1 /* $NetBSD: n_log.c,v 1.8 2014/10/10 20:58:09 martin Exp $ */
2 /*
3 * Copyright (c) 1992, 1993
4 * The Regents of the University of California. All rights reserved.
5 *
6 * Redistribution and use in source and binary forms, with or without
7 * modification, are permitted provided that the following conditions
8 * are met:
9 * 1. Redistributions of source code must retain the above copyright
10 * notice, this list of conditions and the following disclaimer.
11 * 2. Redistributions in binary form must reproduce the above copyright
12 * notice, this list of conditions and the following disclaimer in the
13 * documentation and/or other materials provided with the distribution.
14 * 3. Neither the name of the University nor the names of its contributors
15 * may be used to endorse or promote products derived from this software
16 * without specific prior written permission.
17 *
18 * THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND
19 * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
20 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
21 * ARE DISCLAIMED. IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE
22 * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
23 * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
24 * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
25 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
26 * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
27 * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
28 * SUCH DAMAGE.
29 */
30
31 #ifndef lint
32 #if 0
33 static char sccsid[] = "@(#)log.c 8.2 (Berkeley) 11/30/93";
34 #endif
35 #endif /* not lint */
36
37 #include "../src/namespace.h"
38
39 #include <math.h>
40 #include <errno.h>
41
42 #include "mathimpl.h"
43
44 #ifdef __weak_alias
45 __weak_alias(log, _log);
46 __weak_alias(_logl, _log);
47 __weak_alias(logf, _logf);
48 #endif
49
50 /* Table-driven natural logarithm.
51 *
52 * This code was derived, with minor modifications, from:
53 * Peter Tang, "Table-Driven Implementation of the
54 * Logarithm in IEEE Floating-Point arithmetic." ACM Trans.
55 * Math Software, vol 16. no 4, pp 378-400, Dec 1990).
56 *
57 * Calculates log(2^m*F*(1+f/F)), |f/j| <= 1/256,
58 * where F = j/128 for j an integer in [0, 128].
59 *
60 * log(2^m) = log2_hi*m + log2_tail*m
61 * since m is an integer, the dominant term is exact.
62 * m has at most 10 digits (for subnormal numbers),
63 * and log2_hi has 11 trailing zero bits.
64 *
65 * log(F) = logF_hi[j] + logF_lo[j] is in tabular form in log_table.h
66 * logF_hi[] + 512 is exact.
67 *
68 * log(1+f/F) = 2*f/(2*F + f) + 1/12 * (2*f/(2*F + f))**3 + ...
69 * the leading term is calculated to extra precision in two
70 * parts, the larger of which adds exactly to the dominant
71 * m and F terms.
72 * There are two cases:
73 * 1. when m, j are non-zero (m | j), use absolute
74 * precision for the leading term.
75 * 2. when m = j = 0, |1-x| < 1/256, and log(x) ~= (x-1).
76 * In this case, use a relative precision of 24 bits.
77 * (This is done differently in the original paper)
78 *
79 * Special cases:
80 * 0 return signalling -Inf
81 * neg return signalling NaN
82 * +Inf return +Inf
83 */
84
85 #if defined(__vax__) || defined(tahoe)
86 #define _IEEE 0
87 #define TRUNC(x) x = (double) (float) (x)
88 #else
89 #define _IEEE 1
90 #define endian (((*(int *) &one)) ? 1 : 0)
91 #define TRUNC(x) *(((int *) &x) + endian) &= 0xf8000000
92 #define infnan(x) 0.0
93 #endif
94
95 #define N 128
96
97 /* Table of log(Fj) = logF_head[j] + logF_tail[j], for Fj = 1+j/128.
98 * Used for generation of extend precision logarithms.
99 * The constant 35184372088832 is 2^45, so the divide is exact.
100 * It ensures correct reading of logF_head, even for inaccurate
101 * decimal-to-binary conversion routines. (Everybody gets the
102 * right answer for integers less than 2^53.)
103 * Values for log(F) were generated using error < 10^-57 absolute
104 * with the bc -l package.
