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
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
|
/* Double-precision vector log(x) function.
Copyright (C) 2019 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
<http://www.gnu.org/licenses/>. */
#include <math.h>
#include <stdint.h>
#include <math/math-svid-compat.h>
#include <shlib-compat.h>
#include <libm-alias-double.h>
#include "math_config_dbl.h"
#include <altivec.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
#define LO asuint64 (1.0 - 0x1p-4)
#define HI asuint64 (1.0 + 0x1.09p-4)
/* Top 16 bits of a double. */
static inline uint32_t
top16 (double x)
{
return asuint64 (x) >> 48;
}
vector double
_ZGVbN2v_log (vector double x)
{
vector double w, z, r, r2, r3, y, invc, logc, kd, hi, lo;
vector double rhi, rlo;
vector unsigned long long ix, iz, tmp, top;
vector signed long long k, i;
ix[0] = asuint64 (x[0]);
ix[1] = asuint64 (x[1]);
top[0] = top16 (x[0]);
top[1] = top16 (x[1]);
if (__glibc_unlikely ((ix[0] - LO < HI - LO) && (ix[1] - 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[0] == asuint64 (1.0))
&& (ix[1] == asuint64 (1.0))))
{
y[0] = 0.0;
y[1] = 0.0;
return y;
}
else if (WANT_ROUNDING && __glibc_unlikely (ix[0] == asuint64 (1.0)))
{
y[0] = 0.0;
y[1] = log (x[1]);
return y;
}
else if (WANT_ROUNDING && __glibc_unlikely (ix[1] == asuint64 (1.0)))
{
y[0] = log (x[0]);
y[1] = 0.0;
return y;
}
r = x - (vector double) {1.0, 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 * (vector double) {0x1p27, 0x1p27};
rhi = r + w - w;
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;
}
else if (__glibc_unlikely ((ix[0] - 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[0] == asuint64 (1.0)))
{
y[0] = 0.0;
y[1] = log (x[1]);
return y;
}
y[0] = log (x[0]);
y[1] = log (x[1]);
return y;
}
else if (__glibc_unlikely ((ix[1] - 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[1] == asuint64 (1.0)))
{
y[0] = log (x[0]);
y[1] = 0.0;
return y;
}
y[0] = log (x[0]);
y[1] = log (x[1]);
return y;
}
if (__glibc_unlikely ((top[0] - 0x0010 >= 0x7ff0 - 0x0010)
&& (top[1] - 0x0010 >= 0x7ff0 - 0x0010)))
{
/* x < 0x1p-1022 or inf or nan. */
if ((ix[0] * 2 == 0) || (ix[1] * 2 == 0))
{
y[0] = log (x[0]);
y[1] = log (x[1]);
return y;
}
/* log(inf) == inf. */
if ((ix[0] == asuint64 (INFINITY)) && (ix[1] == asuint64 (INFINITY)))
{
y[0] = x[0];
y[1] = x[1];
return y;
}
else if (ix[0] == asuint64 (INFINITY))
{
y[0] = x[0];
y[1] = log (x[1]);
return y;
}
else if (ix[1] == asuint64 (INFINITY))
{
y[0] = log (x[0]);
y[1] = x[1];
return y;
}
if (((top[0] & 0x8000) || (top[0] & 0x7ff0) == 0x7ff0)
|| ((top[1] & 0x8000) || (top[1] & 0x7ff0) == 0x7ff0))
{
y[0] = log (x[0]);
y[1] = log (x[1]);
return y;
}
/* x is subnormal, normalize it. */
ix[0] = asuint64 (x[0] * 0x1p52);
ix[0] -= 52ULL << 52;
ix[1] = asuint64 (x[1] * 0x1p52);
ix[1] -= 52ULL << 52;
}
else if (__glibc_unlikely (top[0] - 0x0010 >= 0x7ff0 - 0x0010))
{
/* x < 0x1p-1022 or inf or nan. */
if (ix[0] * 2 == 0)
{
y[0] = log (x[0]);
y[1] = log (x[1]);
return y;
}
/* log(inf) == inf. */
if (ix[0] == asuint64 (INFINITY))
{
y[0] = x[0];
y[1] = log (x[1]);
return y;
}
if (((top[0] & 0x8000) || (top[0] & 0x7ff0) == 0x7ff0))
{
y[0] = log (x[0]);
y[1] = log (x[1]);
return y;
}
/* x is subnormal, normalize it. */
ix[0] = asuint64 (x[0] * 0x1p52);
ix[0] -= 52ULL << 52;
}
else if (__glibc_unlikely (top[1] - 0x0010 >= 0x7ff0 - 0x0010))
{
/* x < 0x1p-1022 or inf or nan. */
if (ix[1] * 2 == 0)
{
y[0] = log (x[0]);
y[1] = log (x[1]);
return y;
}
/* log(inf) == inf. */
if (ix[1] == asuint64 (INFINITY))
{
y[0] = log (x[0]);
y[1] = x[1];
return y;
}
if (((top[1] & 0x8000) || (top[1] & 0x7ff0) == 0x7ff0))
{
y[0] = log (x[0]);
y[1] = log (x[1]);
return y;
}
/* x is subnormal, normalize it. */
ix[1] = asuint64 (x[1] * 0x1p52);
ix[1] -= 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 - (vector unsigned long long) {OFF, OFF};
i = (vector signed long)
(tmp >> vec_splats ((unsigned long) (52 - LOG_TABLE_BITS)))
% vec_splats ((unsigned long) N);
k = ((vector signed long long) tmp) >> vec_splats (52ULL);
iz = ix - (tmp & vec_splats (0xfffULL) << vec_splats (52ULL));
invc[0] = T[i[0]].invc;
logc[0] = T[i[0]].logc;
invc[1] = T[i[1]].invc;
logc[1] = T[i[1]].logc;
z[0] = asdouble (iz[0]);
z[1] = asdouble (iz[1]);
/* log(x) = log1p(z/c-1) + log(c) + k*Ln2. */
/* r ~= z/c - 1, |r| < 1/(2*N). */
/* rounding error: 0x1p-55/N + 0x1p-66. */
r = z * invc - (vector double) {1.0, 1.0};
kd = vec_double (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;
}
|
