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
267
268
269
270
271
272
273
274
275
276
277
278
279
280
|
/* logll.c
*
* Natural logarithm for 128-bit long double precision.
*
*
*
* SYNOPSIS:
*
* long double x, y, logl();
*
* y = logl( x );
*
*
*
* DESCRIPTION:
*
* Returns the base e (2.718...) logarithm of x.
*
* The argument is separated into its exponent and fractional
* parts. Use of a lookup table increases the speed of the routine.
* The program uses logarithms tabulated at intervals of 1/128 to
* cover the domain from approximately 0.7 to 1.4.
*
* On the interval [-1/128, +1/128] the logarithm of 1+x is approximated by
* log(1+x) = x - 0.5 x^2 + x^3 P(x) .
*
*
*
* ACCURACY:
*
* Relative error:
* arithmetic domain # trials peak rms
* IEEE 0.875, 1.125 100000 1.2e-34 4.1e-35
* IEEE 0.125, 8 100000 1.2e-34 4.1e-35
*
*
* WARNING:
*
* This program uses integer operations on bit fields of floating-point
* numbers. It does not work with data structures other than the
* structure assumed.
*
*/
/* Copyright 2001 by Stephen L. Moshier <moshier@na-net.ornl.gov>
This 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.
This 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 this library; if not, see
<http://www.gnu.org/licenses/>. */
#include "math_private.h"
/* log(1+x) = x - .5 x^2 + x^3 l(x)
-.0078125 <= x <= +.0078125
peak relative error 1.2e-37 */
static const long double
l3 = 3.333333333333333333333333333333336096926E-1L,
l4 = -2.499999999999999999999999999486853077002E-1L,
l5 = 1.999999999999999999999999998515277861905E-1L,
l6 = -1.666666666666666666666798448356171665678E-1L,
l7 = 1.428571428571428571428808945895490721564E-1L,
l8 = -1.249999999999999987884655626377588149000E-1L,
l9 = 1.111111111111111093947834982832456459186E-1L,
l10 = -1.000000000000532974938900317952530453248E-1L,
l11 = 9.090909090915566247008015301349979892689E-2L,
l12 = -8.333333211818065121250921925397567745734E-2L,
l13 = 7.692307559897661630807048686258659316091E-2L,
l14 = -7.144242754190814657241902218399056829264E-2L,
l15 = 6.668057591071739754844678883223432347481E-2L;
/* Lookup table of ln(t) - (t-1)
t = 0.5 + (k+26)/128)
k = 0, ..., 91 */
static const long double logtbl[92] = {
-5.5345593589352099112142921677820359632418E-2L,
-5.2108257402767124761784665198737642086148E-2L,
-4.8991686870576856279407775480686721935120E-2L,
-4.5993270766361228596215288742353061431071E-2L,
-4.3110481649613269682442058976885699556950E-2L,
-4.0340872319076331310838085093194799765520E-2L,
-3.7682072451780927439219005993827431503510E-2L,
-3.5131785416234343803903228503274262719586E-2L,
-3.2687785249045246292687241862699949178831E-2L,
-3.0347913785027239068190798397055267411813E-2L,
-2.8110077931525797884641940838507561326298E-2L,
-2.5972247078357715036426583294246819637618E-2L,
-2.3932450635346084858612873953407168217307E-2L,
-2.1988775689981395152022535153795155900240E-2L,
-2.0139364778244501615441044267387667496733E-2L,
-1.8382413762093794819267536615342902718324E-2L,
-1.6716169807550022358923589720001638093023E-2L,
-1.5138929457710992616226033183958974965355E-2L,
-1.3649036795397472900424896523305726435029E-2L,
-1.2244881690473465543308397998034325468152E-2L,
-1.0924898127200937840689817557742469105693E-2L,
-9.6875626072830301572839422532631079809328E-3L,
-8.5313926245226231463436209313499745894157E-3L,
-7.4549452072765973384933565912143044991706E-3L,
-6.4568155251217050991200599386801665681310E-3L,
-5.5356355563671005131126851708522185605193E-3L,
-4.6900728132525199028885749289712348829878E-3L,
-3.9188291218610470766469347968659624282519E-3L,
-3.2206394539524058873423550293617843896540E-3L,
-2.5942708080877805657374888909297113032132E-3L,
-2.0385211375711716729239156839929281289086E-3L,
-1.5522183228760777967376942769773768850872E-3L,
-1.1342191863606077520036253234446621373191E-3L,
-7.8340854719967065861624024730268350459991E-4L,
-4.9869831458030115699628274852562992756174E-4L,
-2.7902661731604211834685052867305795169688E-4L,
-1.2335696813916860754951146082826952093496E-4L,
-3.0677461025892873184042490943581654591817E-5L,
#define ZERO logtbl[38]
0.0000000000000000000000000000000000000000E0L,
-3.0359557945051052537099938863236321874198E-5L,
-1.2081346403474584914595395755316412213151E-4L,
-2.7044071846562177120083903771008342059094E-4L,
-4.7834133324631162897179240322783590830326E-4L,
-7.4363569786340080624467487620270965403695E-4L,
