lib/log1p.c

   1 /* Natural logarithm of 1 plus argument.
   2    Copyright (C) 2012-2013 Free Software Foundation, Inc.
   3
   4    This program is free software: you can redistribute it and/or modify
   5    it under the terms of the GNU Lesser General Public License as published by
   6    the Free Software Foundation; either version 3 of the License, or
   7    (at your option) any later version.
   8
   9    This program is distributed in the hope that it will be useful,
  10    but WITHOUT ANY WARRANTY; without even the implied warranty of
  11    MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
  12    GNU Lesser General Public License for more details.
  13
  14    You should have received a copy of the GNU Lesser General Public License
  15    along with this program.  If not, see <http://www.gnu.org/licenses/>.  */
  16
  17 #include <config.h>
  18
  19 /* Specification.  */
  20 #include <math.h>
  21
  22 double
  23 log1p (double x)
  24 {
  25   if (isnand (x))
  26     return x;
  27
  28   if (x <= -1.0)
  29     {
  30       if (x == -1.0)
  31         /* Return -Infinity.  */
  32         return - HUGE_VAL;
  33       else
  34         {
  35           /* Return NaN.  */
  36 #if defined _MSC_VER || (defined __sgi && !defined __GNUC__)
  37           static double zero;
  38           return zero / zero;
  39 #else
  40           return 0.0 / 0.0;
  41 #endif
  42         }
  43     }
  44
  45   if (x < -0.5 || x > 1.0)
  46     return log (1.0 + x);
  47   /* Here -0.5 <= x <= 1.0.  */
  48
  49   if (x == 0.0)
  50     /* Return a zero with the same sign as x.  */
  51     return x;
  52
  53   /* Decompose x into
  54        1 + x = (1 + m/256) * (1 + y)
  55      where
  56        m is an integer, -128 <= m <= 256,
  57        y is a number, |y| <= 1/256.
  58      y is computed as
  59        y = (256 * x - m) / (256 + m).
  60      Then
  61        log(1+x) = log(m/256) + log(1+y)
  62      The first summand is a table lookup.
  63      The second summand is computed
  64        - either through the power series
  65            log(1+y) = y
  66                       - 1/2 * y^2
  67                       + 1/3 * y^3
  68                       - 1/4 * y^4
  69                       + 1/5 * y^5
  70                       - 1/6 * y^6
  71                       + 1/7 * y^7
  72                       - 1/8 * y^8
  73                       + 1/9 * y^9
  74                       - 1/10 * y^10
  75                       + 1/11 * y^11
  76                       - 1/12 * y^12
  77                       + 1/13 * y^13
  78                       - 1/14 * y^14
  79                       + 1/15 * y^15
  80                       - ...
  81        - or as log(1+y) = log((1+z)/(1-z)) = 2 * atanh(z)
  82          where z = y/(2+y)
  83          and atanh(z) is computed through its power series:
  84            atanh(z) = z
  85                       + 1/3 * z^3
  86                       + 1/5 * z^5
  87                       + 1/7 * z^7
  88                       + 1/9 * z^9
  89                       + 1/11 * z^11
  90                       + 1/13 * z^13
  91                       + 1/15 * z^15
  92                       + ...
  93          Since |z| <= 1/511 < 0.002, the relative contribution of the z^9
  94          term is < 1/9*0.002^8 < 2^-60 <= 2^-DBL_MANT_DIG, therefore we
  95          can truncate the series after the z^7 term.  */
  96
  97   {
  98     double m = round (x * 256.0);
  99     double y = ((x * 256.0) - m) / (m + 256.0);
 100     double z = y / (2.0 + y);
 101
 102 /* Coefficients of the power series for atanh(z).  */
 103 #define ATANH_COEFF_1  1.0
 104 #define ATANH_COEFF_3  0.333333333333333333333333333333333333334
 105 #define ATANH_COEFF_5  0.2
 106 #define ATANH_COEFF_7  0.142857142857142857142857142857142857143
 107 #define ATANH_COEFF_9  0.1111111111111111111111111111111111111113
 108 #define ATANH_COEFF_11 0.090909090909090909090909090909090909091
 109 #define ATANH_COEFF_13 0.076923076923076923076923076923076923077
 110 #define ATANH_COEFF_15 0.066666666666666666666666666666666666667
 111
 112     double z2 = z * z;
 113     double atanh_z =
 114       (((ATANH_COEFF_7
 115          * z2 + ATANH_COEFF_5)
 116         * z2 + ATANH_COEFF_3)
 117        * z2 + ATANH_COEFF_1)
 118       * z;
 119
 120     /* log_table[i] = log((i + 128) / 256).
