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
23namespace LIBC_NAMESPACE {
24
25LLVM_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: x * 0x1.0p7f);
89 // Subtract (hi + mid) from x to get lo.
90 double xd = static_cast<double>(fputil::multiply_add(x: kf, y: -0x1.0p-7f, z: 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(x: xd, a0: 0x1p0, a: 0x1.ffffffffff777p-1, a: 0x1.000000000071cp-1,
104 a: 0x1.555566668e5e7p-3, a: 0x1.55555555ef243p-5);
105 return static_cast<float>(exp_hi * exp_mid * exp_lo);
106}
107
108} // namespace LIBC_NAMESPACE
109

source code of libc/src/math/generic/expf.cpp