lib/libm/src/b_log.c

   1 /*      $NetBSD: b_log.c,v 1.1 2012/05/05 17:54:14 christos Exp $       */
   2
   3 /*
   4  * Copyright (c) 1992, 1993
   5  *      The Regents of the University of California.  All rights reserved.
   6  *
   7  * Redistribution and use in source and binary forms, with or without
   8  * modification, are permitted provided that the following conditions
   9  * are met:
  10  * 1. Redistributions of source code must retain the above copyright
  11  *    notice, this list of conditions and the following disclaimer.
  12  * 2. Redistributions in binary form must reproduce the above copyright
  13  *    notice, this list of conditions and the following disclaimer in the
  14  *    documentation and/or other materials provided with the distribution.
  15  * 3. All advertising materials mentioning features or use of this software
  16  *    must display the following acknowledgement:
  17  *      This product includes software developed by the University of
  18  *      California, Berkeley and its contributors.
  19  * 4. Neither the name of the University nor the names of its contributors
  20  *    may be used to endorse or promote products derived from this software
  21  *    without specific prior written permission.
  22  *
  23  * THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND
  24  * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
  25  * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
  26  * ARE DISCLAIMED.  IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE
  27  * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
  28  * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
  29  * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
  30  * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
  31  * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
  32  * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
  33  * SUCH DAMAGE.
  34  */
  35
  36 /* @(#)log.c    8.2 (Berkeley) 11/30/93 */
  37 #include <sys/cdefs.h>
  38 #if 0
  39 __FBSDID("$FreeBSD: release/9.0.0/lib/msun/bsdsrc/b_log.c 176449 2008-02-22 02:26:51Z das $");
  40 #else
  41 __RCSID("$NetBSD: b_log.c,v 1.1 2012/05/05 17:54:14 christos Exp $");
  42 #endif
  43
  44 #include <errno.h>
  45 #include "math.h"
  46 #include "math_private.h"
  47
  48 /* Table-driven natural logarithm.
  49  *
  50  * This code was derived, with minor modifications, from:
  51  *      Peter Tang, "Table-Driven Implementation of the
  52  *      Logarithm in IEEE Floating-Point arithmetic." ACM Trans.
  53  *      Math Software, vol 16. no 4, pp 378-400, Dec 1990).
  54  *
  55  * Calculates log(2^m*F*(1+f/F)), |f/j| <= 1/256,
  56  * where F = j/128 for j an integer in [0, 128].
  57  *
  58  * log(2^m) = log2_hi*m + log2_tail*m
  59  * since m is an integer, the dominant term is exact.
  60  * m has at most 10 digits (for subnormal numbers),
  61  * and log2_hi has 11 trailing zero bits.
  62  *
  63  * log(F) = logF_hi[j] + logF_lo[j] is in tabular form in log_table.h
  64  * logF_hi[] + 512 is exact.
  65  *
  66  * log(1+f/F) = 2*f/(2*F + f) + 1/12 * (2*f/(2*F + f))**3 + ...
  67  * the leading term is calculated to extra precision in two
  68  * parts, the larger of which adds exactly to the dominant
  69  * m and F terms.
  70  * There are two cases:
  71  *      1. when m, j are non-zero (m | j), use absolute
  72  *         precision for the leading term.
  73  *      2. when m = j = 0, |1-x| < 1/256, and log(x) ~= (x-1).
  74  *         In this case, use a relative precision of 24 bits.
  75  * (This is done differently in the original paper)
  76  *
  77  * Special cases:
  78  *      0       return signalling -Inf
  79  *      neg     return signalling NaN
  80  *      +Inf    return +Inf
  81 */
  82
  83 #define N 128
  84
  85 /* Table of log(Fj) = logF_head[j] + logF_tail[j], for Fj = 1+j/128.
  86  * Used for generation of extend precision logarithms.
  87  * The constant 35184372088832 is 2^45, so the divide is exact.
