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
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
|
/* Double-precision log(x) function.
Copyright (c) 2018 Arm Ltd. All rights reserved.
SPDX-License-Identifier: BSD-3-Clause
Redistribution and use in source and binary forms, with or without
modification, are permitted provided that the following conditions
are met:
1. Redistributions of source code must retain the above copyright
notice, this list of conditions and the following disclaimer.
2. Redistributions in binary form must reproduce the above copyright
notice, this list of conditions and the following disclaimer in the
documentation and/or other materials provided with the distribution.
3. The name of the company may not be used to endorse or promote
products derived from this software without specific prior written
permission.
THIS SOFTWARE IS PROVIDED BY Arm LTD ``AS IS AND ANY EXPRESS OR IMPLIED
WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF
MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED.
IN NO EVENT SHALL Arm LTD BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED
TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR
PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF
LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING
NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. */
#include "fdlibm.h"
#if !__OBSOLETE_MATH
#include <math.h>
#include <stdint.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;
}
double
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);
#if LOG_POLY1_ORDER == 10 || LOG_POLY1_ORDER == 11
# define LO asuint64 (1.0 - 0x1p-5)
# define HI asuint64 (1.0 + 0x1.1p-5)
#elif LOG_POLY1_ORDER == 12
# define LO asuint64 (1.0 - 0x1p-4)
# define HI asuint64 (1.0 + 0x1.09p-4)
#endif
if (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 && unlikely (ix == asuint64 (1.0)))
return 0;
r = x - 1.0;
r2 = r * r;
r3 = r * r2;
#if LOG_POLY1_ORDER == 10
/* Worst-case error is around 0.516 ULP. */
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])));
w = B[0] * r2; /* B[0] == -0.5. */
hi = r + w;
y += r - hi + w;
y += hi;
#elif LOG_POLY1_ORDER == 11
/* Worst-case error is around 0.516 ULP. */
y = r3 * (B[1] + r * B[2]
+ r2 * (B[3] + r * B[4] + r2 * B[5]
+ r3 * (B[6] + r * B[7] + r2 * B[8] + r3 * B[9])));
w = B[0] * r2; /* B[0] == -0.5. */
hi = r + w;
y += r - hi + w;
y += hi;
#elif LOG_POLY1_ORDER == 12
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])));
# if N <= 64
/* Worst-case error is around 0.532 ULP. */
w = B[0] * r2; /* B[0] == -0.5. */
hi = r + w;
y += r - hi + w;
y += hi;
# else
/* 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;
# endif
#endif
return y;
}
if (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). */
#if HAVE_FAST_FMA
/* rounding error: 0x1p-55/N. */
r = 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-5:
0.5 + 4.13/N + abs-poly-error*2^57 ULP (+ 0.002 ULP without fma)
Worst case error if |y| > 0x1p-4:
0.5 + 2.06/N + abs-poly-error*2^56 ULP (+ 0.001 ULP without fma). */
#if LOG_POLY_ORDER == 6
y = lo + r2 * A[0] + r * r2 * (A[1] + r * A[2] + r2 * (A[3] + r * A[4])) + hi;
#elif LOG_POLY_ORDER == 7
y = lo
+ r2 * (A[0] + r * A[1] + r2 * (A[2] + r * A[3])
+ r2 * r2 * (A[4] + r * A[5]))
+ hi;
#endif
return y;
}
#endif
|
