/*-
 * SPDX-License-Identifier: BSD-3-Clause
 *
 * 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.
 */

/* 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/F| <= 1/256,
 * where F = j/128 for j an integer in [0, 128].
 *
 * log(2^m) = log2_hi*m + log2_tail*m
 * The leading term is exact, because m is an integer,
 * 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 table below.
 * 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 and 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

/*
 * Coefficients in the polynomial approximation of log(1+f/F).
 * Domain of x is [0,1./256] with 2**(-64.187) precision.
 */
static const double
    A1 =  8.3333333333333329e-02, /* 0x3fb55555, 0x55555555 */
    A2 =  1.2499999999943598e-02, /* 0x3f899999, 0x99991a98 */
    A3 =  2.2321527525957776e-03; /* 0x3f624929, 0xe24e70be */

/*
 * 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 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
};
/*
 * Extra precision variant, returning struct {double a, b;};
 * log(x) = a+b to 63 bits, with 'a' rounded to 24 bits.
 */
static struct Double
__log__D(double x)
{
        int m, j;
        double F, f, g, q, u, v, u1, u2;
        struct Double r;

        /*
         * Argument reduction: 1 <= g < 2; x/2^m = g;
         * y = F*(1 + f/F) for |f| <= 2^-8
         */
        g = frexp(x, &m);
        g *= 2;
        m--;
        if (m == -1022) {
                j = ilogb(g);
                m += j;
                g = ldexp(g, -j);
        }
        j = N * (g - 1) + 0.5;
        F = (1. / 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));
        if (m | j) {
                u1 = u + 513;
                u1 -= 513;
        } else {
                u1 = (float)u;
        }
        u2 = (2 * (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 = (float)(u1 + u2);         /* Only difference is here. */
        r.b = (u1 - r.a) + u2;
        return (r);
}