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log.c

#include "FEATURE/uwin"

#if !_UWIN

void _STUB_log(){}

#else

/*
 * Copyright (c) 1992, 1993
 *    The Regents of the University of California.  All rights reserved.
 *
 * Redistribution and use in source and binary forms, with or without
 * modification, are permitted provided that the following conditions
 * are met:
 * 1. Redistributions of source code must retain the above copyright
 *    notice, this list of conditions and the following disclaimer.
 * 2. Redistributions in binary form must reproduce the above copyright
 *    notice, this list of conditions and the following disclaimer in the
 *    documentation and/or other materials provided with the distribution.
 * 3. Neither the name of the University nor the names of its contributors
 *    may be used to endorse or promote products derived from this software
 *    without specific prior written permission.
 *
 * THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND
 * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
 * ARE DISCLAIMED.  IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE
 * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
 * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
 * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
 * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
 * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
 * SUCH DAMAGE.
 */

#ifndef lint
static char sccsid[] = "@(#)log.c   8.2 (Berkeley) 11/30/93";
#endif /* not lint */

#include <math.h>
#include <errno.h>

#include "mathimpl.h"

/* Table-driven natural logarithm.
 *
 * This code was derived, with minor modifications, from:
 *    Peter Tang, "Table-Driven Implementation of the
 *    Logarithm in IEEE Floating-Point arithmetic." ACM Trans.
 *    Math Software, vol 16. no 4, pp 378-400, Dec 1990).
 *
 * Calculates log(2^m*F*(1+f/F)), |f/j| <= 1/256,
 * where F = j/128 for j an integer in [0, 128].
 *
 * log(2^m) = log2_hi*m + log2_tail*m
 * since m is an integer, the dominant term is exact.
 * m has at most 10 digits (for subnormal numbers),
 * and log2_hi has 11 trailing zero bits.
 *
 * log(F) = logF_hi[j] + logF_lo[j] is in tabular form in log_table.h
 * logF_hi[] + 512 is exact.
 *
 * log(1+f/F) = 2*f/(2*F + f) + 1/12 * (2*f/(2*F + f))**3 + ...
 * the leading term is calculated to extra precision in two
 * parts, the larger of which adds exactly to the dominant
 * m and F terms.
 * There are two cases:
 *    1. when m, j are non-zero (m | j), use absolute
 *       precision for the leading term.
 *    2. when m = j = 0, |1-x| < 1/256, and log(x) ~= (x-1).
 *       In this case, use a relative precision of 24 bits.
 * (This is done differently in the original paper)
 *
 * Special cases:
 *    0     return signalling -Inf
 *    neg   return signalling NaN
 *    +Inf  return +Inf
*/

#if defined(vax) || defined(tahoe)
#define _IEEE           0
#define TRUNC(x)  x = (double) (float) (x)
#else
#define _IEEE           1
#define endian          (((*(int *) &one)) ? 1 : 0)
#define TRUNC(x)  *(((int *) &x) + endian) &= 0xf8000000
#define infnan(x) 0.0
#endif

#define N 128

/* Table of log(Fj) = logF_head[j] + logF_tail[j], for Fj = 1+j/128.
 * Used for generation of extend precision logarithms.
 * The constant 35184372088832 is 2^45, so the divide is exact.
 * It ensures correct reading of logF_head, even for inaccurate
 * decimal-to-binary conversion routines.  (Everybody gets the
 * right answer for integers less than 2^53.)
 * Values for log(F) were generated using error < 10^-57 absolute
 * with the bc -l package.
*/
static double     A1 =    .08333333333333178827;
static double     A2 =    .01250000000377174923;
static double     A3 =   .002232139987919447809;
static double     A4 =  .0004348877777076145742;

