2 * Copyright (c) 1992, 1993
3 * The Regents of the University of California. All rights reserved.
5 * Redistribution and use in source and binary forms, with or without
6 * modification, are permitted provided that the following conditions
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.
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
35 static char sccsid[] = "@(#)log.c 8.2 (Berkeley) 11/30/93";
43 /* Table-driven natural logarithm.
45 * This code was derived, with minor modifications, from:
46 * Peter Tang, "Table-Driven Implementation of the
47 * Logarithm in IEEE Floating-Point arithmetic." ACM Trans.
48 * Math Software, vol 16. no 4, pp 378-400, Dec 1990).
50 * Calculates log(2^m*F*(1+f/F)), |f/j| <= 1/256,
51 * where F = j/128 for j an integer in [0, 128].
53 * log(2^m) = log2_hi*m + log2_tail*m
54 * since m is an integer, the dominant term is exact.
55 * m has at most 10 digits (for subnormal numbers),
56 * and log2_hi has 11 trailing zero bits.
58 * log(F) = logF_hi[j] + logF_lo[j] is in tabular form in log_table.h
59 * logF_hi[] + 512 is exact.
61 * log(1+f/F) = 2*f/(2*F + f) + 1/12 * (2*f/(2*F + f))**3 + ...
62 * the leading term is calculated to extra precision in two
63 * parts, the larger of which adds exactly to the dominant
65 * There are two cases:
66 * 1. when m, j are non-zero (m | j), use absolute
67 * precision for the leading term.
68 * 2. when m = j = 0, |1-x| < 1/256, and log(x) ~= (x-1).
69 * In this case, use a relative precision of 24 bits.
70 * (This is done differently in the original paper)
73 * 0 return signalling -Inf
74 * neg return signalling NaN
78 #if defined(vax) || defined(tahoe)
80 #define TRUNC(x) x = (double) (float) (x)
83 #define endian (((*(int *) &one)) ? 1 : 0)
84 #define TRUNC(x) *(((int *) &x) + endian) &= 0xf8000000
90 /* Table of log(Fj) = logF_head[j] + logF_tail[j], for Fj = 1+j/128.
91 * Used for generation of extend precision logarithms.
92 * The constant 35184372088832 is 2^45, so the divide is exact.
93 * It ensures correct reading of logF_head, even for inaccurate
94 * decimal-to-binary conversion routines. (Everybody gets the
95 * right answer for integers less than 2^53.)
96 * Values for log(F) were generated using error < 10^-57 absolute
97 * with the bc -l package.
99 static double A1 = .08333333333333178827;
100 static double A2 = .01250000000377174923;
101 static double A3 = .002232139987919447809;
102 static double A4 = .0004348877777076145742;
104 static double logF_head[N+1] = {
106 .007782140442060381246,
107 .015504186535963526694,
108 .023167059281547608406,
109 .030771658666765233647,
110 .038318864302141264488,
111 .045809536031242714670,
112 .053244514518837604555,
113 .060624621816486978786,
114 .067950661908525944454,
115 .075223421237524235039,
116 .082443669210988446138,
117 .089612158689760690322,
118 .096729626458454731618,
119 .103796793681567578460,
120 .110814366340264314203,
121 .117783035656430001836,
122 .124703478501032805070,
123 .131576357788617315236,
124 .138402322859292326029,
125 .145182009844575077295,
126 .151916042025732167530,
127 .158605030176659056451,
128 .165249572895390883786,
129 .171850256926518341060,
130 .178407657472689606947,
131 .184922338493834104156,
132 .191394852999565046047,
133 .197825743329758552135,
134 .204215541428766300668,
135 .210564769107350002741,
136 .216873938300523150246,
137 .223143551314024080056,
138 .229374101064877322642,
139 .235566071312860003672,
140 .241719936886966024758,
141 .247836163904594286577,
142 .253915209980732470285,
143 .259957524436686071567,
144 .265963548496984003577,
145 .271933715484010463114,
146 .277868451003087102435,
147 .283768173130738432519,
148 .289633292582948342896,
149 .295464212893421063199,
150 .301261330578199704177,
151 .307025035294827830512,
152 .312755710004239517729,
153 .318453731118097493890,
154 .324119468654316733591,
155 .329753286372579168528,
156 .335355541920762334484,
157 .340926586970454081892,
158 .346466767346100823488,
159 .351976423156884266063,
160 .357455888922231679316,
161 .362905493689140712376,
162 .368325561158599157352,
163 .373716409793814818840,
164 .379078352934811846353,
165 .384411698910298582632,
166 .389716751140440464951,
167 .394993808240542421117,
