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1/*
2 * Copyright (c) 1992, 1993
3 * The Regents of the University of California. All rights reserved.
4 *
5 * Redistribution and use in source and binary forms, with or without
6 * modification, are permitted provided that the following conditions
7 * are met:
8 * 1. Redistributions of source code must retain the above copyright
9 * notice, this list of conditions and the following disclaimer.
10 * 2. Redistributions in binary form must reproduce the above copyright
11 * notice, this list of conditions and the following disclaimer in the
12 * documentation and/or other materials provided with the distribution.
13 * 3. All advertising materials mentioning features or use of this software
14 * must display the following acknowledgement:
15 * This product includes software developed by the University of
16 * California, Berkeley and its contributors.
17 * 4. Neither the name of the University nor the names of its contributors
18 * may be used to endorse or promote products derived from this software
19 * without specific prior written permission.
20 *
21 * THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND
22 * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
23 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
24 * ARE DISCLAIMED. IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE
25 * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
26 * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
27 * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
28 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
29 * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
30 * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
31 * SUCH DAMAGE.
32 */
33
34#ifndef lint
35static char sccsid[] = "@(#)log.c 8.2 (Berkeley) 11/30/93";
36#endif /* not lint */
37#include <sys/cdefs.h>
38__FBSDID("$FreeBSD: head/lib/msun/bsdsrc/b_log.c 138924 2004-12-16 20:40:37Z das $");
38__FBSDID("$FreeBSD: head/lib/msun/bsdsrc/b_log.c 150318 2005-09-19 11:28:19Z bde $");
39
40#include <math.h>
41#include <errno.h>
42
43#include "mathimpl.h"
44
45/* Table-driven natural logarithm.
46 *
47 * This code was derived, with minor modifications, from:
48 * Peter Tang, "Table-Driven Implementation of the
49 * Logarithm in IEEE Floating-Point arithmetic." ACM Trans.
50 * Math Software, vol 16. no 4, pp 378-400, Dec 1990).
51 *
52 * Calculates log(2^m*F*(1+f/F)), |f/j| <= 1/256,
53 * where F = j/128 for j an integer in [0, 128].
54 *
55 * log(2^m) = log2_hi*m + log2_tail*m
56 * since m is an integer, the dominant term is exact.
57 * m has at most 10 digits (for subnormal numbers),
58 * and log2_hi has 11 trailing zero bits.
59 *
60 * log(F) = logF_hi[j] + logF_lo[j] is in tabular form in log_table.h
61 * logF_hi[] + 512 is exact.
62 *
63 * log(1+f/F) = 2*f/(2*F + f) + 1/12 * (2*f/(2*F + f))**3 + ...
64 * the leading term is calculated to extra precision in two
65 * parts, the larger of which adds exactly to the dominant
66 * m and F terms.
67 * There are two cases:
68 * 1. when m, j are non-zero (m | j), use absolute
69 * precision for the leading term.
70 * 2. when m = j = 0, |1-x| < 1/256, and log(x) ~= (x-1).
71 * In this case, use a relative precision of 24 bits.
72 * (This is done differently in the original paper)
73 *
74 * Special cases:
75 * 0 return signalling -Inf
76 * neg return signalling NaN
77 * +Inf return +Inf
78*/
79
80#define endian (((*(int *) &one)) ? 1 : 0)
81#define TRUNC(x) *(((int *) &x) + endian) &= 0xf8000000
82
80#define N 128
81
82/* Table of log(Fj) = logF_head[j] + logF_tail[j], for Fj = 1+j/128.
83 * Used for generation of extend precision logarithms.
84 * The constant 35184372088832 is 2^45, so the divide is exact.
85 * It ensures correct reading of logF_head, even for inaccurate
86 * decimal-to-binary conversion routines. (Everybody gets the
87 * right answer for integers less than 2^53.)
88 * Values for log(F) were generated using error < 10^-57 absolute
89 * with the bc -l package.