105 */
106 static const double A1 = .08333333333333178827;
107 static const double A2 = .01250000000377174923;
108 static const double A3 = .002232139987919447809;
109 static const double A4 = .0004348877777076145742;
110
111 static const double logF_head[N+1] = {
112 0.,
113 .007782140442060381246,
114 .015504186535963526694,
115 .023167059281547608406,
116 .030771658666765233647,
117 .038318864302141264488,
118 .045809536031242714670,
119 .053244514518837604555,
120 .060624621816486978786,
121 .067950661908525944454,
122 .075223421237524235039,
123 .082443669210988446138,
124 .089612158689760690322,
125 .096729626458454731618,
126 .103796793681567578460,
127 .110814366340264314203,
128 .117783035656430001836,
129 .124703478501032805070,
130 .131576357788617315236,
131 .138402322859292326029,
132 .145182009844575077295,
133 .151916042025732167530,
134 .158605030176659056451,
135 .165249572895390883786,
136 .171850256926518341060,
137 .178407657472689606947,
138 .184922338493834104156,
139 .191394852999565046047,
140 .197825743329758552135,
141 .204215541428766300668,
142 .210564769107350002741,
143 .216873938300523150246,
144 .223143551314024080056,
145 .229374101064877322642,
146 .235566071312860003672,
147 .241719936886966024758,
148 .247836163904594286577,
149 .253915209980732470285,
150 .259957524436686071567,
151 .265963548496984003577,
152 .271933715484010463114,
153 .277868451003087102435,
154 .283768173130738432519,
155 .289633292582948342896,
156 .295464212893421063199,
157 .301261330578199704177,
158 .307025035294827830512,
159 .312755710004239517729,
160 .318453731118097493890,
161 .324119468654316733591,
162 .329753286372579168528,
163 .335355541920762334484,
164 .340926586970454081892,
165 .346466767346100823488,
166 .351976423156884266063,
167 .357455888922231679316,
168 .362905493689140712376,
169 .368325561158599157352,
170 .373716409793814818840,
171 .379078352934811846353,
172 .384411698910298582632,
173 .389716751140440464951,
174 .394993808240542421117,
175 .400243164127459749579,
176 .405465108107819105498,
177 .410659924985338875558,
178 .415827895143593195825,
179 .420969294644237379543,
180 .426084395310681429691,
181 .431173464818130014464,
182 .436236766774527495726,
183 .441274560805140936281,
184 .446287102628048160113,
185 .451274644139630254358,
186 .456237433481874177232,
187 .461175715122408291790,
188 .466089729924533457960,
189 .470979715219073113985,
190 .475845904869856894947,
191 .480688529345570714212,
192 .485507815781602403149,
193 .490303988045525329653,
194 .495077266798034543171,
195 .499827869556611403822,
196 .504556010751912253908,
197 .509261901790523552335,
198 .513945751101346104405,
199 .518607764208354637958,
200 .523248143765158602036,
201 .527867089620485785417,
202 .532464798869114019908,
203 .537041465897345915436,
204 .541597282432121573947,
205 .546132437597407260909,
206 .550647117952394182793,
207 .555141507540611200965,
208 .559615787935399566777,
209 .564070138285387656651,
210 .568504735352689749561,
211 .572919753562018740922,
212 .577315365035246941260,
213 .581691739635061821900,
214 .586049045003164792433,
215 .590387446602107957005,
216 .594707107746216934174,
217 .599008189645246602594,
218 .603290851438941899687,
219 .607555250224322662688,
220 .611801541106615331955,
221 .616029877215623855590,
222 .620240409751204424537,
223 .624433288012369303032,
224 .628608659422752680256,
225 .632766669570628437213,
226 .636907462236194987781,
227 .641031179420679109171,
228 .645137961373620782978,
229 .649227946625615004450,
230 .653301272011958644725,
231 .657358072709030238911,
232 .661398482245203922502,
233 .665422632544505177065,
234 .669430653942981734871,
235 .673422675212350441142,
236 .677398823590920073911,
237 .681359224807238206267,
238 .685304003098281100392,
239 .689233281238557538017,