 121        Computed in GNU clisp through
 122          (setf (long-float-digits) 128)
 123          (setq a 0L0)
 124          (setf (long-float-digits) 256)
 125          (dotimes (i 385)
 126            (format t "        ~D,~%"
 127                    (float (log (* (/ (+ i 128) 256) 1L0)) a)))  */
 128     static const double log_table[385] =
 129       {
 130         -0.693147180559945309417232121458176568075,
 131         -0.6853650401178903604697692213970398044,
 132         -0.677642994023980055266378075415729732197,
 133         -0.669980121278410931188432960495886651496,
 134         -0.662375521893191621046203913861404403985,
 135         -0.65482831625780871022347679633437927773,
 136         -0.647337644528651106250552853843513225963,
 137         -0.639902666041133026551361927671647791137,
 138         -0.632522558743510466836625989417756304788,
 139         -0.625196518651437560022666843685547154042,
 140         -0.617923759322357783718626781474514153438,
 141         -0.61070351134887071814907205278986876216,
 142         -0.60353502187025817679728065207969203929,
 143         -0.59641755410139419712166106497071313106,
 144         -0.58935038687830174459117031769420187977,
 145         -0.582332814219655195222425952134964639978,
 146         -0.575364144903561854878438011987654863008,
 147         -0.568443702058988073553825606077313299585,
 148         -0.561570822771226036828515992768693405624,
 149         -0.554744857700826173731906247856527380683,
 150         -0.547965170715447412135297057717612244552,
 151         -0.541231138534103334345428696561292056747,
 152         -0.534542150383306725323860946832334992828,
 153         -0.527897607664638146541620672180936254347,
 154         -0.52129692363328608707713317540302930314,
 155         -0.514739523087127012297831879947234599722,
 156         -0.50822484206593331675332852879892694707,
 157         -0.50175232756031585480793331389686769463,
 158         -0.495321437230025429054660050261215099,
 159         -0.488931639131254417913411735261937295862,
 160         -0.482582411452595671747679308725825054355,
 161         -0.476273242259330949798142595713829069596,
 162         -0.470003629245735553650937031148342064701,
 163         -0.463773079495099479425751396412036696525,
 164         -0.457581109247178400339643902517133157939,
 165         -0.451427243672800141272924605544662667972,
 166         -0.445311016655364052636629355711651820077,
 167         -0.43923197057898186527990882355156990061,
 168         -0.4331896561230192424451526269158655235,
 169         -0.427183632062807368078106194920633178807,
 170         -0.421213465076303550585562626925177406092,
 171         -0.415278729556489003230882088534775334993,
 172         -0.409379007429300711070330899107921801414,
 173         -0.403513887976902632538339065932507598071,
 174         -0.397682967666109433030550215403212372894,
 175         -0.391885849981783528404356583224421075418,
 176         -0.386122145265033447342107580922798666387,
 177         -0.380391470556048421030985561769857535915,
 178         -0.374693449441410693606984907867576972481,
 179         -0.369027711905733333326561361023189215893,
 180         -0.363393894187477327602809309537386757124,
 181         -0.357791638638807479160052541644010369001,
 182         -0.352220593589352099112142921677820359633,
 183         -0.346680413213736728498769933032403617363,
 184         -0.341170757402767124761784665198737642087,
 185         -0.33569129163814153519122263131727209364,
 186         -0.330241686870576856279407775480686721935,
 187         -0.324821619401237656369001967407777741178,
 188         -0.31943077076636122859621528874235306143,
 189         -0.314068827624975851026378775827156709194,
 190         -0.308735481649613269682442058976885699557,
 191         -0.303430429419920096046768517454655701024,
 192         -0.298153372319076331310838085093194799765,
 193         -0.292904016432932602487907019463045397996,
 194         -0.287682072451780927439219005993827431504,
 195         -0.282487255574676923482925918282353780414,
 196         -0.277319285416234343803903228503274262719,
 197         -0.272177885915815673288364959951380595626,
 198         -0.267062785249045246292687241862699949179,
 199         -0.261973715741573968558059642502581569596,
 200         -0.256910413785027239068190798397055267412,
 201         -0.251872619755070079927735679796875342712,
