gonzui

    1: /*      $OpenBSD: b_log__D.c,v 1.4 2009/10/27 23:59:29 deraadt 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 "math.h"
   32: #include "math_private.h"
   33: 
   34: /* Table-driven natural logarithm.
   35:  *
   36:  * This code was derived, with minor modifications, from:
   37:  *      Peter Tang, "Table-Driven Implementation of the
   38:  *      Logarithm in IEEE Floating-Point arithmetic." ACM Trans.
   39:  *      Math Software, vol 16. no 4, pp 378-400, Dec 1990).
   40:  *
   41:  * Calculates log(2^m*F*(1+f/F)), |f/j| <= 1/256,
   42:  * where F = j/128 for j an integer in [0, 128].
   43:  *
   44:  * log(2^m) = log2_hi*m + log2_tail*m
   45:  * since m is an integer, the dominant term is exact.
   46:  * m has at most 10 digits (for subnormal numbers),
   47:  * and log2_hi has 11 trailing zero bits.
   48:  *
   49:  * log(F) = logF_hi[j] + logF_lo[j] is in tabular form in log_table.h
   50:  * logF_hi[] + 512 is exact.
   51:  *
   52:  * log(1+f/F) = 2*f/(2*F + f) + 1/12 * (2*f/(2*F + f))**3 + ...
   53:  * the leading term is calculated to extra precision in two
   54:  * parts, the larger of which adds exactly to the dominant
   55:  * m and F terms.
   56:  * There are two cases:
   57:  *      1. when m, j are non-zero (m | j), use absolute
   58:  *         precision for the leading term.
   59:  *      2. when m = j = 0, |1-x| < 1/256, and log(x) ~= (x-1).
   60:  *         In this case, use a relative precision of 24 bits.
   61:  * (This is done differently in the original paper)
   62:  *
   63:  * Special cases:
   64:  *      0    return signalling -Inf
   65:  *      neg  return signalling NaN
   66:  *      +Inf return +Inf
   67: */
   68: 
   69: #define N 128
   70: 
   71: /* Table of log(Fj) = logF_head[j] + logF_tail[j], for Fj = 1+j/128.
   72:  * Used for generation of extend precision logarithms.
   73:  * The constant 35184372088832 is 2^45, so the divide is exact.
   74:  * It ensures correct reading of logF_head, even for inaccurate
   75:  * decimal-to-binary conversion routines.  (Everybody gets the
   76:  * right answer for integers less than 2^53.)
   77:  * Values for log(F) were generated using error < 10^-57 absolute
   78:  * with the bc -l package.
   79: */
   80: static const double     A1 =          .08333333333333178827;
   81: static const double     A2 =          .01250000000377174923;
   82: static const double     A3 =         .002232139987919447809;
   83: static const double     A4 =        .0004348877777076145742;
   84: 
   85: static const double logF_head[N+1] = {
   86:         0.,
   87:         .007782140442060381246,
   88:         .015504186535963526694,
   89:         .023167059281547608406,
   90:         .030771658666765233647,
   91:         .038318864302141264488,
   92:         .045809536031242714670,
   93:         .053244514518837604555,
   94:         .060624621816486978786,
   95:         .067950661908525944454,
   96:         .075223421237524235039,
   97:         .082443669210988446138,
   98:         .089612158689760690322,
   99:         .096729626458454731618,
  100:         .103796793681567578460,
  101:         .110814366340264314203,
  102:         .117783035656430001836,
  103:         .124703478501032805070,
  104:         .131576357788617315236,
  105:         .138402322859292326029,
  106:         .145182009844575077295,
  107:         .151916042025732167530,
  108:         .158605030176659056451,
  109:         .165249572895390883786,
  110:         .171850256926518341060,
  111:         .178407657472689606947,
  112:         .184922338493834104156,
  113:         .191394852999565046047,
  114:         .197825743329758552135,
  115:         .204215541428766300668,
  116:         .210564769107350002741,
  117:         .216873938300523150246,
  118:         .223143551314024080056,
  119:         .229374101064877322642,
  120:         .235566071312860003672,
  121:         .241719936886966024758,
  122:         .247836163904594286577,
  123:         .253915209980732470285,
  124:         .259957524436686071567,
  125:         .265963548496984003577,
  126:         .271933715484010463114,
  127:         .277868451003087102435,
  128:         .283768173130738432519,
  129:         .289633292582948342896,
  130:         .295464212893421063199,
  131:         .301261330578199704177,
  132:         .307025035294827830512,
  133:         .312755710004239517729,
  134:         .318453731118097493890,
  135:         .324119468654316733591,
  136:         .329753286372579168528,
  137:         .335355541920762334484,
  138:         .340926586970454081892,
  139:         .346466767346100823488,
  140:         .351976423156884266063,
  141:         .357455888922231679316,
  142:         .362905493689140712376,
  143:         .368325561158599157352,
  144:         .373716409793814818840,
  145:         .379078352934811846353,
  146:         .384411698910298582632,
  147:         .389716751140440464951,
  148:         .394993808240542421117,
  149:         .400243164127459749579,
  150:         .405465108107819105498,