  88  * It ensures correct reading of logF_head, even for inaccurate
  89  * decimal-to-binary conversion routines.  (Everybody gets the
  90  * right answer for integers less than 2^53.)
  91  * Values for log(F) were generated using error < 10^-57 absolute
  92  * with the bc -l package.
  93 */
  94 static const double     A1 =      .08333333333333178827;
  95 static const double     A2 =      .01250000000377174923;
  96 static const double     A3 =     .002232139987919447809;
  97 static const double     A4 =    .0004348877777076145742;
  98
  99 static const double logF_head[N+1] = {
 100         0.,
 101         .007782140442060381246,
 102         .015504186535963526694,
 103         .023167059281547608406,
 104         .030771658666765233647,
 105         .038318864302141264488,
 106         .045809536031242714670,
 107         .053244514518837604555,
 108         .060624621816486978786,
 109         .067950661908525944454,
 110         .075223421237524235039,
 111         .082443669210988446138,
 112         .089612158689760690322,
 113         .096729626458454731618,
 114         .103796793681567578460,
 115         .110814366340264314203,
 116         .117783035656430001836,
 117         .124703478501032805070,
 118         .131576357788617315236,
 119         .138402322859292326029,
 120         .145182009844575077295,
 121         .151916042025732167530,
 122         .158605030176659056451,
 123         .165249572895390883786,
 124         .171850256926518341060,
 125         .178407657472689606947,
 126         .184922338493834104156,
 127         .191394852999565046047,
 128         .197825743329758552135,
 129         .204215541428766300668,
 130         .210564769107350002741,
 131         .216873938300523150246,
 132         .223143551314024080056,
 133         .229374101064877322642,
 134         .235566071312860003672,
 135         .241719936886966024758,
 136         .247836163904594286577,
 137         .253915209980732470285,
 138         .259957524436686071567,
 139         .265963548496984003577,
 140         .271933715484010463114,
 141         .277868451003087102435,
 142         .283768173130738432519,
 143         .289633292582948342896,
 144         .295464212893421063199,
 145         .301261330578199704177,
 146         .307025035294827830512,
 147         .312755710004239517729,
 148         .318453731118097493890,
 149         .324119468654316733591,
 150         .329753286372579168528,
 151         .335355541920762334484,
 152         .340926586970454081892,
 153         .346466767346100823488,
 154         .351976423156884266063,
 155         .357455888922231679316,
 156         .362905493689140712376,
 157         .368325561158599157352,
 158         .373716409793814818840,
 159         .379078352934811846353,
 160         .384411698910298582632,
 161         .389716751140440464951,
 162         .394993808240542421117,
 163         .400243164127459749579,
 164         .405465108107819105498,
 165         .410659924985338875558,
 166         .415827895143593195825,
 167         .420969294644237379543,
 168         .426084395310681429691,
 169         .431173464818130014464,
 170         .436236766774527495726,
 171         .441274560805140936281,
 172         .446287102628048160113,
 173         .451274644139630254358,
 174         .456237433481874177232,
 175         .461175715122408291790,
 176         .466089729924533457960,
 177         .470979715219073113985,
 178         .475845904869856894947,
 179         .480688529345570714212,
 180         .485507815781602403149,
 181         .490303988045525329653,
 182         .495077266798034543171,
 183         .499827869556611403822,
 184         .504556010751912253908,
 185         .509261901790523552335,
 186         .513945751101346104405,
 187         .518607764208354637958,
 188         .523248143765158602036,
 189         .527867089620485785417,
 190         .532464798869114019908,
 191         .537041465897345915436,
 192         .541597282432121573947,
 193         .546132437597407260909,
 194         .550647117952394182793,
 195         .555141507540611200965,
 196         .559615787935399566777,
 197         .564070138285387656651,
 198         .568504735352689749561,
