lib/exp2.c

/* Exponential base 2 function.
   Copyright (C) 2012-2026 Free Software Foundation, Inc.

   This file is free software: you can redistribute it and/or modify
   it under the terms of the GNU Lesser General Public License as
   published by the Free Software Foundation, either version 3 of the
   License, or (at your option) any later version.

   This file is distributed in the hope that it will be useful,
   but WITHOUT ANY WARRANTY; without even the implied warranty of
   MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
   GNU Lesser General Public License for more details.

   You should have received a copy of the GNU Lesser General Public License
   along with this program.  If not, see <https://www.gnu.org/licenses/>.  */

#include <config.h>

/* Specification.  */
#include <math.h>

#include <float.h>

/* Best possible approximation of log(2) as a 'double'.  */
#define LOG2 0.693147180559945309417232121458176568075

/* Best possible approximation of 1/log(2) as a 'double'.  */
#define LOG2_INVERSE 1.44269504088896340735992468100189213743

/* Best possible approximation of log(2)/256 as a 'double'.  */
#define LOG2_BY_256 0.00270760617406228636491106297444600221904

/* Best possible approximation of 256/log(2) as a 'double'.  */
#define LOG2_BY_256_INVERSE 369.329930467574632284140718336484387181

double
exp2 (double x)
{
  /* exp2(x) = exp(x*log(2)).
     If we would compute it like this, there would be rounding errors for
     integer or near-integer values of x.  To avoid these, we inline the
     algorithm for exp(), and the multiplication with log(2) cancels a
     division by log(2).  */

  if (isnand (x))
    return x;

  if (x > (double) DBL_MAX_EXP)
    /* x > DBL_MAX_EXP
       hence exp2(x) > 2^DBL_MAX_EXP, overflows to Infinity.  */
    return HUGE_VAL;

  if (x < (double) (DBL_MIN_EXP - 1 - DBL_MANT_DIG))
    /* x < (DBL_MIN_EXP - 1 - DBL_MANT_DIG)
       hence exp2(x) < 2^(DBL_MIN_EXP-1-DBL_MANT_DIG),
       underflows to zero.  */
    return 0.0;

  /* Decompose x into
       x = n + m/256 + y/log(2)
     where
       n is an integer,
       m is an integer, -128 <= m <= 128,
       y is a number, |y| <= log(2)/512 + epsilon = 0.00135...
     Then
       exp2(x) = 2^n * exp(m * log(2)/256) * exp(y)
     The first factor is an ldexpl() call.
     The second factor is a table lookup.
     The third factor is computed
     - either as sinh(y) + cosh(y)
       where sinh(y) is computed through the power series:
         sinh(y) = y + y^3/3! + y^5/5! + ...
       and cosh(y) is computed as hypot(1, sinh(y)),
     - or as exp(2*z) = (1 + tanh(z)) / (1 - tanh(z))
       where z = y/2
       and tanh(z) is computed through its power series:
         tanh(z) = z
                   - 1/3 * z^3
                   + 2/15 * z^5
                   - 17/315 * z^7
                   + 62/2835 * z^9
                   - 1382/155925 * z^11
                   + 21844/6081075 * z^13
                   - 929569/638512875 * z^15
                   + ...
       Since |z| <= log(2)/1024 < 0.0007, the relative contribution of the
       z^7 term is < 0.0007^6 < 2^-60 <= 2^-DBL_MANT_DIG, therefore we can
       truncate the series after the z^5 term.  */

  {
    double nm = round (x * 256.0); /* = 256 * n + m */
    double z = (x * 256.0 - nm) * (LOG2_BY_256 * 0.5);

/* Coefficients of the power series for tanh(z).  */
#define TANH_COEFF_1   1.0
#define TANH_COEFF_3  -0.333333333333333333333333333333333333334
#define TANH_COEFF_5   0.133333333333333333333333333333333333334
#define TANH_COEFF_7  -0.053968253968253968253968253968253968254
#define TANH_COEFF_9   0.0218694885361552028218694885361552028218
#define TANH_COEFF_11 -0.00886323552990219656886323552990219656886
#define TANH_COEFF_13  0.00359212803657248101692546136990581435026
#define TANH_COEFF_15 -0.00145583438705131826824948518070211191904

    double z2 = z * z;
    double tanh_z =
      ((TANH_COEFF_5
        * z2 + TANH_COEFF_3)
       * z2 + TANH_COEFF_1)
      * z;

    double exp_y = (1.0 + tanh_z) / (1.0 - tanh_z);

    int n = (int) round (nm * (1.0 / 256.0));
    int m = (int) nm - 256 * n;