  151:         .410659924985338875558,
  152:         .415827895143593195825,
  153:         .420969294644237379543,
  154:         .426084395310681429691,
  155:         .431173464818130014464,
  156:         .436236766774527495726,
  157:         .441274560805140936281,
  158:         .446287102628048160113,
  159:         .451274644139630254358,
  160:         .456237433481874177232,
  161:         .461175715122408291790,
  162:         .466089729924533457960,
  163:         .470979715219073113985,
  164:         .475845904869856894947,
  165:         .480688529345570714212,
  166:         .485507815781602403149,
  167:         .490303988045525329653,
  168:         .495077266798034543171,
  169:         .499827869556611403822,
  170:         .504556010751912253908,
  171:         .509261901790523552335,
  172:         .513945751101346104405,
  173:         .518607764208354637958,
  174:         .523248143765158602036,
  175:         .527867089620485785417,
  176:         .532464798869114019908,
  177:         .537041465897345915436,
  178:         .541597282432121573947,
  179:         .546132437597407260909,
  180:         .550647117952394182793,
  181:         .555141507540611200965,
  182:         .559615787935399566777,
  183:         .564070138285387656651,
  184:         .568504735352689749561,
  185:         .572919753562018740922,
  186:         .577315365035246941260,
  187:         .581691739635061821900,
  188:         .586049045003164792433,
  189:         .590387446602107957005,
  190:         .594707107746216934174,
  191:         .599008189645246602594,
  192:         .603290851438941899687,
  193:         .607555250224322662688,
  194:         .611801541106615331955,
  195:         .616029877215623855590,
  196:         .620240409751204424537,
  197:         .624433288012369303032,
  198:         .628608659422752680256,
  199:         .632766669570628437213,
  200:         .636907462236194987781,
  201:         .641031179420679109171,
  202:         .645137961373620782978,
  203:         .649227946625615004450,
  204:         .653301272011958644725,
  205:         .657358072709030238911,
  206:         .661398482245203922502,
  207:         .665422632544505177065,
  208:         .669430653942981734871,
  209:         .673422675212350441142,
  210:         .677398823590920073911,
  211:         .681359224807238206267,
  212:         .685304003098281100392,
  213:         .689233281238557538017,
  214:         .693147180560117703862
  215: };
  216: 
  217: static const double logF_tail[N+1] = {
  218:         0.,
  219:         -.00000000000000543229938420049,
  220:          .00000000000000172745674997061,
  221:         -.00000000000001323017818229233,
  222:         -.00000000000001154527628289872,
  223:         -.00000000000000466529469958300,
  224:          .00000000000005148849572685810,
  225:         -.00000000000002532168943117445,
  226:         -.00000000000005213620639136504,
  227:         -.00000000000001819506003016881,
  228:          .00000000000006329065958724544,
  229:          .00000000000008614512936087814,
  230:         -.00000000000007355770219435028,
  231:          .00000000000009638067658552277,
  232:          .00000000000007598636597194141,
  233:          .00000000000002579999128306990,
  234:         -.00000000000004654729747598444,
  235:         -.00000000000007556920687451336,
  236:          .00000000000010195735223708472,
  237:         -.00000000000017319034406422306,
  238:         -.00000000000007718001336828098,
  239:          .00000000000010980754099855238,
  240:         -.00000000000002047235780046195,
  241:         -.00000000000008372091099235912,
  242:          .00000000000014088127937111135,
  243:          .00000000000012869017157588257,
  244:          .00000000000017788850778198106,
  245:          .00000000000006440856150696891,
  246:          .00000000000016132822667240822,
  247:         -.00000000000007540916511956188,
  248:         -.00000000000000036507188831790,
  249:          .00000000000009120937249914984,
  250:          .00000000000018567570959796010,
  251:         -.00000000000003149265065191483,
  252:         -.00000000000009309459495196889,
  253:          .00000000000017914338601329117,
  254:         -.00000000000001302979717330866,
  255:          .00000000000023097385217586939,
  256:          .00000000000023999540484211737,
  257:          .00000000000015393776174455408,
  258:         -.00000000000036870428315837678,
  259:          .00000000000036920375082080089,
  260:         -.00000000000009383417223663699,
  261:          .00000000000009433398189512690,
  262:          .00000000000041481318704258568,
  263:         -.00000000000003792316480209314,
  264:          .00000000000008403156304792424,
  265:         -.00000000000034262934348285429,
  266:          .00000000000043712191957429145,
  267:         -.00000000000010475750058776541,
  268:         -.00000000000011118671389559323,
  269:          .00000000000037549577257259853,
  270:          .00000000000013912841212197565,
  271:          .00000000000010775743037572640,
  272:          .00000000000029391859187648000,
