1 //===-- Single-precision e^x 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/expf.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/FPBits.h" 14 #include "src/__support/FPUtil/PolyEval.h" 15 #include "src/__support/FPUtil/multiply_add.h" 16 #include "src/__support/FPUtil/nearest_integer.h" 17 #include "src/__support/FPUtil/rounding_mode.h" 18 #include "src/__support/common.h" 19 #include "src/__support/macros/optimization.h" // LIBC_UNLIKELY 20 21 #include <errno.h> 22 23 namespace LIBC_NAMESPACE { 24 25 LLVM_LIBC_FUNCTION(float, expf, (float x)) { 26 using FPBits = typename fputil::FPBits<float>; 27 FPBits xbits(x); 28 29 uint32_t x_u = xbits.uintval(); 30 uint32_t x_abs = x_u & 0x7fff'ffffU; 31 32 // Exceptional values 33 if (LIBC_UNLIKELY(x_u == 0xc236'bd8cU)) { // x = -0x1.6d7b18p+5f 34 return 0x1.108a58p-66f - x * 0x1.0p-95f; 35 } 36 37 // When |x| >= 89, |x| < 2^-25, or x is nan 38 if (LIBC_UNLIKELY(x_abs >= 0x42b2'0000U || x_abs <= 0x3280'0000U)) { 39 // |x| < 2^-25 40 if (xbits.get_biased_exponent() <= 101) { 41 return 1.0f + x; 42 } 43 44 // When x < log(2^-150) or nan 45 if (xbits.uintval() >= 0xc2cf'f1b5U) { 46 // exp(-Inf) = 0 47 if (xbits.is_inf()) 48 return 0.0f; 49 // exp(nan) = nan 50 if (xbits.is_nan()) 51 return x; 52 if (fputil::fenv_is_round_up()) 53 return FPBits::min_subnormal().get_val(); 54 fputil::set_errno_if_required(ERANGE); 55 fputil::raise_except_if_required(FE_UNDERFLOW); 56 return 0.0f; 57 } 58 // x >= 89 or nan 59 if (xbits.is_pos() && (xbits.uintval() >= 0x42b2'0000)) { 60 // x is finite 61 if (xbits.uintval() < 0x7f80'0000U) { 62 int rounding = fputil::quick_get_round(); 63 if (rounding == FE_DOWNWARD || rounding == FE_TOWARDZERO) 64 return FPBits::max_normal().get_val(); 65 66 fputil::set_errno_if_required(ERANGE); 67 fputil::raise_except_if_required(FE_OVERFLOW); 68 } 69 // x is +inf or nan 70 return x + FPBits::inf().get_val(); 71 } 72 } 73 // For -104 < x < 89, to compute exp(x), we perform the following range 74 // reduction: find hi, mid, lo such that: 75 // x = hi + mid + lo, in which 76 // hi is an integer, 77 // mid * 2^7 is an integer 78 // -2^(-8) <= lo < 2^-8. 79 // In particular, 80 // hi + mid = round(x * 2^7) * 2^(-7). 81 // Then, 82 // exp(x) = exp(hi + mid + lo) = exp(hi) * exp(mid) * exp(lo). 83 // We store exp(hi) and exp(mid) in the lookup tables EXP_M1 and EXP_M2 84 // respectively. exp(lo) is computed using a degree-4 minimax polynomial 85 // generated by Sollya. 86 87 // x_hi = (hi + mid) * 2^7 = round(x * 2^7). 88 float kf = fputil::nearest_integer(x * 0x1.0p7f); 89 // Subtract (hi + mid) from x to get lo. 90 double xd = static_cast<double>(fputil::multiply_add(kf, -0x1.0p-7f, x)); 91 int x_hi = static_cast<int>(kf); 92 x_hi += 104 << 7; 93 // hi = x_hi >> 7 94 double exp_hi = EXP_M1[x_hi >> 7]; 95 // mid * 2^7 = x_hi & 0x0000'007fU; 96 double exp_mid = EXP_M2[x_hi & 0x7f]; 97 // Degree-4 minimax polynomial generated by Sollya with the following 98 // commands: 99 // > display = hexadecimal; 100 // > Q = fpminimax(expm1(x)/x, 3, [|D...|], [-2^-8, 2^-8]); 101 // > Q; 102 double exp_lo = 103 fputil::polyeval(xd, 0x1p0, 0x1.ffffffffff777p-1, 0x1.000000000071cp-1, 104 0x1.555566668e5e7p-3, 0x1.55555555ef243p-5); 105 return static_cast<float>(exp_hi * exp_mid * exp_lo); 106 } 107 108 } // namespace LIBC_NAMESPACE 109