static double logF_head[N+1] = {
      0.,
      .007782140442060381246,
      .015504186535963526694,
      .023167059281547608406,
      .030771658666765233647,
      .038318864302141264488,
      .045809536031242714670,
      .053244514518837604555,
      .060624621816486978786,
      .067950661908525944454,
      .075223421237524235039,
      .082443669210988446138,
      .089612158689760690322,
      .096729626458454731618,
      .103796793681567578460,
      .110814366340264314203,
      .117783035656430001836,
      .124703478501032805070,
      .131576357788617315236,
      .138402322859292326029,
      .145182009844575077295,
      .151916042025732167530,
      .158605030176659056451,
      .165249572895390883786,
      .171850256926518341060,
      .178407657472689606947,
      .184922338493834104156,
      .191394852999565046047,
      .197825743329758552135,
      .204215541428766300668,
      .210564769107350002741,
      .216873938300523150246,
      .223143551314024080056,
      .229374101064877322642,
      .235566071312860003672,
      .241719936886966024758,
      .247836163904594286577,
      .253915209980732470285,
      .259957524436686071567,
      .265963548496984003577,
      .271933715484010463114,
      .277868451003087102435,
      .283768173130738432519,
      .289633292582948342896,
      .295464212893421063199,
      .301261330578199704177,
      .307025035294827830512,
      .312755710004239517729,
      .318453731118097493890,
      .324119468654316733591,
      .329753286372579168528,
      .335355541920762334484,
      .340926586970454081892,
      .346466767346100823488,
      .351976423156884266063,
      .357455888922231679316,
      .362905493689140712376,
      .368325561158599157352,
      .373716409793814818840,
      .379078352934811846353,
      .384411698910298582632,
      .389716751140440464951,
      .394993808240542421117,
      .400243164127459749579,
      .405465108107819105498,
      .410659924985338875558,
      .415827895143593195825,
      .420969294644237379543,
      .426084395310681429691,
      .431173464818130014464,
      .436236766774527495726,
      .441274560805140936281,
      .446287102628048160113,
      .451274644139630254358,
      .456237433481874177232,
      .461175715122408291790,
      .466089729924533457960,
      .470979715219073113985,
      .475845904869856894947,
      .480688529345570714212,
      .485507815781602403149,
      .490303988045525329653,
      .495077266798034543171,
      .499827869556611403822,
      .504556010751912253908,
      .509261901790523552335,
      .513945751101346104405,
      .518607764208354637958,
      .523248143765158602036,
      .527867089620485785417,
      .532464798869114019908,
      .537041465897345915436,
      .541597282432121573947,
      .546132437597407260909,
      .550647117952394182793,
      .555141507540611200965,
      .559615787935399566777,
      .564070138285387656651,
      .568504735352689749561,
      .572919753562018740922,
      .577315365035246941260,
      .581691739635061821900,
      .586049045003164792433,
      .590387446602107957005,
      .594707107746216934174,
      .599008189645246602594,
      .603290851438941899687,
      .607555250224322662688,
      .611801541106615331955,
      .616029877215623855590,
      .620240409751204424537,
      .624433288012369303032,
      .628608659422752680256,
      .632766669570628437213,
      .636907462236194987781,
      .641031179420679109171,
      .645137961373620782978,
      .649227946625615004450,
      .653301272011958644725,
      .657358072709030238911,
      .661398482245203922502,
      .665422632544505177065,
      .669430653942981734871,
      .673422675212350441142,
      .677398823590920073911,
      .681359224807238206267,
      .685304003098281100392,
      .689233281238557538017,
      .693147180560117703862
};