 199         .572919753562018740922,
 200         .577315365035246941260,
 201         .581691739635061821900,
 202         .586049045003164792433,
 203         .590387446602107957005,
 204         .594707107746216934174,
 205         .599008189645246602594,
 206         .603290851438941899687,
 207         .607555250224322662688,
 208         .611801541106615331955,
 209         .616029877215623855590,
 210         .620240409751204424537,
 211         .624433288012369303032,
 212         .628608659422752680256,
 213         .632766669570628437213,
 214         .636907462236194987781,
 215         .641031179420679109171,
 216         .645137961373620782978,
 217         .649227946625615004450,
 218         .653301272011958644725,
 219         .657358072709030238911,
 220         .661398482245203922502,
 221         .665422632544505177065,
 222         .669430653942981734871,
 223         .673422675212350441142,
 224         .677398823590920073911,
 225         .681359224807238206267,
 226         .685304003098281100392,
 227         .689233281238557538017,
 228         .693147180560117703862
 229 };
 230
 231 static const double logF_tail[N+1] = {
 232         0.,
 233         -.00000000000000543229938420049,
 234          .00000000000000172745674997061,
 235         -.00000000000001323017818229233,
 236         -.00000000000001154527628289872,
 237         -.00000000000000466529469958300,
 238          .00000000000005148849572685810,
 239         -.00000000000002532168943117445,
 240         -.00000000000005213620639136504,
 241         -.00000000000001819506003016881,
 242          .00000000000006329065958724544,
 243          .00000000000008614512936087814,
 244         -.00000000000007355770219435028,
 245          .00000000000009638067658552277,
 246          .00000000000007598636597194141,
 247          .00000000000002579999128306990,
 248         -.00000000000004654729747598444,
 249         -.00000000000007556920687451336,
 250          .00000000000010195735223708472,
 251         -.00000000000017319034406422306,
 252         -.00000000000007718001336828098,
 253          .00000000000010980754099855238,
 254         -.00000000000002047235780046195,
 255         -.00000000000008372091099235912,
 256          .00000000000014088127937111135,
 257          .00000000000012869017157588257,
 258          .00000000000017788850778198106,
 259          .00000000000006440856150696891,
 260          .00000000000016132822667240822,
 261         -.00000000000007540916511956188,
 262         -.00000000000000036507188831790,
 263          .00000000000009120937249914984,
 264          .00000000000018567570959796010,
 265         -.00000000000003149265065191483,
 266         -.00000000000009309459495196889,
 267          .00000000000017914338601329117,
 268         -.00000000000001302979717330866,
 269          .00000000000023097385217586939,
 270          .00000000000023999540484211737,
 271          .00000000000015393776174455408,
 272         -.00000000000036870428315837678,
 273          .00000000000036920375082080089,
 274         -.00000000000009383417223663699,
 275          .00000000000009433398189512690,
 276          .00000000000041481318704258568,
 277         -.00000000000003792316480209314,
 278          .00000000000008403156304792424,
 279         -.00000000000034262934348285429,
 280          .00000000000043712191957429145,
 281         -.00000000000010475750058776541,
 282         -.00000000000011118671389559323,
 283          .00000000000037549577257259853,
 284          .00000000000013912841212197565,
 285          .00000000000010775743037572640,
 286          .00000000000029391859187648000,
 287         -.00000000000042790509060060774,
 288          .00000000000022774076114039555,
 289          .00000000000010849569622967912,
 290         -.00000000000023073801945705758,
 291          .00000000000015761203773969435,
 292          .00000000000003345710269544082,
 293         -.00000000000041525158063436123,
 294          .00000000000032655698896907146,
 295         -.00000000000044704265010452446,
 296          .00000000000034527647952039772,
 297         -.00000000000007048962392109746,
 298          .00000000000011776978751369214,
