sysdeps/generic/log.c

   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
  35 static char sccsid[] = "@(#)log.c       8.2 (Berkeley) 11/30/93";
  36 #endif /* not lint */
  37
  38 #include <math.h>
  39 #include <errno.h>
  40
  41 #include "mathimpl.h"
  42
  43 /* Table-driven natural logarithm.
  44  *
  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).
  49  *
  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].
  52  *
  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.
  57  *
  58  * log(F) = logF_hi[j] + logF_lo[j] is in tabular form in log_table.h
  59  * logF_hi[] + 512 is exact.
  60  *
  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
  64  * m and F terms.
  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)
  71  *
  72  * Special cases:
  73  *      0       return signalling -Inf
  74  *      neg     return signalling NaN
  75  *      +Inf    return +Inf
  76 */
  77
  78 #if defined(vax) || defined(tahoe)
  79 #define _IEEE           0
  80 #define TRUNC(x)        x = (double) (float) (x)
  81 #else
  82 #define _IEEE           1
  83 #define endian          (((*(int *) &one)) ? 1 : 0)
  84 #define TRUNC(x)        *(((int *) &x) + endian) &= 0xf8000000
  85 #define infnan(x)       0.0
  86 #endif
  87
  88 #define N 128
  89
  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.
  98 */
  99 static double   A1 =      .08333333333333178827;
 100 static double   A2 =      .01250000000377174923;
 101 static double   A3 =     .002232139987919447809;
 102 static double   A4 =    .0004348877777076145742;
 103
 104 static double logF_head[N+1] = {
 105         0.,
 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
 234 };
 235
 236 static double logF_tail[N+1] = {
 237         0.,
 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
 366 };
 367
 368 double
 369 #ifdef _ANSI_SOURCE
 370 log(double x)
 371 #else
 372 log(x) double x;
 373 #endif
 374 {
 375         int m, j;
 376         double F, f, g, q, u, u2, v, zero = 0.0, one = 1.0;
 377         volatile double u1;
 378
 379         /* Catch special cases */
 380         if (x <= 0)
 381                 if (_IEEE && x == zero) /* log(0) = -Inf */
 382                         return (-one/zero);
 383                 else if (_IEEE)         /* log(neg) = NaN */
 384                         return (zero/zero);
 385                 else if (x == zero)     /* NOT REACHED IF _IEEE */
 386                         return (infnan(-ERANGE));
 387                 else
 388                         return (infnan(EDOM));
 389         else if (!finite(x))
 390                 if (_IEEE)              /* x = NaN, Inf */
 391                         return (x+x);
 392                 else
 393                         return (infnan(ERANGE));
 394
 395         /* Argument reduction: 1 <= g < 2; x/2^m = g;   */
 396         /* y = F*(1 + f/F) for |f| <= 2^-8              */
 397
 398         m = logb(x);
 399         g = ldexp(x, -m);
 400         if (_IEEE && m == -1022) {
 401                 j = logb(g), m += j;
 402                 g = ldexp(g, -j);
 403         }
 404         j = N*(g-1) + .5;
 405         F = (1.0/N) * j + 1;    /* F*128 is an integer in [128, 512] */
 406         f = g - F;
 407
 408         /* Approximate expansion for log(1+f/F) ~= u + q */
 409         g = 1/(2*F+f);
 410         u = 2*f*g;
 411         v = u*u;
 412         q = u*v*(A1 + v*(A2 + v*(A3 + v*A4)));
 413
 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
 417     */
 418         if (m | j)
 419                 u1 = u + 513, u1 -= 513;
 420
 421     /* case 2:  |1-x| < 1/256. The m- and j- dependent terms are zero;
 422      *          u1 = u to 24 bits.
 423     */
 424         else
 425                 u1 = u, TRUNC(u1);
 426         u2 = (2.0*(f - F*u1) - u1*f) * g;
 427                         /* u1 + u2 = 2f/(2F+f) to extra precision.      */
 428
 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)                                             */
 432
 433         u1 += m*logF_head[N] + logF_head[j];            /* exact */
 434         u2 = (u2 + logF_tail[j]) + q;                   /* tiny */
 435         u2 += logF_tail[N]*m;
 436         return (u1 + u2);
 437 }
 438
 439 /*
 440  * Extra precision variant, returning struct {double a, b;};
 441  * log(x) = a+b to 63 bits, with a is rounded to 26 bits.
 442  */
 443 struct Double
 444 #ifdef _ANSI_SOURCE
 445 __log__D(double x)
 446 #else
 447 __log__D(x) double x;
 448 #endif
 449 {
 450         int m, j;
 451         double F, f, g, q, u, v, u2, one = 1.0;
 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 }