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/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, expf, (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 values 31 if (unlikely(x_u == 0xc236'bd8cU)) { // x = -0x1.6d7b18p+5f 32 return 0x1.108a58p-66f - x * 0x1.0p-95f; 33 } 34 35 // When |x| >= 89, |x| < 2^-25, or x is nan 36 if (unlikely(x_abs >= 0x42b2'0000U || x_abs <= 0x3280'0000U)) { 37 // |x| < 2^-25 38 if (xbits.get_unbiased_exponent() <= 101) { 39 return 1.0f + x; 40 } 41 42 // When x < log(2^-150) or nan 43 if (xbits.uintval() >= 0xc2cf'f1b5U) { 44 // exp(-Inf) = 0 45 if (xbits.is_inf()) 46 return 0.0f; 47 // exp(nan) = nan 48 if (xbits.is_nan()) 49 return x; 50 if (fputil::get_round() == FE_UPWARD) 51 return static_cast<float>(FPBits(FPBits::MIN_SUBNORMAL)); 52 errno = ERANGE; 53 return 0.0f; 54 } 55 // x >= 89 or nan 56 if (!xbits.get_sign() && (xbits.uintval() >= 0x42b2'0000)) { 57 // x is finite 58 if (xbits.uintval() < 0x7f80'0000U) { 59 int rounding = fputil::get_round(); 60 if (rounding == FE_DOWNWARD || rounding == FE_TOWARDZERO) 61 return static_cast<float>(FPBits(FPBits::MAX_NORMAL)); 62 63 errno = ERANGE; 64 } 65 // x is +inf or nan 66 return x + static_cast<float>(FPBits::inf()); 67 } 68 } 69 // For -104 < x < 89, to compute exp(x), we perform the following range 70 // reduction: find hi, mid, lo such that: 71 // x = hi + mid + lo, in which 72 // hi is an integer, 73 // mid * 2^7 is an integer 74 // -2^(-8) <= lo < 2^-8. 75 // In particular, 76 // hi + mid = round(x * 2^7) * 2^(-7). 77 // Then, 78 // exp(x) = exp(hi + mid + lo) = exp(hi) * exp(mid) * exp(lo). 79 // We store exp(hi) and exp(mid) in the lookup tables EXP_M1 and EXP_M2 80 // respectively. exp(lo) is computed using a degree-4 minimax polynomial 81 // generated by Sollya. 82 83 // x_hi = (hi + mid) * 2^7 = round(x * 2^7). 84 // The default rounding mode for float-to-int conversion in C++ is 85 // round-toward-zero. To make it round-to-nearest, we add (-1)^sign(x) * 0.5 86 // before conversion. 87 int x_hi = static_cast<int>(x * 0x1.0p7f + (xbits.get_sign() ? -0.5f : 0.5f)); 88 // Subtract (hi + mid) from x to get lo. 89 x -= static_cast<float>(x_hi) * 0x1.0p-7f; 90 double xd = static_cast<double>(x); 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 __llvm_libc 108