168 .400243164127459749579,
169 .405465108107819105498,
170 .410659924985338875558,
171 .415827895143593195825,
172 .420969294644237379543,
173 .426084395310681429691,
174 .431173464818130014464,
175 .436236766774527495726,
176 .441274560805140936281,
177 .446287102628048160113,
178 .451274644139630254358,
179 .456237433481874177232,
180 .461175715122408291790,
181 .466089729924533457960,
182 .470979715219073113985,
183 .475845904869856894947,
184 .480688529345570714212,
185 .485507815781602403149,
186 .490303988045525329653,
187 .495077266798034543171,
188 .499827869556611403822,
189 .504556010751912253908,
190 .509261901790523552335,
191 .513945751101346104405,
192 .518607764208354637958,
193 .523248143765158602036,
194 .527867089620485785417,
195 .532464798869114019908,
196 .537041465897345915436,
197 .541597282432121573947,
198 .546132437597407260909,
199 .550647117952394182793,
200 .555141507540611200965,
201 .559615787935399566777,
202 .564070138285387656651,
203 .568504735352689749561,
204 .572919753562018740922,
205 .577315365035246941260,
206 .581691739635061821900,
207 .586049045003164792433,
208 .590387446602107957005,
209 .594707107746216934174,
210 .599008189645246602594,
211 .603290851438941899687,
212 .607555250224322662688,
213 .611801541106615331955,
214 .616029877215623855590,
215 .620240409751204424537,
216 .624433288012369303032,
217 .628608659422752680256,
218 .632766669570628437213,
219 .636907462236194987781,
220 .641031179420679109171,
221 .645137961373620782978,
222 .649227946625615004450,
223 .653301272011958644725,
224 .657358072709030238911,
225 .661398482245203922502,
226 .665422632544505177065,
227 .669430653942981734871,
228 .673422675212350441142,
229 .677398823590920073911,
230 .681359224807238206267,
231 .685304003098281100392,
232 .689233281238557538017,
233 .693147180560117703862
236 static double logF_tail[N+1] = {
238 -.00000000000000543229938420049,
239 .00000000000000172745674997061,
240 -.00000000000001323017818229233,
241 -.00000000000001154527628289872,
242 -.00000000000000466529469958300,
243 .00000000000005148849572685810,
244 -.00000000000002532168943117445,
245 -.00000000000005213620639136504,
246 -.00000000000001819506003016881,
247 .00000000000006329065958724544,
248 .00000000000008614512936087814,
249 -.00000000000007355770219435028,
250 .00000000000009638067658552277,
251 .00000000000007598636597194141,
252 .00000000000002579999128306990,
253 -.00000000000004654729747598444,
254 -.00000000000007556920687451336,
255 .00000000000010195735223708472,
256 -.00000000000017319034406422306,
257 -.00000000000007718001336828098,
258 .00000000000010980754099855238,
259 -.00000000000002047235780046195,
260 -.00000000000008372091099235912,
261 .00000000000014088127937111135,
262 .00000000000012869017157588257,
263 .00000000000017788850778198106,
264 .00000000000006440856150696891,
265 .00000000000016132822667240822,
266 -.00000000000007540916511956188,
267 -.00000000000000036507188831790,
268 .00000000000009120937249914984,
269 .00000000000018567570959796010,
270 -.00000000000003149265065191483,
271 -.00000000000009309459495196889,
272 .00000000000017914338601329117,
273 -.00000000000001302979717330866,
274 .00000000000023097385217586939,
275 .00000000000023999540484211737,
276 .00000000000015393776174455408,
277 -.00000000000036870428315837678,
278 .00000000000036920375082080089,
279 -.00000000000009383417223663699,
280 .00000000000009433398189512690,
281 .00000000000041481318704258568,
282 -.00000000000003792316480209314,
283 .00000000000008403156304792424,
284 -.00000000000034262934348285429,
285 .00000000000043712191957429145,
286 -.00000000000010475750058776541,
287 -.00000000000011118671389559323,
288 .00000000000037549577257259853,
289 .00000000000013912841212197565,
290 .00000000000010775743037572640,
291 .00000000000029391859187648000,
292 -.00000000000042790509060060774,
293 .00000000000022774076114039555,
294 .00000000000010849569622967912,
295 -.00000000000023073801945705758,
296 .00000000000015761203773969435,
297 .00000000000003345710269544082,
298 -.00000000000041525158063436123,
299 .00000000000032655698896907146,
300 -.00000000000044704265010452446,
301 .00000000000034527647952039772,