90*/
91static double A1 = .08333333333333178827;
92static double A2 = .01250000000377174923;
93static double A3 = .002232139987919447809;
94static double A4 = .0004348877777076145742;
95
96static double logF_head[N+1] = {
97 0.,
98 .007782140442060381246,
99 .015504186535963526694,
100 .023167059281547608406,
101 .030771658666765233647,
102 .038318864302141264488,
103 .045809536031242714670,
104 .053244514518837604555,
105 .060624621816486978786,
106 .067950661908525944454,
107 .075223421237524235039,
108 .082443669210988446138,
109 .089612158689760690322,
110 .096729626458454731618,
111 .103796793681567578460,
112 .110814366340264314203,
113 .117783035656430001836,
114 .124703478501032805070,
115 .131576357788617315236,
116 .138402322859292326029,
117 .145182009844575077295,
118 .151916042025732167530,
119 .158605030176659056451,
120 .165249572895390883786,
121 .171850256926518341060,
122 .178407657472689606947,
123 .184922338493834104156,
124 .191394852999565046047,
125 .197825743329758552135,
126 .204215541428766300668,
127 .210564769107350002741,
128 .216873938300523150246,
129 .223143551314024080056,
130 .229374101064877322642,
131 .235566071312860003672,
132 .241719936886966024758,
133 .247836163904594286577,
134 .253915209980732470285,
135 .259957524436686071567,
136 .265963548496984003577,
137 .271933715484010463114,
138 .277868451003087102435,
139 .283768173130738432519,
140 .289633292582948342896,
141 .295464212893421063199,
142 .301261330578199704177,
143 .307025035294827830512,
144 .312755710004239517729,
145 .318453731118097493890,
146 .324119468654316733591,
147 .329753286372579168528,
148 .335355541920762334484,
149 .340926586970454081892,
150 .346466767346100823488,
151 .351976423156884266063,
152 .357455888922231679316,
153 .362905493689140712376,
154 .368325561158599157352,
155 .373716409793814818840,
156 .379078352934811846353,
157 .384411698910298582632,
158 .389716751140440464951,
159 .394993808240542421117,
160 .400243164127459749579,
161 .405465108107819105498,
162 .410659924985338875558,
163 .415827895143593195825,
164 .420969294644237379543,
165 .426084395310681429691,
166 .431173464818130014464,
167 .436236766774527495726,
168 .441274560805140936281,
169 .446287102628048160113,
170 .451274644139630254358,
171 .456237433481874177232,
172 .461175715122408291790,
173 .466089729924533457960,
174 .470979715219073113985,
175 .475845904869856894947,
176 .480688529345570714212,
177 .485507815781602403149,
178 .490303988045525329653,
179 .495077266798034543171,
180 .499827869556611403822,
181 .504556010751912253908,
182 .509261901790523552335,
183 .513945751101346104405,
184 .518607764208354637958,
185 .523248143765158602036,
186 .527867089620485785417,
187 .532464798869114019908,
188 .537041465897345915436,
189 .541597282432121573947,
190 .546132437597407260909,
191 .550647117952394182793,
192 .555141507540611200965,
193 .559615787935399566777,
194 .564070138285387656651,
195 .568504735352689749561,
196 .572919753562018740922,
197 .577315365035246941260,
198 .581691739635061821900,
199 .586049045003164792433,
200 .590387446602107957005,
201 .594707107746216934174,
202 .599008189645246602594,
203 .603290851438941899687,
204 .607555250224322662688,
205 .611801541106615331955,
206 .616029877215623855590,
207 .620240409751204424537,
208 .624433288012369303032,
209 .628608659422752680256,
210 .632766669570628437213,
211 .636907462236194987781,
212 .641031179420679109171,
213 .645137961373620782978,
214 .649227946625615004450,
215 .653301272011958644725,
216 .657358072709030238911,
217 .661398482245203922502,
218 .665422632544505177065,
219 .669430653942981734871,
220 .673422675212350441142,
221 .677398823590920073911,
222 .681359224807238206267,
223 .685304003098281100392,
224 .689233281238557538017,
