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