newlib/libm/ld80/b_logl.c

   1 /*-
   2  * SPDX-License-Identifier: BSD-3-Clause
   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. Neither the name of the University nor the names of its contributors
  16  *    may be used to endorse or promote products derived from this software
  17  *    without specific prior written permission.
  18  *
  19  * THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND
  20  * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
  21  * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
  22  * ARE DISCLAIMED.  IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE
  23  * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
  24  * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
  25  * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
  26  * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
  27  * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
  28  * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
  29  * SUCH DAMAGE.
  30  */
  31
  32 /*
  33  * See bsdsrc/b_log.c for implementation details.
  34  *
  35  * bsdrc/b_log.c converted to long double by Steven G. Kargl.
  36  */
  37
  38 #define N 128
  39
  40 /*
  41  * Coefficients in the polynomial approximation of log(1+f/F).
  42  * Domain of x is [0,1./256] with 2**(-84.48) precision.
  43  */
  44 static const union IEEEl2bits
  45     a1u = LD80C(0xaaaaaaaaaaaaaaab,    -4,  8.33333333333333333356e-02L),
  46     a2u = LD80C(0xcccccccccccccd29,    -7,  1.25000000000000000781e-02L),
  47     a3u = LD80C(0x9249249241ed3764,    -9,  2.23214285711721994134e-03L),
  48     a4u = LD80C(0xe38e959e1e7e01cf,   -12,  4.34030476540000360640e-04L);
  49 #define A1      (a1u.e)
  50 #define A2      (a2u.e)
  51 #define A3      (a3u.e)
  52 #define A4      (a4u.e)
  53
  54 /*
  55  * Table of log(Fj) = logF_head[j] + logF_tail[j], for Fj = 1+j/128.
  56  * Used for generation of extend precision logarithms.
  57  * The constant 35184372088832 is 2^45, so the divide is exact.
  58  * It ensures correct reading of logF_head, even for inaccurate
  59  * decimal-to-binary conversion routines.  (Everybody gets the
  60  * right answer for integers less than 2^53.)
  61  * Values for log(F) were generated using error < 10^-57 absolute
  62  * with the bc -l package.
  63  */
  64
  65 static double logF_head[N+1] = {
  66         0.,
  67         .007782140442060381246,
  68         .015504186535963526694,
  69         .023167059281547608406,
  70         .030771658666765233647,
  71         .038318864302141264488,
  72         .045809536031242714670,
  73         .053244514518837604555,
  74         .060624621816486978786,
  75         .067950661908525944454,
  76         .075223421237524235039,
  77         .082443669210988446138,
  78         .089612158689760690322,
  79         .096729626458454731618,
  80         .103796793681567578460,
  81         .110814366340264314203,
  82         .117783035656430001836,
  83         .124703478501032805070,
  84         .131576357788617315236,
  85         .138402322859292326029,
  86         .145182009844575077295,
  87         .151916042025732167530,
  88         .158605030176659056451,
  89         .165249572895390883786,
  90         .171850256926518341060,
  91         .178407657472689606947,
  92         .184922338493834104156,
  93         .191394852999565046047,
  94         .197825743329758552135,
  95         .204215541428766300668,
  96         .210564769107350002741,
  97         .216873938300523150246,
  98         .223143551314024080056,
  99         .229374101064877322642,
 100         .235566071312860003672,
 101         .241719936886966024758,
 102         .247836163904594286577,
 103         .253915209980732470285,
 104         .259957524436686071567,
 105         .265963548496984003577,
 106         .271933715484010463114,
 107         .277868451003087102435,
 108         .283768173130738432519,
 109         .289633292582948342896,
 110         .295464212893421063199,
 111         .301261330578199704177,
 112         .307025035294827830512,
 113         .312755710004239517729,
 114         .318453731118097493890,
 115         .324119468654316733591,
 116         .329753286372579168528,
 117         .335355541920762334484,
 118         .340926586970454081892,
 119         .346466767346100823488,
 120         .351976423156884266063,
 121         .357455888922231679316,
 122         .362905493689140712376,
 123         .368325561158599157352,
 124         .373716409793814818840,
 125         .379078352934811846353,
 126         .384411698910298582632,
 127         .389716751140440464951,
 128         .394993808240542421117,
 129         .400243164127459749579,
 130         .405465108107819105498,
 131         .410659924985338875558,
 132         .415827895143593195825,
 133         .420969294644237379543,