 202         -0.2468600779315257978846419408385075613265,
 203         -0.24187253642048672427253973837916408939,
 204         -0.2369097470783577150364265832942468196375,
 205         -0.2319714654377751430492321958603212094726,
 206         -0.2270574506353460848586128739534071682175,
 207         -0.222167465341154296870334265401817316702,
 208         -0.2173012756899813951520225351537951559,
 209         -0.212458651214193401740613666010165016867,
 210         -0.2076393647782445016154410442673876674964,
 211         -0.202843192514751471266885961812429707545,
 212         -0.1980699137620937948192675366153429027185,
 213         -0.193319311003495979595900706211132426563,
 214         -0.188591169807550022358923589720001638093,
 215         -0.183885278770137362613157202229852743197,
 216         -0.179201429457710992616226033183958974965,
 217         -0.174539416351899677264255125093377869519,
 218         -0.169899036795397472900424896523305726435,
 219         -0.165280090939102924303339903679875604517,
 220         -0.160682381690473465543308397998034325468,
 221         -0.156105714663061654850502877304344269052,
 222         -0.1515498981272009378406898175577424691056,
 223         -0.1470147429618096590348349122269674042104,
 224         -0.142500062607283030157283942253263107981,
 225         -0.1380056730194437167017517619422725179055,
 226         -0.1335313926245226231463436209313499745895,
 227         -0.129077042275142343345847831367985856258,
 228         -0.124642445207276597338493356591214304499,
 229         -0.1202274269981598003244753948319154994493,
 230         -0.115831815525121705099120059938680166568,
 231         -0.1114554409253228268966213677328042273655,
 232         -0.1070981355563671005131126851708522185606,
 233         -0.1027597339577689347753154133345778104976,
 234         -0.098440072813252519902888574928971234883,
 235         -0.094138990913861910035632096996525066015,
 236         -0.0898563291218610470766469347968659624282,
 237         -0.0855919303354035139161469686670511961825,
 238         -0.0813456394539524058873423550293617843895,
 239         -0.077117303344431289769666193261475917783,
 240         -0.072906770808087780565737488890929711303,
 241         -0.0687138925480518083746933774035034481663,
 242         -0.064538521137571171672923915683992928129,
 243         -0.0603805109889074798714456529545968095868,
 244         -0.0562397183228760777967376942769773768851,
 245         -0.0521160011390140183616307870527840213665,
 246         -0.0480092191863606077520036253234446621373,
 247         -0.0439192339348354905263921515528654458042,
 248         -0.0398459085471996706586162402473026835046,
 249         -0.0357891078515852792753420982122404025613,
 250         -0.0317486983145803011569962827485256299276,
 251         -0.0277245480148548604671395114515163869272,
 252         -0.0237165266173160421183468505286730579517,
 253         -0.0197245053477785891192717326571593033246,
 254         -0.015748356968139168607549511460828269521,
 255         -0.0117879557520422404691605618900871263399,
 256         -0.0078431774610258928731840424909435816546,
 257         -0.00391389932113632909231778364357266484272,
 258         0.0,
 259         0.00389864041565732301393734309584290701073,
 260         0.00778214044205494894746290006113676367813,
 261         0.01165061721997527413559144280921434893315,
 262         0.0155041865359652541508540460424468358779,
 263         0.01934296284313093463590553454155047018545,
 264         0.0231670592815343782287991609622899165794,
 265         0.0269765876982020757480692925396595457815,
 266         0.0307716586667536883710282075967721640917,
 267         0.0345523815066597334073715005898328652816,
 268         0.038318864302136599193755325123797290346,
 269         0.042071213920687054375203805926962379448,
 270         0.045809536031294203166679267614663342114,
 271         0.049533935122276630882096208829824573267,
 272         0.0532445145188122828658701937865287769396,
 273         0.0569413764001384247590131015404494943015,
 274         0.0606246218164348425806061320404202632862,
 275         0.0642943507053972572162284502656114944857,
 276         0.0679506619085077493945652777726294140346,
 277         0.071593653187008817925605272752092034269,
 278         0.075223421237587525698605339983662414637,
