1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
|
//===-- Single-precision 2^x function -------------------------------------===//
//
// Part of the LLVM Project, under the Apache License v2.0 with LLVM Exceptions.
// See https://llvm.org/LICENSE.txt for license information.
// SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception
//
//===----------------------------------------------------------------------===//
#include "src/math/exp2f.h"
#include "src/__support/FPUtil/FEnvImpl.h"
#include "src/__support/FPUtil/FPBits.h"
#include "src/__support/FPUtil/PolyEval.h"
#include "src/__support/FPUtil/except_value_utils.h"
#include "src/__support/FPUtil/multiply_add.h"
#include "src/__support/FPUtil/nearest_integer.h"
#include "src/__support/FPUtil/rounding_mode.h"
#include "src/__support/common.h"
#include "src/__support/macros/optimization.h" // LIBC_UNLIKELY
#include "src/__support/macros/properties/cpu_features.h"
#include <errno.h>
#include "explogxf.h"
namespace LIBC_NAMESPACE {
constexpr uint32_t EXVAL1 = 0x3b42'9d37U;
constexpr uint32_t EXVAL2 = 0xbcf3'a937U;
constexpr uint32_t EXVAL_MASK = EXVAL1 & EXVAL2;
LLVM_LIBC_FUNCTION(float, exp2f, (float x)) {
using FPBits = typename fputil::FPBits<float>;
FPBits xbits(x);
uint32_t x_u = xbits.uintval();
uint32_t x_abs = x_u & 0x7fff'ffffU;
// When |x| >= 128, or x is nan, or |x| <= 2^-5
if (LIBC_UNLIKELY(x_abs >= 0x4300'0000U || x_abs <= 0x3d00'0000U)) {
// |x| <= 2^-5
if (x_abs <= 0x3d00'0000) {
// |x| < 2^-25
if (LIBC_UNLIKELY(x_abs <= 0x3280'0000U)) {
return 1.0f + x;
}
// Check exceptional values.
if (LIBC_UNLIKELY((x_u & EXVAL_MASK) == EXVAL_MASK)) {
if (LIBC_UNLIKELY(x_u == EXVAL1)) { // x = 0x1.853a6ep-9f
return fputil::round_result_slightly_down(0x1.00870ap+0f);
} else if (LIBC_UNLIKELY(x_u == EXVAL2)) { // x = -0x1.e7526ep-6f
return fputil::round_result_slightly_down(0x1.f58d62p-1f);
}
}
// Minimax polynomial generated by Sollya with:
// > P = fpminimax((2^x - 1)/x, 5, [|D...|], [-2^-5, 2^-5]);
constexpr double COEFFS[] = {
0x1.62e42fefa39f3p-1, 0x1.ebfbdff82c57bp-3, 0x1.c6b08d6f2d7aap-5,
0x1.3b2ab6fc92f5dp-7, 0x1.5d897cfe27125p-10, 0x1.43090e61e6af1p-13};
double xd = static_cast<double>(x);
double xsq = xd * xd;
double c0 = fputil::multiply_add(xd, COEFFS[1], COEFFS[0]);
double c1 = fputil::multiply_add(xd, COEFFS[3], COEFFS[2]);
double c2 = fputil::multiply_add(xd, COEFFS[5], COEFFS[4]);
double p = fputil::polyeval(xsq, c0, c1, c2);
double r = fputil::multiply_add(p, xd, 1.0);
return static_cast<float>(r);
}
// x >= 128
if (!xbits.get_sign()) {
// x is finite
if (x_u < 0x7f80'0000U) {
int rounding = fputil::quick_get_round();
if (rounding == FE_DOWNWARD || rounding == FE_TOWARDZERO)
return static_cast<float>(FPBits(FPBits::MAX_NORMAL));
fputil::set_errno_if_required(ERANGE);
fputil::raise_except_if_required(FE_OVERFLOW);
}
// x is +inf or nan
return x + FPBits::inf().get_val();
}
// x <= -150
if (x_u >= 0xc316'0000U) {
// exp(-Inf) = 0
if (xbits.is_inf())
return 0.0f;
// exp(nan) = nan
if (xbits.is_nan())
return x;
if (fputil::fenv_is_round_up())
return FPBits(FPBits::MIN_SUBNORMAL).get_val();
if (x != 0.0f) {
fputil::set_errno_if_required(ERANGE);
fputil::raise_except_if_required(FE_UNDERFLOW);
}
return 0.0f;
}
}
// For -150 < x < 128, to compute 2^x, we perform the following range
// reduction: find hi, mid, lo such that:
// x = hi + mid + lo, in which
// hi is an integer,
// 0 <= mid * 2^5 < 32 is an integer
// -2^(-6) <= lo <= 2^-6.
// In particular,
// hi + mid = round(x * 2^5) * 2^(-5).
// Then,
// 2^x = 2^(hi + mid + lo) = 2^hi * 2^mid * 2^lo.
// 2^mid is stored in the lookup table of 32 elements.
// 2^lo is computed using a degree-5 minimax polynomial
// generated by Sollya.
// We perform 2^hi * 2^mid by simply add hi to the exponent field
// of 2^mid.
// kf = (hi + mid) * 2^5 = round(x * 2^5)
float kf;
int k;
#ifdef LIBC_TARGET_CPU_HAS_NEAREST_INT
kf = fputil::nearest_integer(x * 32.0f);
k = static_cast<int>(kf);
#else
constexpr float HALF[2] = {0.5f, -0.5f};
k = static_cast<int>(fputil::multiply_add(x, 32.0f, HALF[x < 0.0f]));
kf = static_cast<float>(k);
#endif // LIBC_TARGET_CPU_HAS_NEAREST_INT
// dx = lo = x - (hi + mid) = x - kf * 2^(-5)
double dx = fputil::multiply_add(-0x1.0p-5f, kf, x);
// hi = floor(kf * 2^(-4))
// exp_hi = shift hi to the exponent field of double precision.
int64_t exp_hi =
static_cast<int64_t>(static_cast<uint64_t>(k >> ExpBase::MID_BITS)
<< fputil::FloatProperties<double>::MANTISSA_WIDTH);
// mh = 2^hi * 2^mid
// mh_bits = bit field of mh
int64_t mh_bits = ExpBase::EXP_2_MID[k & ExpBase::MID_MASK] + exp_hi;
double mh = fputil::FPBits<double>(uint64_t(mh_bits)).get_val();
// Degree-5 polynomial approximating (2^x - 1)/x generating by Sollya with:
// > P = fpminimax((2^x - 1)/x, 5, [|D...|], [-1/32. 1/32]);
constexpr double COEFFS[5] = {0x1.62e42fefa39efp-1, 0x1.ebfbdff8131c4p-3,
0x1.c6b08d7061695p-5, 0x1.3b2b1bee74b2ap-7,
0x1.5d88091198529p-10};
double dx_sq = dx * dx;
double c1 = fputil::multiply_add(dx, COEFFS[0], 1.0);
double c2 = fputil::multiply_add(dx, COEFFS[2], COEFFS[1]);
double c3 = fputil::multiply_add(dx, COEFFS[4], COEFFS[3]);
double p = fputil::multiply_add(dx_sq, c3, c2);
// 2^x = 2^(hi + mid + lo)
// = 2^(hi + mid) * 2^lo
// ~ mh * (1 + lo * P(lo))
// = mh + (mh*lo) * P(lo)
return static_cast<float>(fputil::multiply_add(p, dx_sq * mh, c1 * mh));
}
} // namespace LIBC_NAMESPACE
|