static double logF_tail[N+1] = {
      0.,
      -.00000000000000543229938420049,
       .00000000000000172745674997061,
      -.00000000000001323017818229233,
      -.00000000000001154527628289872,
      -.00000000000000466529469958300,
       .00000000000005148849572685810,
      -.00000000000002532168943117445,
      -.00000000000005213620639136504,
      -.00000000000001819506003016881,
       .00000000000006329065958724544,
       .00000000000008614512936087814,
      -.00000000000007355770219435028,
       .00000000000009638067658552277,
       .00000000000007598636597194141,
       .00000000000002579999128306990,
      -.00000000000004654729747598444,
      -.00000000000007556920687451336,
       .00000000000010195735223708472,
      -.00000000000017319034406422306,
      -.00000000000007718001336828098,
       .00000000000010980754099855238,
      -.00000000000002047235780046195,
      -.00000000000008372091099235912,
       .00000000000014088127937111135,
       .00000000000012869017157588257,
       .00000000000017788850778198106,
       .00000000000006440856150696891,
       .00000000000016132822667240822,
      -.00000000000007540916511956188,
      -.00000000000000036507188831790,
       .00000000000009120937249914984,
       .00000000000018567570959796010,
      -.00000000000003149265065191483,
      -.00000000000009309459495196889,
       .00000000000017914338601329117,
      -.00000000000001302979717330866,
       .00000000000023097385217586939,
       .00000000000023999540484211737,
       .00000000000015393776174455408,
      -.00000000000036870428315837678,
       .00000000000036920375082080089,
      -.00000000000009383417223663699,
       .00000000000009433398189512690,
       .00000000000041481318704258568,
      -.00000000000003792316480209314,
       .00000000000008403156304792424,
      -.00000000000034262934348285429,
       .00000000000043712191957429145,
      -.00000000000010475750058776541,
      -.00000000000011118671389559323,
       .00000000000037549577257259853,
       .00000000000013912841212197565,
       .00000000000010775743037572640,
       .00000000000029391859187648000,
      -.00000000000042790509060060774,
       .00000000000022774076114039555,
       .00000000000010849569622967912,
      -.00000000000023073801945705758,
       .00000000000015761203773969435,
       .00000000000003345710269544082,
      -.00000000000041525158063436123,
       .00000000000032655698896907146,
      -.00000000000044704265010452446,
       .00000000000034527647952039772,
      -.00000000000007048962392109746,
       .00000000000011776978751369214,
      -.00000000000010774341461609578,
       .00000000000021863343293215910,
       .00000000000024132639491333131,
       .00000000000039057462209830700,
      -.00000000000026570679203560751,
       .00000000000037135141919592021,
      -.00000000000017166921336082431,
      -.00000000000028658285157914353,
      -.00000000000023812542263446809,
       .00000000000006576659768580062,
      -.00000000000028210143846181267,
       .00000000000010701931762114254,
       .00000000000018119346366441110,
       .00000000000009840465278232627,
      -.00000000000033149150282752542,
      -.00000000000018302857356041668,
      -.00000000000016207400156744949,
       .00000000000048303314949553201,
      -.00000000000071560553172382115,
       .00000000000088821239518571855,
      -.00000000000030900580513238244,
      -.00000000000061076551972851496,
       .00000000000035659969663347830,
       .00000000000035782396591276383,
      -.00000000000046226087001544578,
       .00000000000062279762917225156,
       .00000000000072838947272065741,
       .00000000000026809646615211673,
      -.00000000000010960825046059278,
       .00000000000002311949383800537,
      -.00000000000058469058005299247,
      -.00000000000002103748251144494,
      -.00000000000023323182945587408,
      -.00000000000042333694288141916,
      -.00000000000043933937969737844,
       .00000000000041341647073835565,
       .00000000000006841763641591466,
       .00000000000047585534004430641,
       .00000000000083679678674757695,
      -.00000000000085763734646658640,
       .00000000000021913281229340092,
      -.00000000000062242842536431148,
      -.00000000000010983594325438430,
       .00000000000065310431377633651,
      -.00000000000047580199021710769,
      -.00000000000037854251265457040,
       .00000000000040939233218678664,
       .00000000000087424383914858291,
       .00000000000025218188456842882,
      -.00000000000003608131360422557,
      -.00000000000050518555924280902,
       .00000000000078699403323355317,
      -.00000000000067020876961949060,
       .00000000000016108575753932458,
       .00000000000058527188436251509,
      -.00000000000035246757297904791,
      -.00000000000018372084495629058,
       .00000000000088606689813494916,
       .00000000000066486268071468700,
       .00000000000063831615170646519,
       .00000000000025144230728376072,
      -.00000000000017239444525614834
};

#if !_lib_log

extern double
#ifdef _ANSI_SOURCE
log(double x)
#else
log(x) double x;
#endif
{
      int m, j;
      double F, f, g, q, u, u2, v, zero = 0.0, one = 1.0;
      volatile double u1;