 279         0.078840061707776024531540577859198294559,
 280         0.082443669211074591268160068668307805914,
 281         0.086034337341803153381797826721996075141,
 282         0.0896121586896871326199514693784845287854,
 283         0.093177224854183289768781353027759396216,
 284         0.096729626458551112295571056487463437015,
 285         0.1002694531636751493081301751297276601964,
 286         0.1037967936816435648260618037639746883066,
 287         0.1073117357890880506671750303711543368066,
 288         0.1108143663402901141948061693232119280986,
 289         0.1143047712800586336342591448151747734094,
 290         0.1177830356563834545387941094705217050686,
 291         0.1212492436328696851612122640808405265723,
 292         0.1247034785009572358634065153808632684918,
 293         0.128145822691930038174109886961074873852,
 294         0.1315763577887192725887161286894831624516,
 295         0.134995164537504830601983291147085645626,
 296         0.138402322859119135685325873601649187393,
 297         0.1417979118602573498789527352804727189846,
 298         0.1451820098444978972819350637405643235226,
 299         0.1485546943231371429098223170672938691604,
 300         0.151916042025841975071803424896884511328,
 301         0.1552661289111239515223833017101021786436,
 302         0.1586050301766385840933711746258415752456,
 303         0.161932820269313253240338285123614220592,
 304         0.165249572895307162875611449277240313729,
 305         0.1685553610298066669415865321701023169345,
 306         0.171850256926659222340098946055147264935,
 307         0.1751343321278491480142914649863898412374,
 308         0.1784076574728182971194002415109419683545,
 309         0.181670303107634678260605595617079739242,
 310         0.184922338494011992663903592659249621006,
 311         0.1881638324181829868259905803105539806714,
 312         0.191394852999629454609298807561308873447,
 313         0.194615467699671658858138593767269731516,
 314         0.1978257433299198803625720711969614690756,
 315         0.201025746060590741340908337591797808969,
 316         0.204215541428690891503820386196239272214,
 317         0.2073951943460705871587455788490062338536,
 318         0.210564769107349637669552812732351513721,
 319         0.2137243293977181388619051976331987647734,
 320         0.216873938300614359619089525744347498479,
 321         0.220013658305282095907358638661628360712,
 322         0.2231435513142097557662950903098345033745,
 323         0.226263678650453389361787082280390161607,
 324         0.229374101064845829991480725046139871551,
 325         0.232474878743094064920705078095567528222,
 326         0.235566071312766909077588218941043410137,
 327         0.2386477378501750099171491363522813392526,
 328         0.241719936887145168144307515913513900104,
 329         0.244782726417690916434704717466314811104,
 330         0.247836163904581256780602765746524747999,
 331         0.25088030628580941658844644154994089393,
 332         0.253915209980963444137323297906606667466,
 333         0.256940930897500425446759867911224262093,
 334         0.259957524436926066972079494542311044577,
 335         0.26296504550088135182072917321108602859,
 336         0.265963548497137941339125926537543389269,
 337         0.268953087345503958932974357924497845489,
 338         0.271933715483641758831669494532999161983,
 339         0.274905485872799249167009582983018668293,
 340         0.277868451003456306186350032923401233082,
 341         0.280822662900887784639519758873134832073,
 342         0.28376817313064459834690122235025476666,
 343         0.286705032803954314653250930842073965668,
 344         0.289633292583042676878893055525668970004,
 345         0.292553002686377439978201258664126644308,
 346         0.295464212893835876386681906054964195182,
 347         0.298366972551797281464900430293496918012,
 348         0.301261330578161781012875538233755492657,
 349         0.304147335467296717015819874720446989991,
 350         0.30702503529491186207512454053537790169,
 351         0.309894477722864687861624550833227164546,
 352         0.31275571000389688838624655968831903216,
 353         0.315608778986303334901366180667483174144,
 354         0.318453731118534615810247213590599595595,
 355         0.321290612453734292057863145522557457887,
 356         0.324119468654211976090670760434987352183,
 357         0.326940344995853320592356894073809191681,