240 .693147180560117703862
241 };
242
243 static const double logF_tail[N+1] = {
244 0.,
245 -.00000000000000543229938420049,
246 .00000000000000172745674997061,
247 -.00000000000001323017818229233,
248 -.00000000000001154527628289872,
249 -.00000000000000466529469958300,
250 .00000000000005148849572685810,
251 -.00000000000002532168943117445,
252 -.00000000000005213620639136504,
253 -.00000000000001819506003016881,
254 .00000000000006329065958724544,
255 .00000000000008614512936087814,
256 -.00000000000007355770219435028,
257 .00000000000009638067658552277,
258 .00000000000007598636597194141,
259 .00000000000002579999128306990,
260 -.00000000000004654729747598444,
261 -.00000000000007556920687451336,
262 .00000000000010195735223708472,
263 -.00000000000017319034406422306,
264 -.00000000000007718001336828098,
265 .00000000000010980754099855238,
266 -.00000000000002047235780046195,
267 -.00000000000008372091099235912,
268 .00000000000014088127937111135,
269 .00000000000012869017157588257,
270 .00000000000017788850778198106,
271 .00000000000006440856150696891,
272 .00000000000016132822667240822,
273 -.00000000000007540916511956188,
274 -.00000000000000036507188831790,
275 .00000000000009120937249914984,
276 .00000000000018567570959796010,
277 -.00000000000003149265065191483,
278 -.00000000000009309459495196889,
279 .00000000000017914338601329117,
280 -.00000000000001302979717330866,
281 .00000000000023097385217586939,
282 .00000000000023999540484211737,
283 .00000000000015393776174455408,
284 -.00000000000036870428315837678,
285 .00000000000036920375082080089,
286 -.00000000000009383417223663699,
287 .00000000000009433398189512690,
288 .00000000000041481318704258568,
289 -.00000000000003792316480209314,
290 .00000000000008403156304792424,
291 -.00000000000034262934348285429,
292 .00000000000043712191957429145,
293 -.00000000000010475750058776541,
294 -.00000000000011118671389559323,
295 .00000000000037549577257259853,
296 .00000000000013912841212197565,
297 .00000000000010775743037572640,
298 .00000000000029391859187648000,
299 -.00000000000042790509060060774,
300 .00000000000022774076114039555,
301 .00000000000010849569622967912,
302 -.00000000000023073801945705758,
303 .00000000000015761203773969435,
304 .00000000000003345710269544082,
305 -.00000000000041525158063436123,
306 .00000000000032655698896907146,
307 -.00000000000044704265010452446,
308 .00000000000034527647952039772,
309 -.00000000000007048962392109746,
310 .00000000000011776978751369214,
311 -.00000000000010774341461609578,
312 .00000000000021863343293215910,
313 .00000000000024132639491333131,
314 .00000000000039057462209830700,
315 -.00000000000026570679203560751,
316 .00000000000037135141919592021,
317 -.00000000000017166921336082431,
318 -.00000000000028658285157914353,
319 -.00000000000023812542263446809,
320 .00000000000006576659768580062,
321 -.00000000000028210143846181267,
322 .00000000000010701931762114254,
323 .00000000000018119346366441110,
324 .00000000000009840465278232627,
325 -.00000000000033149150282752542,
326 -.00000000000018302857356041668,
327 -.00000000000016207400156744949,
328 .00000000000048303314949553201,
329 -.00000000000071560553172382115,
330 .00000000000088821239518571855,
331 -.00000000000030900580513238244,
332 -.00000000000061076551972851496,
333 .00000000000035659969663347830,
334 .00000000000035782396591276383,
335 -.00000000000046226087001544578,
336 .00000000000062279762917225156,
337 .00000000000072838947272065741,
338 .00000000000026809646615211673,
339 -.00000000000010960825046059278,
340 .00000000000002311949383800537,
341 -.00000000000058469058005299247,
342 -.00000000000002103748251144494,
343 -.00000000000023323182945587408,
344 -.00000000000042333694288141916,
345 -.00000000000043933937969737844,
346 .00000000000041341647073835565,
347 .00000000000006841763641591466,