 134         .426084395310681429691,
 135         .431173464818130014464,
 136         .436236766774527495726,
 137         .441274560805140936281,
 138         .446287102628048160113,
 139         .451274644139630254358,
 140         .456237433481874177232,
 141         .461175715122408291790,
 142         .466089729924533457960,
 143         .470979715219073113985,
 144         .475845904869856894947,
 145         .480688529345570714212,
 146         .485507815781602403149,
 147         .490303988045525329653,
 148         .495077266798034543171,
 149         .499827869556611403822,
 150         .504556010751912253908,
 151         .509261901790523552335,
 152         .513945751101346104405,
 153         .518607764208354637958,
 154         .523248143765158602036,
 155         .527867089620485785417,
 156         .532464798869114019908,
 157         .537041465897345915436,
 158         .541597282432121573947,
 159         .546132437597407260909,
 160         .550647117952394182793,
 161         .555141507540611200965,
 162         .559615787935399566777,
 163         .564070138285387656651,
 164         .568504735352689749561,
 165         .572919753562018740922,
 166         .577315365035246941260,
 167         .581691739635061821900,
 168         .586049045003164792433,
 169         .590387446602107957005,
 170         .594707107746216934174,
 171         .599008189645246602594,
 172         .603290851438941899687,
 173         .607555250224322662688,
 174         .611801541106615331955,
 175         .616029877215623855590,
 176         .620240409751204424537,
 177         .624433288012369303032,
 178         .628608659422752680256,
 179         .632766669570628437213,
 180         .636907462236194987781,
 181         .641031179420679109171,
 182         .645137961373620782978,
 183         .649227946625615004450,
 184         .653301272011958644725,
 185         .657358072709030238911,
 186         .661398482245203922502,
 187         .665422632544505177065,
 188         .669430653942981734871,
 189         .673422675212350441142,
 190         .677398823590920073911,
 191         .681359224807238206267,
 192         .685304003098281100392,
 193         .689233281238557538017,
 194         .693147180560117703862
 195 };
 196
 197 static double logF_tail[N+1] = {
 198         0.,
 199         -.00000000000000543229938420049,
 200          .00000000000000172745674997061,
 201         -.00000000000001323017818229233,
 202         -.00000000000001154527628289872,
 203         -.00000000000000466529469958300,
 204          .00000000000005148849572685810,
 205         -.00000000000002532168943117445,
 206         -.00000000000005213620639136504,
 207         -.00000000000001819506003016881,
 208          .00000000000006329065958724544,
 209          .00000000000008614512936087814,
 210         -.00000000000007355770219435028,
 211          .00000000000009638067658552277,
 212          .00000000000007598636597194141,
 213          .00000000000002579999128306990,
 214         -.00000000000004654729747598444,
 215         -.00000000000007556920687451336,
 216          .00000000000010195735223708472,
 217         -.00000000000017319034406422306,
 218         -.00000000000007718001336828098,
 219          .00000000000010980754099855238,
 220         -.00000000000002047235780046195,
 221         -.00000000000008372091099235912,
 222          .00000000000014088127937111135,
 223          .00000000000012869017157588257,
 224          .00000000000017788850778198106,
 225          .00000000000006440856150696891,
 226          .00000000000016132822667240822,
 227         -.00000000000007540916511956188,
 228         -.00000000000000036507188831790,
 229          .00000000000009120937249914984,
 230          .00000000000018567570959796010,
 231         -.00000000000003149265065191483,
 232         -.00000000000009309459495196889,
 233          .00000000000017914338601329117,
 234         -.00000000000001302979717330866,
 235          .00000000000023097385217586939,
 236          .00000000000023999540484211737,
 237          .00000000000015393776174455408,
 238         -.00000000000036870428315837678,
 239          .00000000000036920375082080089,
 240         -.00000000000009383417223663699,
 241          .00000000000009433398189512690,
 242          .00000000000041481318704258568,
 243         -.00000000000003792316480209314,
 244          .00000000000008403156304792424,
 245         -.00000000000034262934348285429,
 246          .00000000000043712191957429145,
 247         -.00000000000010475750058776541,
 248         -.00000000000011118671389559323,
 249          .00000000000037549577257259853,
 250          .00000000000013912841212197565,
 251          .00000000000010775743037572640,
 252          .00000000000029391859187648000,
 253         -.00000000000042790509060060774,
