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
|
/* Double-precision log(x) function.
Copyright (C) 2018-2020 Free Software Foundation, Inc.
This file is part of the GNU C Library.
The GNU C Library is free software; you can redistribute it and/or
modify it under the terms of the GNU Lesser General Public
License as published by the Free Software Foundation; either
version 2.1 of the License, or (at your option) any later version.
The GNU C Library is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
Lesser General Public License for more details.
You should have received a copy of the GNU Lesser General Public
License along with the GNU C Library; if not, see
<https://www.gnu.org/licenses/>. */
#include <math.h>
#include <stdint.h>
#include <math-svid-compat.h>
#include <shlib-compat.h>
#include <libm-alias-double.h>
#include "math_config.h"
#define T __log_data.tab
#define T2 __log_data.tab2
#define B __log_data.poly1
#define A __log_data.poly
#define Ln2hi __log_data.ln2hi
#define Ln2lo __log_data.ln2lo
#define N (1 << LOG_TABLE_BITS)
#define OFF 0x3fe6000000000000
/* Top 16 bits of a double. */
static inline uint32_t
top16 (double x)
{
return asuint64 (x) >> 48;
}
#ifndef SECTION
# define SECTION
#endif
double
SECTION
__log (double x)
{
/* double_t for better performance on targets with FLT_EVAL_METHOD==2. */
double_t w, z, r, r2, r3, y, invc, logc, kd, hi, lo;
uint64_t ix, iz, tmp;
uint32_t top;
int k, i;
ix = asuint64 (x);
top = top16 (x);
#define LO asuint64 (1.0 - 0x1p-4)
#define HI asuint64 (1.0 + 0x1.09p-4)
if (__glibc_unlikely (ix - LO < HI - LO))
{
/* Handle close to 1.0 inputs separately. */
/* Fix sign of zero with downward rounding when x==1. */
if (WANT_ROUNDING && __glibc_unlikely (ix == asuint64 (1.0)))
return 0;
r = x - 1.0;
r2 = r * r;
r3 = r * r2;
y = r3 * (B[1] + r * B[2] + r2 * B[3]
+ r3 * (B[4] + r * B[5] + r2 * B[6]
+ r3 * (B[7] + r * B[8] + r2 * B[9] + r3 * B[10])));
/* Worst-case error is around 0.507 ULP. */
w = r * 0x1p27;
double_t rhi = r + w - w;
double_t rlo = r - rhi;
w = rhi * rhi * B[0]; /* B[0] == -0.5. */
hi = r + w;
lo = r - hi + w;
lo += B[0] * rlo * (rhi + r);
y += lo;
y += hi;
return y;
}
if (__glibc_unlikely (top - 0x0010 >= 0x7ff0 - 0x0010))
{
/* x < 0x1p-1022 or inf or nan. */
if (ix * 2 == 0)
return __math_divzero (1);
if (ix == asuint64 (INFINITY)) /* log(inf) == inf. */
return x;
if ((top & 0x8000) || (top & 0x7ff0) == 0x7ff0)
return __math_invalid (x);
/* x is subnormal, normalize it. */
ix = asuint64 (x * 0x1p52);
ix -= 52ULL << 52;
}
/* x = 2^k z; where z is in range [OFF,2*OFF) and exact.
The range is split into N subintervals.
The ith subinterval contains z and c is near its center. */
tmp = ix - OFF;
i = (tmp >> (52 - LOG_TABLE_BITS)) % N;
k = (int64_t) tmp >> 52; /* arithmetic shift */
iz = ix - (tmp & 0xfffULL << 52);
invc = T[i].invc;
logc = T[i].logc;
z = asdouble (iz);
/* log(x) = log1p(z/c-1) + log(c) + k*Ln2. */
/* r ~= z/c - 1, |r| < 1/(2*N). */
#ifdef __FP_FAST_FMA
/* rounding error: 0x1p-55/N. */
r = __builtin_fma (z, invc, -1.0);
#else
/* rounding error: 0x1p-55/N + 0x1p-66. */
r = (z - T2[i].chi - T2[i].clo) * invc;
#endif
kd = (double_t) k;
/* hi + lo = r + log(c) + k*Ln2. */
w = kd * Ln2hi + logc;
hi = w + r;
lo = w - hi + r + kd * Ln2lo;
/* log(x) = lo + (log1p(r) - r) + hi. */
r2 = r * r; /* rounding error: 0x1p-54/N^2. */
/* Worst case error if |y| > 0x1p-4: 0.519 ULP (0.520 ULP without fma).
0.5 + 2.06/N + abs-poly-error*2^56 ULP (+ 0.001 ULP without fma). */
y = lo + r2 * A[0] + r * r2 * (A[1] + r * A[2] + r2 * (A[3] + r * A[4])) + hi;
return y;
}
#ifndef __log
strong_alias (__log, __ieee754_log)
strong_alias (__log, __log_finite)
# if LIBM_SVID_COMPAT
versioned_symbol (libm, __log, log, GLIBC_2_29);
libm_alias_double_other (__log, log)
# else
libm_alias_double (__log, log)
# endif
#endif
|