    /* exp_table[i] = exp((i - 128) * log(2)/256).
       Computed in GNU clisp through
         (setf (long-float-digits) 128)
         (setq a 0L0)
         (setf (long-float-digits) 256)
         (dotimes (i 257)
           (format t "        ~D,~%"
                   (float (exp (* (/ (- i 128) 256) (log 2L0))) a)))  */
    static const double exp_table[257] =
      {
        0.707106781186547524400844362104849039284,
        0.709023942160207598920563322257676190836,
        0.710946301084582779904674297352120049962,
        0.71287387205274715340350157671438300618,
        0.714806669195985005617532889137569953044,
        0.71674470668389442125974978427737336719,
        0.71868799872449116280161304224785251353,
        0.720636559564312831364255957304947586072,
        0.72259040348852331001850312073583545284,
        0.724549544821017490259402705487111270714,
        0.726513997924526282423036245842287293786,
        0.728483777200721910815451524818606761737,
        0.730458897090323494325651445155310766577,
        0.732439372073202913296664682112279175616,
        0.734425216668490963430822513132890712652,
        0.736416445434683797507470506133110286942,
        0.738413072969749655693453740187024961962,
        0.740415113911235885228829945155951253966,
        0.742422582936376250272386395864403155277,
        0.744435494762198532693663597314273242753,
        0.746453864145632424600321765743336770838,
        0.748477705883617713391824861712720862423,
        0.750507034813212760132561481529764324813,
        0.752541865811703272039672277899716132493,
        0.75458221379671136988300977551659676571,
        0.756628093726304951096818488157633113612,
        0.75867952059910734940489114658718937343,
        0.760736509454407291763130627098242426467,
        0.762799075372269153425626844758470477304,
        0.76486723347364351194254345936342587308,
        0.766940998920478000900300751753859329456,
        0.769020386915828464216738479594307884331,
        0.771105412703970411806145931045367420652,
        0.773196091570510777431255778146135325272,
        0.77529243884249997956151370535341912283,
        0.777394469888544286059157168801667390437,
        0.779502200118918483516864044737428940745,
        0.781615644985678852072965367573877941354,
        0.783734819982776446532455855478222575498,
        0.78585974064617068462428149076570281356,
        0.787990422553943243227635080090952504452,
        0.790126881326412263402248482007960521995,
        0.79226913262624686505993407346567890838,
        0.794417192158581972116898048814333564685,
        0.796571075671133448968624321559534367934,
        0.798730798954313549131410147104316569576,
        0.800896377841346676896923120795476813684,
        0.803067828208385462848443946517563571584,
        0.805245165974627154089760333678700291728,
        0.807428407102430320039984581575729114268,
        0.809617567597431874649880866726368203972,
        0.81181266350866441589760797777344082227,
        0.814013710928673883424109261007007338614,
        0.816220725993637535170713864466769240053,
        0.818433724883482243883852017078007231025,
        0.82065272382200311435413206848451310067,
        0.822877739076982422259378362362911222833,
        0.825108786960308875483586738272485101678,
        0.827345883828097198786118571797909120834,
        0.829589046080808042697824787210781231927,
        0.831838290163368217523168228488195222638,
        0.834093632565291253329796170708536192903,
        0.836355089820798286809404612069230711295,
        0.83862267850893927589613232455870870518,
        0.84089641525371454303112547623321489504,
        0.84317631672419664796432298771385230143,
        0.84546239963465259098692866759361830709,
        0.84775468074466634749045860363936420312,
        0.850053176859261734750681286748751167545,
        0.852357904829025611837203530384718316326,
        0.854668881550231413551897437515331498025,
        0.856986123964963019301812477839166009452,
        0.859309649061238957814672188228156252257,
        0.861639473873136948607517116872358729753,
        0.863975615480918781121524414614366207052,
        0.866318091011155532438509953514163469652,
        0.868666917636853124497101040936083380124,
        0.871022112577578221729056715595464682243,
        0.873383693099584470038708278290226842228,
        0.875751676515939078050995142767930296012,