 358         0.329753286372467981814422811920789810952,
 359         0.332558337300076601412275626573419425269,
 360         0.335355541921137830257179579814166199074,
 361         0.338144944008716397710235913939267433111,
 362         0.340926586970593210305089199780356208443,
 363         0.34370051385331844468019789211029452987,
 364         0.346466767346208580918462188425772950712,
 365         0.349225389785288304181275421187371759687,
 366         0.35197642315717818465544745625943892599,
 367         0.354719909102929028355011218999317665826,
 368         0.357455888921803774226009490140904474434,
 369         0.360184403575007796281574967493016620926,
 370         0.362905493689368453137824345977489846141,
 371         0.365619199560964711319396875217046453067,
 372         0.368325561158707653048230154050398826898,
 373         0.371024618127872663911964910806824955394,
 374         0.373716409793584080821016832715823506644,
 375         0.376400975164253065997877633436251593315,
 376         0.379078352934969458390853345631019858882,
 377         0.38174858149084833985966626493567607862,
 378         0.384411698910332039734790062481290868519,
 379         0.387067742968448287898902502261817665695,
 380         0.38971675114002521337046360400352086705,
 381         0.392358760602863872479379611988215363485,
 382         0.39499380824086897810639403636498176831,
 383         0.397621930647138489104829072973405554918,
 384         0.40024316412701270692932510199513117008,
 385         0.402857544701083514655197565487057707577,
 386         0.405465108108164381978013115464349136572,
 387         0.408065889808221748430198682969084124381,
 388         0.410659924985268385934306203175822787661,
 389         0.41324724855021933092547601552548590025,
 390         0.415827895143710965613328892954902305356,
 391         0.418401899138883817510763261966760106515,
 392         0.42096929464412963612886716150679597245,
 393         0.423530115505803295718430478017910109426,
 394         0.426084395310900063124544879595476618897,
 395         0.428632167389698760206812276426639053152,
 396         0.43117346481837134085917247895559499848,
 397         0.433708320421559393435847903042186017095,
 398         0.436236766774918070349041323061121300663,
 399         0.438758836207627937745575058511446738878,
 400         0.441274560804875229489496441661301225362,
 401         0.443783972410300981171768440588146426918,
 402         0.446287102628419511532590180619669006749,
 403         0.448783982827006710512822115683937186274,
 404         0.451274644139458585144692383079012478686,
 405         0.453759117467120506644794794442263270651,
 406         0.456237433481587594380805538163929748437,
 407         0.458709622626976664843883309250877913511,
 408         0.461175715122170166367999925597855358603,
 409         0.463635740963032513092182277331163919118,
 410         0.466089729924599224558619247504769399859,
 411         0.468537711563239270375665237462973542708,
 412         0.470979715218791012546897856056359251373,
 413         0.473415770016672131372578393236978550606,
 414         0.475845904869963914265209586304381412175,
 415         0.478270148481470280383546145497464809096,
 416         0.480688529345751907676618455448011551209,
 417         0.48310107575113582273837458485214554795,
 418         0.485507815781700807801791077190788900579,
 419         0.487908777319238973246173184132656942487,
 420         0.490303988045193838150346159645746860531,
 421         0.492693475442575255695076950020077845328,
 422         0.495077266797851514597964584842833665358,
 423         0.497455389202818942250859256731684928918,
 424         0.499827869556449329821331415247044141512,
 425         0.502194734566715494273584171951812573586,
 426         0.504556010752395287058308531738174929982,
 427         0.506911724444854354113196312660089270034,
 428         0.509261901789807946804074919228323824878,
 429         0.51160656874906207851888487520338193135,
 430         0.51394575110223431680100608827421759311,
 431         0.51627947444845449617281928478756106467,
 432         0.518607764208045632152976996364798698556,
 433         0.520930645624185312409809834659637709188,
 434         0.52324814376454783651680722493487084164,
 435         0.525560283522927371382427602307131424923,
 436         0.527867089620842385113892217778300963557,