348 .00000000000047585534004430641,
349 .00000000000083679678674757695,
350 -.00000000000085763734646658640,
351 .00000000000021913281229340092,
352 -.00000000000062242842536431148,
353 -.00000000000010983594325438430,
354 .00000000000065310431377633651,
355 -.00000000000047580199021710769,
356 -.00000000000037854251265457040,
357 .00000000000040939233218678664,
358 .00000000000087424383914858291,
359 .00000000000025218188456842882,
360 -.00000000000003608131360422557,
361 -.00000000000050518555924280902,
362 .00000000000078699403323355317,
363 -.00000000000067020876961949060,
364 .00000000000016108575753932458,
365 .00000000000058527188436251509,
366 -.00000000000035246757297904791,
367 -.00000000000018372084495629058,
368 .00000000000088606689813494916,
369 .00000000000066486268071468700,
370 .00000000000063831615170646519,
371 .00000000000025144230728376072,
372 -.00000000000017239444525614834
373 };
374
375 double
log(double x)376 log(double x)
377 {
378 int m, j;
379 double F, f, g, q, u, u2, v, zero = 0.0, one = 1.0;
380 volatile double u1;
381
382 /* Catch special cases */
383 if (x <= 0) {
384 if (_IEEE && x == zero) /* log(0) = -Inf */
385 return (-one/zero);
386 else if (_IEEE) /* log(neg) = NaN */
387 return (zero/zero);
388 else if (x == zero) /* NOT REACHED IF _IEEE */
389 return (infnan(-ERANGE));
390 else
391 return (infnan(EDOM));
392 } else if (!finite(x)) {
393 if (_IEEE) /* x = NaN, Inf */
394 return (x+x);
395 else
396 return (infnan(ERANGE));
397 }
398
399 /* Argument reduction: 1 <= g < 2; x/2^m = g; */
400 /* y = F*(1 + f/F) for |f| <= 2^-8 */
401
402 m = logb(x);
403 g = ldexp(x, -m);
404 if (_IEEE && m == -1022) {
405 j = logb(g), m += j;
406 g = ldexp(g, -j);
407 }
408 j = N*(g-1) + .5;
409 F = (1.0/N) * j + 1; /* F*128 is an integer in [128, 512] */
410 f = g - F;
411
412 /* Approximate expansion for log(1+f/F) ~= u + q */
413 g = 1/(2*F+f);
414 u = 2*f*g;
415 v = u*u;
416 q = u*v*(A1 + v*(A2 + v*(A3 + v*A4)));
417
418 /* case 1: u1 = u rounded to 2^-43 absolute. Since u < 2^-8,
419 * u1 has at most 35 bits, and F*u1 is exact, as F has < 8 bits.
420 * It also adds exactly to |m*log2_hi + log_F_head[j] | < 750
421 */
422 if (m | j)
423 u1 = u + 513, u1 -= 513;
424
425 /* case 2: |1-x| < 1/256. The m- and j- dependent terms are zero;
426 * u1 = u to 24 bits.
427 */
428 else
429 u1 = u, TRUNC(u1);
430 u2 = (2.0*(f - F*u1) - u1*f) * g;
431 /* u1 + u2 = 2f/(2F+f) to extra precision. */
432
433 /* log(x) = log(2^m*F*(1+f/F)) = */
434 /* (m*log2_hi+logF_head[j]+u1) + (m*log2_lo+logF_tail[j]+q); */
435 /* (exact) + (tiny) */
436
437 u1 += m*logF_head[N] + logF_head[j]; /* exact */
438 u2 = (u2 + logF_tail[j]) + q; /* tiny */
439 u2 += logF_tail[N]*m;
440 return (u1 + u2);
441 }
442
443 /*
444 * Extra precision variant, returning struct {double a, b;};
445 * log(x) = a+b to 63 bits, with a is rounded to 26 bits.
446 */
447 struct Double
__log__D(double x)448 __log__D(double x)
449 {
450 int m, j;
451 double F, f, g, q, u, v, u2;
452 volatile double u1;
453 struct Double r;
454
455 /* Argument reduction: 1 <= g < 2; x/2^m = g; */
456 /* y = F*(1 + f/F) for |f| <= 2^-8 */
457
458 m = logb(x);
459 g = ldexp(x, -m);
460 if (_IEEE && m == -1022) {
461 j = logb(g), m += j;
462 g = ldexp(g, -j);
463 }
464 j = N*(g-1) + .5;
465 F = (1.0/N) * j + 1;
466 f = g - F;
467
468 g = 1/(2*F+f);
469 u = 2*f*g;
470 v = u*u;
471 q = u*v*(A1 + v*(A2 + v*(A3 + v*A4)));
472 if (m | j)
473 u1 = u + 513, u1 -= 513;
474 else
475 u1 = u, TRUNC(u1);
476 u2 = (2.0*(f - F*u1) - u1*f) * g;
477
478 u1 += m*logF_head[N] + logF_head[j];
479
480 u2 += logF_tail[j]; u2 += q;
481 u2 += logF_tail[N]*m;
482 r.a = u1 + u2; /* Only difference is here */
483 TRUNC(r.a);
484 r.b = (u1 - r.a) + u2;
485 return (r);
486 }
487
488 float
logf(float x)489 logf(float x)
490 {
491 return(log((double)x));
492 }
493