        0.878126080186649741556080309687656610647,
        0.880506921518791912081045787323636256171,
        0.882894217966636410521691124969260937028,
        0.885287987031777386769987907431242017412,
        0.88768824626326062627527960009966160388,
        0.89009501325771220447985955243623523504,
        0.892508305659467490072110281986409916153,
        0.8949281411607004980029443898876582985,
        0.897354537501553593213851621063890907178,
        0.899787512470267546027427696662514569756,
        0.902227083903311940153838631655504844215,
        0.904673269685515934269259325789226871994,
        0.907126087750199378124917300181170171233,
        0.909585556079304284147971563828178746372,
        0.91205169270352665549806275316460097744,
        0.914524515702448671545983912696158354092,
        0.91700404320467123174354159479414442804,
        0.919490293387946858856304371174663918816,
        0.921983284479312962533570386670938449637,
        0.92448303475522546419252726694739603678,
        0.92698956254169278419622653516884831976,
        0.929502886214410192307650717745572682403,
        0.932023024198894522404814545597236289343,
        0.934549994970619252444512104439799143264,
        0.93708381705514995066499947497722326722,
        0.93962450902828008902058735120448448827,
        0.942172089516167224843810351983745154882,
        0.944726577195469551733539267378681531548,
        0.947287990793482820670109326713462307376,
        0.949856349088277632361251759806996099924,
        0.952431670908837101825337466217860725517,
        0.955013975135194896221170529572799135168,
        0.957603280698573646936305635147915443924,
        0.960199606581523736948607188887070611744,
        0.962802971818062464478519115091191368377,
        0.965413395493813583952272948264534783197,
        0.968030896746147225299027952283345762418,
        0.970655494764320192607710617437589705184,
        0.973287208789616643172102023321302921373,
        0.97592605811548914795551023340047499377,
        0.978572062087700134509161125813435745597,
        0.981225240104463713381244885057070325016,
        0.983885611616587889056366801238014683926,
        0.98655319612761715646797006813220671315,
        0.989228013193975484129124959065583667775,
        0.99191008242510968492991311132615581644,
        0.994599423483633175652477686222166314457,
        0.997296056085470126257659913847922601123,
        1.0,
        1.00271127505020248543074558845036204047,
        1.0054299011128028213513839559347998147,
        1.008155898118417515783094890817201039276,
        1.01088928605170046002040979056186052439,
        1.013630084951489438840258929063939929597,
        1.01637831491095303794049311378629406276,
        1.0191339960777379496848780958207928794,
        1.02189714865411667823448013478329943978,
        1.02466779289713564514828907627081492763,
        1.0274459491187636965388611939222137815,
        1.030231637686041012871707902453904567093,
        1.033024879021228422500108283970460918086,
        1.035825693601957120029983209018081371844,
        1.03863410196137879061243669795463973258,
        1.04145012468831614126454607901189312648,
        1.044273782427413840321966478739929008784,
        1.04710509587928986612990725022711224056,
        1.04994408580068726608203812651590790906,
        1.05279077300462632711989120298074630319,
        1.05564517836055715880834132515293865216,
        1.058507322794512690105772109683716645074,
        1.061377227289262080950567678003883726294,
        1.06425491288446454978861125700158022068,
        1.06714040067682361816952112099280916261,
        1.0700337118202417735424119367576235685,
        1.072934867525975551385035450873827585343,
        1.075843889062791037803228648476057074063,
        1.07876079775711979374068003743848295849,
        1.081685614993215201942115594422531125643,
        1.08461836221330923781610517190661434161,
        1.087559060917769665346797830944039707867,
        1.09050773266525765920701065576070797899,
        1.09346439907288585422822014625044716208,
        1.096429081816376823386138295859248481766,
        1.09940180263022198546369696823882990404,
        1.10238258330784094355641420942564685751,
        1.10537144570174125558827469625695031104,
        1.108368411723678638009423649426619850137,
        1.111373503344817603850149254228916637444,
        1.1143867425958925363088129569196030678,
        1.11740815156736919905457996308578026665,