      /* Catch special cases */
      if (x <= 0)
            if (_IEEE && x == zero) /* log(0) = -Inf */
                  return (-one/zero);
            else if (_IEEE)         /* log(neg) = NaN */
                  return (zero/zero);
            else if (x == zero)     /* NOT REACHED IF _IEEE */
                  return (infnan(-ERANGE));
            else
                  return (infnan(EDOM));
      else if (!finite(x))
            if (_IEEE)        /* x = NaN, Inf */
                  return (x+x);
            else
                  return (infnan(ERANGE));
      
      /* Argument reduction: 1 <= g < 2; x/2^m = g;   */
      /* y = F*(1 + f/F) for |f| <= 2^-8        */

      m = logb(x);
      g = ldexp(x, -m);
      if (_IEEE && m == -1022) {
            j = logb(g), m += j;
            g = ldexp(g, -j);
      }
      j = N*(g-1) + .5;
      F = (1.0/N) * j + 1;    /* F*128 is an integer in [128, 512] */
      f = g - F;

      /* Approximate expansion for log(1+f/F) ~= u + q */
      g = 1/(2*F+f);
      u = 2*f*g;
      v = u*u;
      q = u*v*(A1 + v*(A2 + v*(A3 + v*A4)));

    /* case 1: u1 = u rounded to 2^-43 absolute.  Since u < 2^-8,
     *             u1 has at most 35 bits, and F*u1 is exact, as F has < 8 bits.
     *         It also adds exactly to |m*log2_hi + log_F_head[j] | < 750
    */
      if (m | j)
            u1 = u + 513, u1 -= 513;

    /* case 2:    |1-x| < 1/256. The m- and j- dependent terms are zero;
     *            u1 = u to 24 bits.
    */
      else
            u1 = u, TRUNC(u1);
      u2 = (2.0*(f - F*u1) - u1*f) * g;
                  /* u1 + u2 = 2f/(2F+f) to extra precision.      */

      /* log(x) = log(2^m*F*(1+f/F)) =                      */
      /* (m*log2_hi+logF_head[j]+u1) + (m*log2_lo+logF_tail[j]+q);      */
      /* (exact) + (tiny)                                   */

      u1 += m*logF_head[N] + logF_head[j];            /* exact */
      u2 = (u2 + logF_tail[j]) + q;             /* tiny */
      u2 += logF_tail[N]*m;
      return (u1 + u2);
}

#endif

/*
 * Extra precision variant, returning struct {double a, b;};
 * log(x) = a+b to 63 bits, with a is rounded to 26 bits.
 */
struct Double
#ifdef _ANSI_SOURCE
__log__D(double x)
#else
__log__D(x) double x;
#endif
{
      int m, j;
      double F, f, g, q, u, v, u2, one = 1.0;
      volatile double u1;
      struct Double r;

      /* Argument reduction: 1 <= g < 2; x/2^m = g;   */
      /* y = F*(1 + f/F) for |f| <= 2^-8        */

      m = (int)logb(x);
      g = ldexp(x, -m);
      if (_IEEE && m == -1022) {
            j = (int)logb(g), m += j;
            g = ldexp(g, -j);
      }
      j = (int)(N*(g-1) + .5);
      F = (1.0/N) * j + 1;
      f = g - F;

      g = 1/(2*F+f);
      u = 2*f*g;
      v = u*u;
      q = u*v*(A1 + v*(A2 + v*(A3 + v*A4)));
      if (m | j)
            u1 = u + 513, u1 -= 513;
      else
            u1 = u, TRUNC(u1);
      u2 = (2.0*(f - F*u1) - u1*f) * g;

      u1 += m*logF_head[N] + logF_head[j];

      u2 +=  logF_tail[j]; u2 += q;
      u2 += logF_tail[N]*m;
      r.a = u1 + u2;                /* Only difference is here */
      TRUNC(r.a);
      r.b = (u1 - r.a) + u2;
      return (r);
}

#endif

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