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