 299         -.00000000000010774341461609578,
 300          .00000000000021863343293215910,
 301          .00000000000024132639491333131,
 302          .00000000000039057462209830700,
 303         -.00000000000026570679203560751,
 304          .00000000000037135141919592021,
 305         -.00000000000017166921336082431,
 306         -.00000000000028658285157914353,
 307         -.00000000000023812542263446809,
 308          .00000000000006576659768580062,
 309         -.00000000000028210143846181267,
 310          .00000000000010701931762114254,
 311          .00000000000018119346366441110,
 312          .00000000000009840465278232627,
 313         -.00000000000033149150282752542,
 314         -.00000000000018302857356041668,
 315         -.00000000000016207400156744949,
 316          .00000000000048303314949553201,
 317         -.00000000000071560553172382115,
 318          .00000000000088821239518571855,
 319         -.00000000000030900580513238244,
 320         -.00000000000061076551972851496,
 321          .00000000000035659969663347830,
 322          .00000000000035782396591276383,
 323         -.00000000000046226087001544578,
 324          .00000000000062279762917225156,
 325          .00000000000072838947272065741,
 326          .00000000000026809646615211673,
 327         -.00000000000010960825046059278,
 328          .00000000000002311949383800537,
 329         -.00000000000058469058005299247,
 330         -.00000000000002103748251144494,
 331         -.00000000000023323182945587408,
 332         -.00000000000042333694288141916,
 333         -.00000000000043933937969737844,
 334          .00000000000041341647073835565,
 335          .00000000000006841763641591466,
 336          .00000000000047585534004430641,
 337          .00000000000083679678674757695,
 338         -.00000000000085763734646658640,
 339          .00000000000021913281229340092,
 340         -.00000000000062242842536431148,
 341         -.00000000000010983594325438430,
 342          .00000000000065310431377633651,
 343         -.00000000000047580199021710769,
 344         -.00000000000037854251265457040,
 345          .00000000000040939233218678664,
 346          .00000000000087424383914858291,
 347          .00000000000025218188456842882,
 348         -.00000000000003608131360422557,
 349         -.00000000000050518555924280902,
 350          .00000000000078699403323355317,
 351         -.00000000000067020876961949060,
 352          .00000000000016108575753932458,
 353          .00000000000058527188436251509,
 354         -.00000000000035246757297904791,
 355         -.00000000000018372084495629058,
 356          .00000000000088606689813494916,
 357          .00000000000066486268071468700,
 358          .00000000000063831615170646519,
 359          .00000000000025144230728376072,
 360         -.00000000000017239444525614834
 361 };
 362
 363 /*
 364  * Extra precision variant, returning struct {double a, b;};
 365  * log(x) = a+b to 63 bits, with a rounded to 26 bits.
 366  */
 367 struct Double
 368 __log__D(double x)
 369 {
 370         int m, j;
 371         double F, f, g, q, u, v, u2;
 372         volatile double u1;
 373         struct Double r;
 374
 375         /* Argument reduction: 1 <= g < 2; x/2^m = g;   */
 376         /* y = F*(1 + f/F) for |f| <= 2^-8              */
 377
 378         m = logb(x);
 379         g = ldexp(x, -m);
 380         if (m == -1022) {
 381                 j = logb(g), m += j;
 382                 g = ldexp(g, -j);
 383         }
 384         j = N*(g-1) + .5;
 385         F = (1.0/N) * j + 1;
 386         f = g - F;
 387
 388         g = 1/(2*F+f);
 389         u = 2*f*g;
 390         v = u*u;
 391         q = u*v*(A1 + v*(A2 + v*(A3 + v*A4)));
 392         if (m | j)
 393                 u1 = u + 513, u1 -= 513;
 394         else
 395                 u1 = u, TRUNC(u1);
 396         u2 = (2.0*(f - F*u1) - u1*f) * g;
 397
 398         u1 += m*logF_head[N] + logF_head[j];
 399
 400         u2 +=  logF_tail[j]; u2 += q;
 401         u2 += logF_tail[N]*m;
 402         r.a = u1 + u2;                  /* Only difference is here */
 403         TRUNC(r.a);
 404         r.b = (u1 - r.a) + u2;
 405         return (r);
 406 }