b_log__D.c

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/*  $OpenBSD: b_log__D.c,v 1.4 2009/10/27 23:59:29 deraadt Exp $    */
/*
 * 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.
 */

#include "math.h"
#include "math_private.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
*/

#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 const double A1 =      .08333333333333178827;
static const double A2 =      .01250000000377174923;
static const double A3 =     .002232139987919447809;
static const double A4 =    .0004348877777076145742;

static const 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 const 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
};

/*
 * Extra precision variant, returning struct {double a, b;};
 * log(x) = a+b to 63 bits, with a rounded to 26 bits.
 */
struct Double
__log__D(double x)
{
    int m, j;
    double F, f, g, q, u, v, u2;
    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 = logb(x);
    g = ldexp(x, -m);
    if (m == -1022) {
        j = logb(g);
        m += j;
        g = ldexp(g, -j);
    }
    j = 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);
}