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/common.h" 17 18 #include <errno.h> 19 20 namespace __llvm_libc { 21 22 INLINE_FMA 23 LLVM_LIBC_FUNCTION(float, expm1f, (float x)) { 24 using FPBits = typename fputil::FPBits<float>; 25 FPBits xbits(x); 26 27 uint32_t x_u = xbits.uintval(); 28 uint32_t x_abs = x_u & 0x7fff'ffffU; 29 30 // Exceptional value 31 if (unlikely(x_u == 0x3e35'bec5U)) { // x = 0x1.6b7d8ap-3f 32 int round_mode = fputil::get_round(); 33 if (round_mode == FE_TONEAREST || round_mode == FE_UPWARD) 34 return 0x1.8dbe64p-3f; 35 return 0x1.8dbe62p-3f; 36 } 37 38 // When |x| > 25*log(2), or nan 39 if (unlikely(x_abs >= 0x418a'a123U)) { 40 // x < log(2^-25) 41 if (xbits.get_sign()) { 42 // exp(-Inf) = 0 43 if (xbits.is_inf()) 44 return -1.0f; 45 // exp(nan) = nan 46 if (xbits.is_nan()) 47 return x; 48 int round_mode = fputil::get_round(); 49 if (round_mode == FE_UPWARD || round_mode == FE_TOWARDZERO) 50 return -0x1.ffff'fep-1f; // -1.0f + 0x1.0p-24f 51 return -1.0f; 52 } else { 53 // x >= 89 or nan 54 if (xbits.uintval() >= 0x42b2'0000) { 55 if (xbits.uintval() < 0x7f80'0000U) { 56 int rounding = fputil::get_round(); 57 if (rounding == FE_DOWNWARD || rounding == FE_TOWARDZERO) 58 return static_cast<float>(FPBits(FPBits::MAX_NORMAL)); 59 60 errno = ERANGE; 61 } 62 return x + static_cast<float>(FPBits::inf()); 63 } 64 } 65 } 66 67 // |x| < 2^-4 68 if (x_abs < 0x3d80'0000U) { 69 // |x| < 2^-25 70 if (x_abs < 0x3300'0000U) { 71 // x = -0.0f 72 if (unlikely(xbits.uintval() == 0x8000'0000U)) 73 return x; 74 // When |x| < 2^-25, the relative error of the approximation e^x - 1 ~ x 75 // is: 76 // |(e^x - 1) - x| / |e^x - 1| < |x^2| / |x| 77 // = |x| 78 // < 2^-25 79 // < epsilon(1)/2. 80 // So the correctly rounded values of expm1(x) are: 81 // = x + eps(x) if rounding mode = FE_UPWARD, 82 // or (rounding mode = FE_TOWARDZERO and x is negative), 83 // = x otherwise. 84 // To simplify the rounding decision and make it more efficient, we use 85 // fma(x, x, x) ~ x + x^2 instead. 86 return fputil::multiply_add(x, x, x); 87 } 88 89 // 2^-25 <= |x| < 2^-4 90 double xd = static_cast<double>(x); 91 double xsq = xd * xd; 92 // Degree-8 minimax polynomial generated by Sollya with: 93 // > display = hexadecimal; 94 // > P = fpminimax((expm1(x) - x)/x^2, 6, [|D...|], [-2^-4, 2^-4]); 95 double r = 96 fputil::polyeval(xd, 0x1p-1, 0x1.55555555557ddp-3, 0x1.55555555552fap-5, 97 0x1.111110fcd58b7p-7, 0x1.6c16c1717660bp-10, 98 0x1.a0241f0006d62p-13, 0x1.a01e3f8d3c06p-16); 99 return static_cast<float>(fputil::multiply_add(r, xsq, xd)); 100 } 101 102 // For -18 < x < 89, to compute expm1(x), we perform the following range 103 // reduction: find hi, mid, lo such that: 104 // x = hi + mid + lo, in which 105 // hi is an integer, 106 // mid * 2^7 is an integer 107 // -2^(-8) <= lo < 2^-8. 108 // In particular, 109 // hi + mid = round(x * 2^7) * 2^(-7). 110 // Then, 111 // expm1(x) = exp(hi + mid + lo) - 1 = exp(hi) * exp(mid) * exp(lo) - 1. 112 // We store exp(hi) and exp(mid) in the lookup tables EXP_M1 and EXP_M2 113 // respectively. exp(lo) is computed using a degree-4 minimax polynomial 114 // generated by Sollya. 115 116 // x_hi = hi + mid. 117 int x_hi = static_cast<int>(x * 0x1.0p7f + (xbits.get_sign() ? -0.5f : 0.5f)); 118 // Subtract (hi + mid) from x to get lo. 119 x -= static_cast<float>(x_hi) * 0x1.0p-7f; 120 double xd = static_cast<double>(x); 121 x_hi += 104 << 7; 122 // hi = x_hi >> 7 123 double exp_hi = EXP_M1[x_hi >> 7]; 124 // lo = x_hi & 0x0000'007fU; 125 double exp_mid = EXP_M2[x_hi & 0x7f]; 126 double exp_hi_mid = exp_hi * exp_mid; 127 // Degree-4 minimax polynomial generated by Sollya with the following 128 // commands: 129 // > display = hexadecimal; 130 // > Q = fpminimax(expm1(x)/x, 3, [|D...|], [-2^-8, 2^-8]); 131 // > Q; 132 double exp_lo = 133 fputil::polyeval(xd, 0x1.0p0, 0x1.ffffffffff777p-1, 0x1.000000000071cp-1, 134 0x1.555566668e5e7p-3, 0x1.55555555ef243p-5); 135 return static_cast<float>(fputil::multiply_add(exp_hi_mid, exp_lo, -1.0)); 136 } 137 138 } // namespace __llvm_libc 139