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/config.h" 20 #include "src/__support/macros/optimization.h" // LIBC_UNLIKELY 21 22 #include <errno.h> 23 24 namespace LIBC_NAMESPACE_DECL { 25 26 LLVM_LIBC_FUNCTION(float, expf, (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 values 34 if (LIBC_UNLIKELY(x_u == 0xc236'bd8cU)) { // x = -0x1.6d7b18p+5f 35 return 0x1.108a58p-66f - x * 0x1.0p-95f; 36 } 37 38 // When |x| >= 89, |x| < 2^-25, or x is nan 39 if (LIBC_UNLIKELY(x_abs >= 0x42b2'0000U || x_abs <= 0x3280'0000U)) { 40 // |x| < 2^-25 41 if (xbits.get_biased_exponent() <= 101) { 42 return 1.0f + x; 43 } 44 45 // When x < log(2^-150) or nan 46 if (xbits.uintval() >= 0xc2cf'f1b5U) { 47 // exp(-Inf) = 0 48 if (xbits.is_inf()) 49 return 0.0f; 50 // exp(nan) = nan 51 if (xbits.is_nan()) 52 return x; 53 if (fputil::fenv_is_round_up()) 54 return FPBits::min_subnormal().get_val(); 55 fputil::set_errno_if_required(ERANGE); 56 fputil::raise_except_if_required(FE_UNDERFLOW); 57 return 0.0f; 58 } 59 // x >= 89 or nan 60 if (xbits.is_pos() && (xbits.uintval() >= 0x42b2'0000)) { 61 // x is finite 62 if (xbits.uintval() < 0x7f80'0000U) { 63 int rounding = fputil::quick_get_round(); 64 if (rounding == FE_DOWNWARD || rounding == FE_TOWARDZERO) 65 return FPBits::max_normal().get_val(); 66 67 fputil::set_errno_if_required(ERANGE); 68 fputil::raise_except_if_required(FE_OVERFLOW); 69 } 70 // x is +inf or nan 71 return x + FPBits::inf().get_val(); 72 } 73 } 74 // For -104 < x < 89, to compute exp(x), we perform the following range 75 // reduction: find hi, mid, lo such that: 76 // x = hi + mid + lo, in which 77 // hi is an integer, 78 // mid * 2^7 is an integer 79 // -2^(-8) <= lo < 2^-8. 80 // In particular, 81 // hi + mid = round(x * 2^7) * 2^(-7). 82 // Then, 83 // exp(x) = exp(hi + mid + lo) = exp(hi) * exp(mid) * exp(lo). 84 // We store exp(hi) and exp(mid) in the lookup tables EXP_M1 and EXP_M2 85 // respectively. exp(lo) is computed using a degree-4 minimax polynomial 86 // generated by Sollya. 87 88 // x_hi = (hi + mid) * 2^7 = round(x * 2^7). 89 float kf = fputil::nearest_integer(x * 0x1.0p7f); 90 // Subtract (hi + mid) from x to get lo. 91 double xd = static_cast<double>(fputil::multiply_add(kf, -0x1.0p-7f, x)); 92 int x_hi = static_cast<int>(kf); 93 x_hi += 104 << 7; 94 // hi = x_hi >> 7 95 double exp_hi = EXP_M1[x_hi >> 7]; 96 // mid * 2^7 = x_hi & 0x0000'007fU; 97 double exp_mid = EXP_M2[x_hi & 0x7f]; 98 // Degree-4 minimax polynomial generated by Sollya with the following 99 // commands: 100 // > display = hexadecimal; 101 // > Q = fpminimax(expm1(x)/x, 3, [|D...|], [-2^-8, 2^-8]); 102 // > Q; 103 double exp_lo = 104 fputil::polyeval(xd, 0x1p0, 0x1.ffffffffff777p-1, 0x1.000000000071cp-1, 105 0x1.555566668e5e7p-3, 0x1.55555555ef243p-5); 106 return static_cast<float>(exp_hi * exp_mid * exp_lo); 107 } 108 109 } // namespace LIBC_NAMESPACE_DECL 110