  273:         -.00000000000042790509060060774,
  274:          .00000000000022774076114039555,
  275:          .00000000000010849569622967912,
  276:         -.00000000000023073801945705758,
  277:          .00000000000015761203773969435,
  278:          .00000000000003345710269544082,
  279:         -.00000000000041525158063436123,
  280:          .00000000000032655698896907146,
  281:         -.00000000000044704265010452446,
  282:          .00000000000034527647952039772,
  283:         -.00000000000007048962392109746,
  284:          .00000000000011776978751369214,
  285:         -.00000000000010774341461609578,
  286:          .00000000000021863343293215910,
  287:          .00000000000024132639491333131,
  288:          .00000000000039057462209830700,
  289:         -.00000000000026570679203560751,
  290:          .00000000000037135141919592021,
  291:         -.00000000000017166921336082431,
  292:         -.00000000000028658285157914353,
  293:         -.00000000000023812542263446809,
  294:          .00000000000006576659768580062,
  295:         -.00000000000028210143846181267,
  296:          .00000000000010701931762114254,
  297:          .00000000000018119346366441110,
  298:          .00000000000009840465278232627,
  299:         -.00000000000033149150282752542,
  300:         -.00000000000018302857356041668,
  301:         -.00000000000016207400156744949,
  302:          .00000000000048303314949553201,
  303:         -.00000000000071560553172382115,
  304:          .00000000000088821239518571855,
  305:         -.00000000000030900580513238244,
  306:         -.00000000000061076551972851496,
  307:          .00000000000035659969663347830,
  308:          .00000000000035782396591276383,
  309:         -.00000000000046226087001544578,
  310:          .00000000000062279762917225156,
  311:          .00000000000072838947272065741,
  312:          .00000000000026809646615211673,
  313:         -.00000000000010960825046059278,
  314:          .00000000000002311949383800537,
  315:         -.00000000000058469058005299247,
  316:         -.00000000000002103748251144494,
  317:         -.00000000000023323182945587408,
  318:         -.00000000000042333694288141916,
  319:         -.00000000000043933937969737844,
  320:          .00000000000041341647073835565,
  321:          .00000000000006841763641591466,
  322:          .00000000000047585534004430641,
  323:          .00000000000083679678674757695,
  324:         -.00000000000085763734646658640,
  325:          .00000000000021913281229340092,
  326:         -.00000000000062242842536431148,
  327:         -.00000000000010983594325438430,
  328:          .00000000000065310431377633651,
  329:         -.00000000000047580199021710769,
  330:         -.00000000000037854251265457040,
  331:          .00000000000040939233218678664,
  332:          .00000000000087424383914858291,
  333:          .00000000000025218188456842882,
  334:         -.00000000000003608131360422557,
  335:         -.00000000000050518555924280902,
  336:          .00000000000078699403323355317,
  337:         -.00000000000067020876961949060,
  338:          .00000000000016108575753932458,
  339:          .00000000000058527188436251509,
  340:         -.00000000000035246757297904791,
  341:         -.00000000000018372084495629058,
  342:          .00000000000088606689813494916,
  343:          .00000000000066486268071468700,
  344:          .00000000000063831615170646519,
  345:          .00000000000025144230728376072,
  346:         -.00000000000017239444525614834
  347: };
  348: 
  349: /*
  350:  * Extra precision variant, returning struct {double a, b;};
  351:  * log(x) = a+b to 63 bits, with a rounded to 26 bits.
  352:  */
  353: struct Double
  354: __log__D(double x)
  355: {
  356:         int m, j;
  357:         double F, f, g, q, u, v, u2;
  358:         volatile double u1;
  359:         struct Double r;
  360: 
  361:         /* Argument reduction: 1 <= g < 2; x/2^m = g;  */
  362:         /* y = F*(1 + f/F) for |f| <= 2^-8             */
  363: 
  364:         m = logb(x);
  365:         g = ldexp(x, -m);
  366:         if (m == -1022) {
  367:                 j = logb(g);
  368:                 m += j;
  369:                 g = ldexp(g, -j);
  370:         }
  371:         j = N*(g-1) + .5;
  372:         F = (1.0/N) * j + 1;
  373:         f = g - F;
  374: 
  375:         g = 1/(2*F+f);
  376:         u = 2*f*g;
  377:         v = u*u;
  378:         q = u*v*(A1 + v*(A2 + v*(A3 + v*A4)));
  379:         if (m | j) {
  380:                 u1 = u + 513;
  381:                 u1 -= 513;
  382:         }
  383:         else {
  384:                 u1 = u;
  385:                 TRUNC(u1);
  386:         }
  387:         u2 = (2.0*(f - F*u1) - u1*f) * g;
  388: 
  389:         u1 += m*logF_head[N] + logF_head[j];
  390: 
  391:         u2 +=  logF_tail[j]; u2 += q;
  392:         u2 += logF_tail[N]*m;
  393:         r.a = u1 + u2;                 /* Only difference is here */
  394:         TRUNC(r.a);
  395:         r.b = (u1 - r.a) + u2;
  396:         return (r);
  397: }