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