225 .693147180560117703862
226};
227
228static double logF_tail[N+1] = {
229 0.,
230 -.00000000000000543229938420049,
231 .00000000000000172745674997061,
232 -.00000000000001323017818229233,
233 -.00000000000001154527628289872,
234 -.00000000000000466529469958300,
235 .00000000000005148849572685810,
236 -.00000000000002532168943117445,
237 -.00000000000005213620639136504,
238 -.00000000000001819506003016881,
239 .00000000000006329065958724544,
240 .00000000000008614512936087814,
241 -.00000000000007355770219435028,
242 .00000000000009638067658552277,
243 .00000000000007598636597194141,
244 .00000000000002579999128306990,
245 -.00000000000004654729747598444,
246 -.00000000000007556920687451336,
247 .00000000000010195735223708472,
248 -.00000000000017319034406422306,
249 -.00000000000007718001336828098,
250 .00000000000010980754099855238,
251 -.00000000000002047235780046195,
252 -.00000000000008372091099235912,
253 .00000000000014088127937111135,
254 .00000000000012869017157588257,
255 .00000000000017788850778198106,
256 .00000000000006440856150696891,
257 .00000000000016132822667240822,
258 -.00000000000007540916511956188,
259 -.00000000000000036507188831790,
260 .00000000000009120937249914984,
261 .00000000000018567570959796010,
262 -.00000000000003149265065191483,
263 -.00000000000009309459495196889,
264 .00000000000017914338601329117,
265 -.00000000000001302979717330866,
266 .00000000000023097385217586939,
267 .00000000000023999540484211737,
268 .00000000000015393776174455408,
269 -.00000000000036870428315837678,
270 .00000000000036920375082080089,
271 -.00000000000009383417223663699,
272 .00000000000009433398189512690,
273 .00000000000041481318704258568,
274 -.00000000000003792316480209314,
275 .00000000000008403156304792424,
276 -.00000000000034262934348285429,
277 .00000000000043712191957429145,
278 -.00000000000010475750058776541,
279 -.00000000000011118671389559323,
280 .00000000000037549577257259853,
281 .00000000000013912841212197565,
282 .00000000000010775743037572640,
283 .00000000000029391859187648000,
284 -.00000000000042790509060060774,
285 .00000000000022774076114039555,
286 .00000000000010849569622967912,
287 -.00000000000023073801945705758,
288 .00000000000015761203773969435,
289 .00000000000003345710269544082,
290 -.00000000000041525158063436123,
291 .00000000000032655698896907146,
292 -.00000000000044704265010452446,
293 .00000000000034527647952039772,
294 -.00000000000007048962392109746,
295 .00000000000011776978751369214,
296 -.00000000000010774341461609578,
297 .00000000000021863343293215910,
298 .00000000000024132639491333131,
299 .00000000000039057462209830700,
300 -.00000000000026570679203560751,
301 .00000000000037135141919592021,
302 -.00000000000017166921336082431,
303 -.00000000000028658285157914353,
304 -.00000000000023812542263446809,
305 .00000000000006576659768580062,
306 -.00000000000028210143846181267,
307 .00000000000010701931762114254,
308 .00000000000018119346366441110,
309 .00000000000009840465278232627,
310 -.00000000000033149150282752542,
311 -.00000000000018302857356041668,
312 -.00000000000016207400156744949,
313 .00000000000048303314949553201,
314 -.00000000000071560553172382115,
315 .00000000000088821239518571855,
316 -.00000000000030900580513238244,
317 -.00000000000061076551972851496,
318 .00000000000035659969663347830,
319 .00000000000035782396591276383,
320 -.00000000000046226087001544578,
321 .00000000000062279762917225156,
322 .00000000000072838947272065741,
323 .00000000000026809646615211673,
324 -.00000000000010960825046059278,
325 .00000000000002311949383800537,
326 -.00000000000058469058005299247,
327 -.00000000000002103748251144494,
328 -.00000000000023323182945587408,
329 -.00000000000042333694288141916,
330 -.00000000000043933937969737844,