-1.0654639687057968333207323853366578860679E-3L,
-1.4429854811877171341298062134712230604279E-3L,
-1.8753781835651574193938679595797367137975E-3L,
-2.3618380914922506054347222273705859653658E-3L,
-2.9015787624124743013946600163375853631299E-3L,
-3.4938307889254087318399313316921940859043E-3L,
-4.1378413103128673800485306215154712148146E-3L,
-4.8328735414488877044289435125365629849599E-3L,
-5.5782063183564351739381962360253116934243E-3L,
-6.3731336597098858051938306767880719015261E-3L,
-7.2169643436165454612058905294782949315193E-3L,
-8.1090214990427641365934846191367315083867E-3L,
-9.0486422112807274112838713105168375482480E-3L,
-1.0035177140880864314674126398350812606841E-2L,
-1.1067990155502102718064936259435676477423E-2L,
-1.2146457974158024928196575103115488672416E-2L,
-1.3269969823361415906628825374158424754308E-2L,
-1.4437927104692837124388550722759686270765E-2L,
-1.5649743073340777659901053944852735064621E-2L,
-1.6904842527181702880599758489058031645317E-2L,
-1.8202661505988007336096407340750378994209E-2L,
-1.9542647000370545390701192438691126552961E-2L,
-2.0924256670080119637427928803038530924742E-2L,
-2.2346958571309108496179613803760727786257E-2L,
-2.3810230892650362330447187267648486279460E-2L,
-2.5313561699385640380910474255652501521033E-2L,
-2.6856448685790244233704909690165496625399E-2L,
-2.8438398935154170008519274953860128449036E-2L,
-3.0058928687233090922411781058956589863039E-2L,
-3.1717563112854831855692484086486099896614E-2L,
-3.3413836095418743219397234253475252001090E-2L,
-3.5147290019036555862676702093393332533702E-2L,
-3.6917475563073933027920505457688955423688E-2L,
-3.8723951502862058660874073462456610731178E-2L,
-4.0566284516358241168330505467000838017425E-2L,
-4.2444048996543693813649967076598766917965E-2L,
-4.4356826869355401653098777649745233339196E-2L,
-4.6304207416957323121106944474331029996141E-2L,
-4.8285787106164123613318093945035804818364E-2L,
-5.0301169421838218987124461766244507342648E-2L,
-5.2349964705088137924875459464622098310997E-2L,
-5.4431789996103111613753440311680967840214E-2L,
-5.6546268881465384189752786409400404404794E-2L,
-5.8693031345788023909329239565012647817664E-2L,
-6.0871713627532018185577188079210189048340E-2L,
-6.3081958078862169742820420185833800925568E-2L,
-6.5323413029406789694910800219643791556918E-2L,
-6.7595732653791419081537811574227049288168E-2L
};
/* ln(2) = ln2a + ln2b with extended precision. */
static const long double
ln2a = 6.93145751953125e-1L,
ln2b = 1.4286068203094172321214581765680755001344E-6L;
long double
__ieee754_logl(long double x)
{
long double z, y, w;
ieee854_long_double_shape_type u, t;
unsigned int m;
int k, e;
u.value = x;
m = u.parts32.w0;
/* Check for IEEE special cases. */
k = m & 0x7fffffff;
/* log(0) = -infinity. */
if ((k | u.parts32.w1 | u.parts32.w2 | u.parts32.w3) == 0)
{
return -0.5L / ZERO;
}
/* log ( x < 0 ) = NaN */
if (m & 0x80000000)
{
return (x - x) / ZERO;
}
/* log (infinity or NaN) */
if (k >= 0x7fff0000)
{
return x + x;
}
/* Extract exponent and reduce domain to 0.703125 <= u < 1.40625 */
e = (int) (m >> 16) - (int) 0x3ffe;
m &= 0xffff;
u.parts32.w0 = m | 0x3ffe0000;
m |= 0x10000;
/* Find lookup table index k from high order bits of the significand. */
if (m < 0x16800)
{
k = (m - 0xff00) >> 9;
/* t is the argument 0.5 + (k+26)/128
of the nearest item to u in the lookup table. */
t.parts32.w0 = 0x3fff0000 + (k << 9);
t.parts32.w1 = 0;
t.parts32.w2 = 0;
t.parts32.w3 = 0;
u.parts32.w0 += 0x10000;
e -= 1;
k += 64;
}
else
{
k = (m - 0xfe00) >> 10;
t.parts32.w0 = 0x3ffe0000 + (k << 10);
t.parts32.w1 = 0;
t.parts32.w2 = 0;
t.parts32.w3 = 0;
}
/* On this interval the table is not used due to cancellation error. */
if ((x <= 1.0078125L) && (x >= 0.9921875L))
{
z = x - 1.0L;
k = 64;
t.value = 1.0L;
e = 0;
}
else
{
/* log(u) = log( t u/t ) = log(t) + log(u/t)
log(t) is tabulated in the lookup table.
Express log(u/t) = log(1+z), where z = u/t - 1 = (u-t)/t.
cf. Cody & Waite. */
z = (u.value - t.value) / t.value;
}
/* Series expansion of log(1+z). */
w = z * z;
y = ((((((((((((l15 * z
+ l14) * z
+ l13) * z
+ l12) * z
+ l11) * z
+ l10) * z
+ l9) * z
+ l8) * z
+ l7) * z
+ l6) * z
+ l5) * z
+ l4) * z
+ l3) * z * w;
y -= 0.5 * w;
y += e * ln2b; /* Base 2 exponent offset times ln(2). */
y += z;
y += logtbl[k-26]; /* log(t) - (t-1) */
y += (t.value - 1.0L);
y += e * ln2a;
return y;
}
strong_alias (__ieee754_logl, __logl_finite)
|