302 -.00000000000007048962392109746,
303 .00000000000011776978751369214,
304 -.00000000000010774341461609578,
305 .00000000000021863343293215910,
306 .00000000000024132639491333131,
307 .00000000000039057462209830700,
308 -.00000000000026570679203560751,
309 .00000000000037135141919592021,
310 -.00000000000017166921336082431,
311 -.00000000000028658285157914353,
312 -.00000000000023812542263446809,
313 .00000000000006576659768580062,
314 -.00000000000028210143846181267,
315 .00000000000010701931762114254,
316 .00000000000018119346366441110,
317 .00000000000009840465278232627,
318 -.00000000000033149150282752542,
319 -.00000000000018302857356041668,
320 -.00000000000016207400156744949,
321 .00000000000048303314949553201,
322 -.00000000000071560553172382115,
323 .00000000000088821239518571855,
324 -.00000000000030900580513238244,
325 -.00000000000061076551972851496,
326 .00000000000035659969663347830,
327 .00000000000035782396591276383,
328 -.00000000000046226087001544578,
329 .00000000000062279762917225156,
330 .00000000000072838947272065741,
331 .00000000000026809646615211673,
332 -.00000000000010960825046059278,
333 .00000000000002311949383800537,
334 -.00000000000058469058005299247,
335 -.00000000000002103748251144494,
336 -.00000000000023323182945587408,
337 -.00000000000042333694288141916,
338 -.00000000000043933937969737844,
339 .00000000000041341647073835565,
340 .00000000000006841763641591466,
341 .00000000000047585534004430641,
342 .00000000000083679678674757695,
343 -.00000000000085763734646658640,
344 .00000000000021913281229340092,
345 -.00000000000062242842536431148,
346 -.00000000000010983594325438430,
347 .00000000000065310431377633651,
348 -.00000000000047580199021710769,
349 -.00000000000037854251265457040,
350 .00000000000040939233218678664,
351 .00000000000087424383914858291,
352 .00000000000025218188456842882,
353 -.00000000000003608131360422557,
354 -.00000000000050518555924280902,
355 .00000000000078699403323355317,
356 -.00000000000067020876961949060,
357 .00000000000016108575753932458,
358 .00000000000058527188436251509,
359 -.00000000000035246757297904791,
360 -.00000000000018372084495629058,
361 .00000000000088606689813494916,
362 .00000000000066486268071468700,
363 .00000000000063831615170646519,
364 .00000000000025144230728376072,
365 -.00000000000017239444525614834
376 double F, f, g, q, u, u2, v, zero = 0.0, one = 1.0;
379 /* Catch special cases */
381 if (_IEEE && x == zero) /* log(0) = -Inf */
383 else if (_IEEE) /* log(neg) = NaN */
385 else if (x == zero) /* NOT REACHED IF _IEEE */
386 return (infnan(-ERANGE));
388 return (infnan(EDOM));
390 if (_IEEE) /* x = NaN, Inf */
393 return (infnan(ERANGE));
395 /* Argument reduction: 1 <= g < 2; x/2^m = g; */
396 /* y = F*(1 + f/F) for |f| <= 2^-8 */
400 if (_IEEE && m == -1022) {
405 F = (1.0/N) * j + 1; /* F*128 is an integer in [128, 512] */
408 /* Approximate expansion for log(1+f/F) ~= u + q */
412 q = u*v*(A1 + v*(A2 + v*(A3 + v*A4)));
414 /* case 1: u1 = u rounded to 2^-43 absolute. Since u < 2^-8,
415 * u1 has at most 35 bits, and F*u1 is exact, as F has < 8 bits.
416 * It also adds exactly to |m*log2_hi + log_F_head[j] | < 750
419 u1 = u + 513, u1 -= 513;
421 /* case 2: |1-x| < 1/256. The m- and j- dependent terms are zero;
426 u2 = (2.0*(f - F*u1) - u1*f) * g;
427 /* u1 + u2 = 2f/(2F+f) to extra precision. */
429 /* log(x) = log(2^m*F*(1+f/F)) = */
430 /* (m*log2_hi+logF_head[j]+u1) + (m*log2_lo+logF_tail[j]+q); */
431 /* (exact) + (tiny) */
433 u1 += m*logF_head[N] + logF_head[j]; /* exact */
434 u2 = (u2 + logF_tail[j]) + q; /* tiny */
435 u2 += logF_tail[N]*m;
440 * Extra precision variant, returning struct {double a, b;};
441 * log(x) = a+b to 63 bits, with a is rounded to 26 bits.
447 __log__D(x) double x;
451 double F, f, g, q, u, v, u2, one = 1.0;
455 /* Argument reduction: 1 <= g < 2; x/2^m = g; */
456 /* y = F*(1 + f/F) for |f| <= 2^-8 */
460 if (_IEEE && m == -1022) {
471 q = u*v*(A1 + v*(A2 + v*(A3 + v*A4)));
473 u1 = u + 513, u1 -= 513;
476 u2 = (2.0*(f - F*u1) - u1*f) * g;
478 u1 += m*logF_head[N] + logF_head[j];
480 u2 += logF_tail[j]; u2 += q;
481 u2 += logF_tail[N]*m;
482 r.a = u1 + u2; /* Only difference is here */
484 r.b = (u1 - r.a) + u2;