 437         0.530168586609121617841419630845212405063,
 438         0.532464798869471843873923723460142242606,
 439         0.534755750616027675477923292032637111077,
 440         0.537041465896883654566729244153832299024,
 441         0.539321968595608874655355158077341155752,
 442         0.54159728243274437157654230390043409897,
 443         0.543867430967283517663338989065998323965,
 444         0.546132437598135650382397209231209163864,
 445         0.548392325565573162748150286179863158565,
 446         0.550647117952662279259948179204913460093,
 447         0.552896837686677737580717902230624314327,
 448         0.55514150754050159271548035951590405017,
 449         0.557381150134006357049816540361233647898,
 450         0.559615787935422686270888500526826593487,
 451         0.561845443262691817915664819160697456814,
 452         0.564070138284802966071384290090190711817,
 453         0.566289895023115872590849979337124343595,
 454         0.568504735352668712078738764866962263577,
 455         0.5707146810034715448536245647415894503,
 456         0.572919753561785509092756726626261068625,
 457         0.575119974471387940421742546569273429365,
 458         0.577315365034823604318112061519496401506,
 459         0.579505946414642223855274409488070989814,
 460         0.58169173963462248252061075372537234071,
 461         0.583872765580982679097413356975291104927,
 462         0.586049045003578208904119436287324349516,
 463         0.588220598517086043034868221609113995052,
 464         0.590387446602176374641916708123598757576,
 465         0.59254960960667159874199020959329739696,
 466         0.594707107746692789514343546529205333192,
 467         0.59685996110779383658731192302565801002,
 468         0.59900818964608339938160002446165150206,
 469         0.601151813189334836191674317068856441547,
 470         0.603290851438084262340585186661310605647,
 471         0.6054253239667168894375677681414899356,
 472         0.607555250224541795501085152791125371894,
 473         0.609680649536855273481833501660588408785,
 474         0.611801541105992903529889766428814783686,
 475         0.613917944012370492196929119645563790777,
 476         0.616029877215514019647565928196700650293,
 477         0.618137359555078733872689126674816271683,
 478         0.620240409751857528851494632567246856773,
 479         0.62233904640877874159710264120869663505,
 480         0.62443328801189350104253874405467311991,
 481         0.626523152931352759778820859734204069282,
 482         0.628608659422374137744308205774183639946,
 483         0.6306898256261987050837261409313532241,
 484         0.63276666957103782954578646850357975849,
 485         0.634839209173010211969493840510489008123,
 486         0.63690746223706923162049442718119919119,
 487         0.63897144645792072137962398326473680873,
 488         0.64103117942093129105560133440539254671,
 489         0.643086678603027315392053859585132960477,
 490         0.645137961373584701665228496134731905937,
 491         0.647185044995309550122320631377863036675,
 492         0.64922794662510981889083996990531112227,
 493         0.651266683314958103396333353349672108398,
 494         0.653301272012745638758615881210873884572,
 495         0.65533172956312763209494967856962559648,
 496         0.657358072708360030141890023245936165513,
 497         0.659380318089127826115336413370955804038,
 498         0.661398482245365008260235838709650938148,
 499         0.66341258161706625109695030429080128179,
 500         0.665422632545090448950092610006660181147,
 501         0.667428651271956189947234166318980478403,
 502         0.669430653942629267298885270929503510123,
 503         0.67142865660530232331713904200189252584,
 504         0.67342267521216672029796038880101726475,
 505         0.67541272562017673108090414397019748722,
 506         0.677398823591806140809682609997348298556,
 507         0.67938098479579735014710062847376425181,
 508         0.681359224807903068948071559568089441735,
 509         0.683333559111620688164363148387750369654,
 510         0.68530400309891941654404807896723298642,
 511         0.687270572070960267497006884394346103924,
 512         0.689233281238808980324914337814603903233,
 513         0.691192145724141958859604629216309755938,
 514         0.693147180559945309417232121458176568075
 515       };
 516     return log_table[128 + (int)m] + 2.0 * atanh_z;
 517   }
 518 }