1 /* $NetBSD: n_log.c,v 1.9 2024/07/16 14:52:50 riastradh 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 #include <sys/cdefs.h>
32 __RCSID("$NetBSD: n_log.c,v 1.9 2024/07/16 14:52:50 riastradh Exp $");
33
34 #ifndef lint
35 #if 0
36 static char sccsid[] = "@(#)log.c 8.2 (Berkeley) 11/30/93";
37 #endif
38 #endif /* not lint */
39
40 #include "namespace.h"
41
42 #include <math.h>
43 #include <errno.h>
44
45 #include "mathimpl.h"
46
47 __weak_alias(logl, _logl)
48 __strong_alias(_logl, _log)
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
__weak_alias(log,_log)375 __weak_alias(log, _log)
376 double
377 log(double x)
378 {
379 int m, j;
380 double F, f, g, q, u, u2, v, zero = 0.0, one = 1.0;
381 volatile double u1;
382
383 /* Catch special cases */
384 if (x <= 0) {
385 if (_IEEE && x == zero) /* log(0) = -Inf */
386 return (-one/zero);
387 else if (_IEEE) /* log(neg) = NaN */
388 return (zero/zero);
389 else if (x == zero) /* NOT REACHED IF _IEEE */
390 return (infnan(-ERANGE));
391 else
392 return (infnan(EDOM));
393 } else if (!finite(x)) {
394 if (_IEEE) /* x = NaN, Inf */
395 return (x+x);
396 else
397 return (infnan(ERANGE));
398 }
399
400 /* Argument reduction: 1 <= g < 2; x/2^m = g; */
401 /* y = F*(1 + f/F) for |f| <= 2^-8 */
402
403 m = logb(x);
404 g = ldexp(x, -m);
405 if (_IEEE && m == -1022) {
406 j = logb(g), m += j;
407 g = ldexp(g, -j);
408 }
409 j = N*(g-1) + .5;
410 F = (1.0/N) * j + 1; /* F*128 is an integer in [128, 512] */
411 f = g - F;
412
413 /* Approximate expansion for log(1+f/F) ~= u + q */
414 g = 1/(2*F+f);
415 u = 2*f*g;
416 v = u*u;
417 q = u*v*(A1 + v*(A2 + v*(A3 + v*A4)));
418
419 /* case 1: u1 = u rounded to 2^-43 absolute. Since u < 2^-8,
420 * u1 has at most 35 bits, and F*u1 is exact, as F has < 8 bits.
421 * It also adds exactly to |m*log2_hi + log_F_head[j] | < 750
422 */
423 if (m | j)
424 u1 = u + 513, u1 -= 513;
425
426 /* case 2: |1-x| < 1/256. The m- and j- dependent terms are zero;
427 * u1 = u to 24 bits.
428 */
429 else
430 u1 = u, TRUNC(u1);
431 u2 = (2.0*(f - F*u1) - u1*f) * g;
432 /* u1 + u2 = 2f/(2F+f) to extra precision. */
433
434 /* log(x) = log(2^m*F*(1+f/F)) = */
435 /* (m*log2_hi+logF_head[j]+u1) + (m*log2_lo+logF_tail[j]+q); */
436 /* (exact) + (tiny) */
437
438 u1 += m*logF_head[N] + logF_head[j]; /* exact */
439 u2 = (u2 + logF_tail[j]) + q; /* tiny */
440 u2 += logF_tail[N]*m;
441 return (u1 + u2);
442 }
443
444 /*
445 * Extra precision variant, returning struct {double a, b;};
446 * log(x) = a+b to 63 bits, with a is rounded to 26 bits.
447 */
448 struct Double
__log__D(double x)449 __log__D(double x)
450 {
451 int m, j;
452 double F, f, g, q, u, v, u2;
453 volatile double u1;
454 struct Double r;
455
456 /* Argument reduction: 1 <= g < 2; x/2^m = g; */
457 /* y = F*(1 + f/F) for |f| <= 2^-8 */
458
459 m = logb(x);
460 g = ldexp(x, -m);
461 if (_IEEE && m == -1022) {
462 j = logb(g), m += j;
463 g = ldexp(g, -j);
464 }
465 j = N*(g-1) + .5;
466 F = (1.0/N) * j + 1;
467 f = g - F;
468
469 g = 1/(2*F+f);
470 u = 2*f*g;
471 v = u*u;
472 q = u*v*(A1 + v*(A2 + v*(A3 + v*A4)));
473 if (m | j)
474 u1 = u + 513, u1 -= 513;
475 else
476 u1 = u, TRUNC(u1);
477 u2 = (2.0*(f - F*u1) - u1*f) * g;
478
479 u1 += m*logF_head[N] + logF_head[j];
480
481 u2 += logF_tail[j]; u2 += q;
482 u2 += logF_tail[N]*m;
483 r.a = u1 + u2; /* Only difference is here */
484 TRUNC(r.a);
485 r.b = (u1 - r.a) + u2;
486 return (r);
487 }
488
__weak_alias(logf,_logf)489 __weak_alias(logf, _logf)
490 float
491 logf(float x)
492 {
493 return(log((double)x));
494 }
495