 254          .00000000000022774076114039555,
 255          .00000000000010849569622967912,
 256         -.00000000000023073801945705758,
 257          .00000000000015761203773969435,
 258          .00000000000003345710269544082,
 259         -.00000000000041525158063436123,
 260          .00000000000032655698896907146,
 261         -.00000000000044704265010452446,
 262          .00000000000034527647952039772,
 263         -.00000000000007048962392109746,
 264          .00000000000011776978751369214,
 265         -.00000000000010774341461609578,
 266          .00000000000021863343293215910,
 267          .00000000000024132639491333131,
 268          .00000000000039057462209830700,
 269         -.00000000000026570679203560751,
 270          .00000000000037135141919592021,
 271         -.00000000000017166921336082431,
 272         -.00000000000028658285157914353,
 273         -.00000000000023812542263446809,
 274          .00000000000006576659768580062,
 275         -.00000000000028210143846181267,
 276          .00000000000010701931762114254,
 277          .00000000000018119346366441110,
 278          .00000000000009840465278232627,
 279         -.00000000000033149150282752542,
 280         -.00000000000018302857356041668,
 281         -.00000000000016207400156744949,
 282          .00000000000048303314949553201,
 283         -.00000000000071560553172382115,
 284          .00000000000088821239518571855,
 285         -.00000000000030900580513238244,
 286         -.00000000000061076551972851496,
 287          .00000000000035659969663347830,
 288          .00000000000035782396591276383,
 289         -.00000000000046226087001544578,
 290          .00000000000062279762917225156,
 291          .00000000000072838947272065741,
 292          .00000000000026809646615211673,
 293         -.00000000000010960825046059278,
 294          .00000000000002311949383800537,
 295         -.00000000000058469058005299247,
 296         -.00000000000002103748251144494,
 297         -.00000000000023323182945587408,
 298         -.00000000000042333694288141916,
 299         -.00000000000043933937969737844,
 300          .00000000000041341647073835565,
 301          .00000000000006841763641591466,
 302          .00000000000047585534004430641,
 303          .00000000000083679678674757695,
 304         -.00000000000085763734646658640,
 305          .00000000000021913281229340092,
 306         -.00000000000062242842536431148,
 307         -.00000000000010983594325438430,
 308          .00000000000065310431377633651,
 309         -.00000000000047580199021710769,
 310         -.00000000000037854251265457040,
 311          .00000000000040939233218678664,
 312          .00000000000087424383914858291,
 313          .00000000000025218188456842882,
 314         -.00000000000003608131360422557,
 315         -.00000000000050518555924280902,
 316          .00000000000078699403323355317,
 317         -.00000000000067020876961949060,
 318          .00000000000016108575753932458,
 319          .00000000000058527188436251509,
 320         -.00000000000035246757297904791,
 321         -.00000000000018372084495629058,
 322          .00000000000088606689813494916,
 323          .00000000000066486268071468700,
 324          .00000000000063831615170646519,
 325          .00000000000025144230728376072,
 326         -.00000000000017239444525614834
 327 };
 328 /*
 329  * Extra precision variant, returning struct {double a, b;};
 330  * log(x) = a + b to 63 bits, with 'a' rounded to 24 bits.
 331  */
 332 static struct Double
 333 __log__D(long double x)
 334 {
 335         int m, j;
 336         long double F, f, g, q, u, v, u1, u2;
 337         struct Double r;
 338
 339         /*
 340          * Argument reduction: 1 <= g < 2; x/2^m = g;
 341          * y = F*(1 + f/F) for |f| <= 2^-8
 342          */
 343         g = frexpl(x, &m);
 344         g *= 2;
 345         m--;
 346         if (m == DBL_MIN_EXP - 1) {
 347                 j = ilogbl(g);
 348                 m += j;
 349                 g = ldexpl(g, -j);
 350         }
 351         j = N * (g - 1) + 0.5L;
 352         F = (1.L / N) * j + 1;
 353         f = g - F;
 354
 355         g = 1 / (2 * F + f);
 356         u = 2 * f * g;
 357         v = u * u;
 358         q = u * v * (A1 + v * (A2 + v * (A3 + v * A4)));
 359         if (m | j) {
 360                 u1 = u + 513;
 361                 u1 -= 513;
 362         } else {
 363                 u1 = (float)u;
 364         }
 365         u2 = (2 * (f - F * u1) - u1 * f) * g;
 366
 367         u1 += m * (long double)logF_head[N] + logF_head[j];
 368
 369         u2 += logF_tail[j];
 370         u2 += q;
 371         u2 += logF_tail[N] * m;
 372         r.a = (float)(u1 + u2);         /* Only difference is here. */
 373         r.b = (u1 - r.a) + u2;
 374         return (r);
 375 }