        1.12043775240960668442900387986631301277,
        1.123475567333019800733729739775321431954,
        1.12652161860824189979479864378703477763,
        1.129575928566288145997264988840249825907,
        1.13263851959871922798707372367762308438,
        1.13570941415780551424039033067611701343,
        1.13878863475669165370383028384151125472,
        1.14187620396956162271229760828788093894,
        1.14497214443180421939441388822291589579,
        1.14807647884017900677879966269734268003,
        1.15118922995298270581775963520198253612,
        1.154310420590216039548221528724806960684,
        1.157440073633751029613085766293796821106,
        1.16057821202749874636945947257609098625,
        1.16372485877757751381357359909218531234,
        1.166880036952481570555516298414089287834,
        1.170043769683250188080259035792738573,
        1.17321608016363724753480435451324538889,
        1.176396991650281276284645728483848641054,
        1.17958652746287594548610056676944051898,
        1.182784710984341029924457204693850757966,
        1.18599156566099383137126564953421556374,
        1.18920711500272106671749997056047591529,
        1.19243138258315122214272755814543101148,
        1.195664392039827374583837049865451975705,
        1.19890616707438048177030255797630020695,
        1.202156731452703142096396957497765876003,
        1.205416109005123825604211432558411335666,
        1.208684323626581577354792255889216998484,
        1.21196139927680119446816891773249304545,
        1.215247359980468878116520251338798457624,
        1.218542229827408361758207148117394510724,
        1.221846032972757516903891841911570785836,
        1.225158793637145437709464594384845353707,
        1.22848053610687000569400895779278184036,
        1.2318112847340759358845566532127948166,
        1.235151063936933305692912507415415760294,
        1.238499898199816567833368865859612431545,
        1.24185781207348404859367746872659560551,
        1.24522483017525793277520496748615267417,
        1.24860097718920473662176609730249554519,
        1.25198627786631627006020603178920359732,
        1.255380757024691089579390657442301194595,
        1.25878443954971644307786044181516261876,
        1.26219735039425070801401025851841645967,
        1.265619514578806324196273999873453036296,
        1.26905095719173322255441908103233800472,
        1.27249170338940275123669204418460217677,
        1.27594177839639210038120243475928938891,
        1.27940120750566922691358797002785254596,
        1.28287001607877828072666978102151405111,
        1.286348229546025533601482208069738348355,
        1.28983587340666581223274729549155218968,
        1.293332973229089436725559789048704304684,
        1.296839554651009665933754117792451159835,
        1.30035564337965065101414056707091779129,
        1.30388126519193589857452364895199736833,
        1.30741644593467724479715157747196172848,
        1.310961211524764341922991786330755849366,
        1.314515587949354658485983613383997794965,
        1.318079601266063994690185647066116617664,
        1.32165327760315751432651181233060922616,
        1.32523664315974129462953709549872167411,
        1.32882972420595439547865089632866510792,
        1.33243254708316144935164337949073577407,
        1.33604513820414577344262790437186975929,
        1.33966752405330300536003066972435257602,
        1.34329973118683526382421714618163087542,
        1.346941786232945835788173713229537282075,
        1.35059371589203439140852219606013396004,
        1.35425554693689272829801474014070280434,
        1.357927306212901046494536695671766697446,
        1.36160902063822475558553593883194147464,
        1.36530071720401181543069836033754285543,
        1.36900242297459061192960113298219283217,
        1.37271416508766836928499785714471721579,
        1.37643597075453010021632280551868696026,
        1.380167867260238095581945274358283464697,
        1.383909881963831954872659527265192818,
        1.387662042298529159042861017950775988896,
        1.39142437577192618714983552956624344668,
        1.395196909966200178275574599249220994716,
        1.398979672538311140209528136715194969206,
        1.40277269122020470637471352433337881711,
        1.40657599381901544248361973255451684411,
        1.410389608217270704414375128268675481145,
        1.41421356237309504880168872420969807857
      };

    return ldexp (exp_table[128 + m] * exp_y, n);
  }
}