331 .00000000000041341647073835565,
332 .00000000000006841763641591466,
333 .00000000000047585534004430641,
334 .00000000000083679678674757695,
335 -.00000000000085763734646658640,
336 .00000000000021913281229340092,
337 -.00000000000062242842536431148,
338 -.00000000000010983594325438430,
339 .00000000000065310431377633651,
340 -.00000000000047580199021710769,
341 -.00000000000037854251265457040,
342 .00000000000040939233218678664,
343 .00000000000087424383914858291,
344 .00000000000025218188456842882,
345 -.00000000000003608131360422557,
346 -.00000000000050518555924280902,
347 .00000000000078699403323355317,
348 -.00000000000067020876961949060,
349 .00000000000016108575753932458,
350 .00000000000058527188436251509,
351 -.00000000000035246757297904791,
352 -.00000000000018372084495629058,
353 .00000000000088606689813494916,
354 .00000000000066486268071468700,
355 .00000000000063831615170646519,
356 .00000000000025144230728376072,
357 -.00000000000017239444525614834
358};
359
360#if 0
361double
362#ifdef _ANSI_SOURCE
363log(double x)
364#else
365log(x) double x;
366#endif
367{
368 int m, j;
369 double F, f, g, q, u, u2, v, zero = 0.0, one = 1.0;
370 volatile double u1;
371
372 /* Catch special cases */
373 if (x <= 0)
374 if (x == zero) /* log(0) = -Inf */
375 return (-one/zero);
376 else /* log(neg) = NaN */
377 return (zero/zero);
378 else if (!finite(x))
379 return (x+x); /* x = NaN, Inf */
380
381 /* Argument reduction: 1 <= g < 2; x/2^m = g; */
382 /* y = F*(1 + f/F) for |f| <= 2^-8 */
383
384 m = logb(x);
385 g = ldexp(x, -m);
386 if (m == -1022) {
387 j = logb(g), m += j;
388 g = ldexp(g, -j);
389 }
390 j = N*(g-1) + .5;
391 F = (1.0/N) * j + 1; /* F*128 is an integer in [128, 512] */
392 f = g - F;
393
394 /* Approximate expansion for log(1+f/F) ~= u + q */
395 g = 1/(2*F+f);
396 u = 2*f*g;
397 v = u*u;
398 q = u*v*(A1 + v*(A2 + v*(A3 + v*A4)));
399
400 /* case 1: u1 = u rounded to 2^-43 absolute. Since u < 2^-8,
401 * u1 has at most 35 bits, and F*u1 is exact, as F has < 8 bits.
402 * It also adds exactly to |m*log2_hi + log_F_head[j] | < 750
403 */
404 if (m | j)
405 u1 = u + 513, u1 -= 513;
406
407 /* case 2: |1-x| < 1/256. The m- and j- dependent terms are zero;
408 * u1 = u to 24 bits.
409 */
410 else
411 u1 = u, TRUNC(u1);
412 u2 = (2.0*(f - F*u1) - u1*f) * g;
413 /* u1 + u2 = 2f/(2F+f) to extra precision. */
414
415 /* log(x) = log(2^m*F*(1+f/F)) = */
416 /* (m*log2_hi+logF_head[j]+u1) + (m*log2_lo+logF_tail[j]+q); */
417 /* (exact) + (tiny) */
418
419 u1 += m*logF_head[N] + logF_head[j]; /* exact */
420 u2 = (u2 + logF_tail[j]) + q; /* tiny */
421 u2 += logF_tail[N]*m;
422 return (u1 + u2);
423}
424#endif
425
426/*
427 * Extra precision variant, returning struct {double a, b;};
428 * log(x) = a+b to 63 bits, with a rounded to 26 bits.
429 */
430struct Double
431#ifdef _ANSI_SOURCE
432__log__D(double x)
433#else
434__log__D(x) double x;
435#endif
436{
437 int m, j;
441 double F, f, g, q, u, v, u2, one = 1.0;
438 double F, f, g, q, u, v, u2;
439 volatile double u1;
440 struct Double r;
441
442 /* Argument reduction: 1 <= g < 2; x/2^m = g; */
443 /* y = F*(1 + f/F) for |f| <= 2^-8 */
444
445 m = logb(x);
446 g = ldexp(x, -m);
447 if (m == -1022) {
448 j = logb(g), m += j;
449 g = ldexp(g, -j);
450 }
451 j = N*(g-1) + .5;
452 F = (1.0/N) * j + 1;
453 f = g - F;
454
455 g = 1/(2*F+f);
456 u = 2*f*g;
457 v = u*u;
458 q = u*v*(A1 + v*(A2 + v*(A3 + v*A4)));
459 if (m | j)
460 u1 = u + 513, u1 -= 513;
461 else
462 u1 = u, TRUNC(u1);
463 u2 = (2.0*(f - F*u1) - u1*f) * g;
464
465 u1 += m*logF_head[N] + logF_head[j];
466
467 u2 += logF_tail[j]; u2 += q;
468 u2 += logF_tail[N]*m;
469 r.a = u1 + u2; /* Only difference is here */
470 TRUNC(r.a);
471 r.b = (u1 - r.a) + u2;
472 return (r);
473}