOWARDZERO and x is 97 // negative), 98 // = x otherwise. 99 // To simplify the rounding decision and make it more efficient, we use 100 // fma(x, x, x) ~ x + x^2 instead. 101 // Note: to use the formula x + x^2 to decide the correct rounding, we 102 // do need fma(x, x, x) to prevent underflow caused by x*x when |x| < 103 // 2^-76. For targets without FMA instructions, we simply use double for 104 // intermediate results as it is more efficient than using an emulated 105 // version of FMA. 106 #if defined(LIBC_TARGET_CPU_HAS_FMA) 107 return fputil::fma(x, x, x); 108 #else 109 double xd = x; 110 return static_cast<float>(fputil::multiply_add(xd, xd, xd)); 111 #endif // LIBC_TARGET_CPU_HAS_FMA 112 } 113 114 constexpr double COEFFS[] = {0x1p-1, 115 0x1.55555555557ddp-3, 116 0x1.55555555552fap-5, 117 0x1.111110fcd58b7p-7, 118 0x1.6c16c1717660bp-10, 119 0x1.a0241f0006d62p-13, 120 0x1.a01e3f8d3c06p-16}; 121 122 // 2^-25 <= |x| < 2^-4 123 double xd = static_cast<double>(x); 124 double xsq = xd * xd; 125 // Degree-8 minimax polynomial generated by Sollya with: 126 // > display = hexadecimal; 127 // > P = fpminimax((expm1(x) - x)/x^2, 6, [|D...|], [-2^-4, 2^-4]); 128 129 double c0 = fputil::multiply_add(xd, COEFFS[1], COEFFS[0]); 130 double c1 = fputil::multiply_add(xd, COEFFS[3], COEFFS[2]); 131 double c2 = fputil::multiply_add(xd, COEFFS[5], COEFFS[4]); 132 133 double r = fputil::polyeval(xsq, c0, c1, c2, COEFFS[6]); 134 return static_cast<float>(fputil::multiply_add(r, xsq, xd)); 135 } 136 137 // For -18 < x < 89, to compute expm1(x), we perform the following range 138 // reduction: find hi, mid, lo such that: 139 // x = hi + mid + lo, in which 140 // hi is an integer, 141 // mid * 2^7 is an integer 142 // -2^(-8) <= lo < 2^-8. 143 // In particular, 144 // hi + mid = round(x * 2^7) * 2^(-7). 145 // Then, 146 // expm1(x) = exp(hi + mid + lo) - 1 = exp(hi) * exp(mid) * exp(lo) - 1. 147 // We store exp(hi) and exp(mid) in the lookup tables EXP_M1 and EXP_M2 148 // respectively. exp(lo) is computed using a degree-4 minimax polynomial 149 // generated by Sollya. 150 151 // x_hi = hi + mid. 152 float kf = fputil::nearest_integer(x * 0x1.0p7f); 153 int x_hi = static_cast<int>(kf); 154 // Subtract (hi + mid) from x to get lo. 155 double xd = static_cast<double>(fputil::multiply_add(kf, -0x1.0p-7f, x)); 156 x_hi += 104 << 7; 157 // hi = x_hi >> 7 158 double exp_hi = EXP_M1[x_hi >> 7]; 159 // lo = x_hi & 0x0000'007fU; 160 double exp_mid = EXP_M2[x_hi & 0x7f]; 161 double exp_hi_mid = exp_hi * exp_mid; 162 // Degree-4 minimax polynomial generated by Sollya with the following 163 // commands: 164 // > display = hexadecimal; 165 // > Q = fpminimax(expm1(x)/x, 3, [|D...|], [-2^-8, 2^-8]); 166 // > Q; 167 double exp_lo = 168 fputil::polyeval(xd, 0x1.0p0, 0x1.ffffffffff777p-1, 0x1.000000000071cp-1, 169 0x1.555566668e5e7p-3, 0x1.55555555ef243p-5); 170 return static_cast<float>(fputil::multiply_add(exp_hi_mid, exp_lo, -1.0)); 171 } 172 173 } // namespace LIBC_NAMESPACE 174