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