--- /dev/null
+/* Exponential base 2 function.
+ Copyright (C) 2012 Free Software Foundation, Inc.
+
+ This program is free software: you can redistribute it and/or modify
+ it under the terms of the GNU General Public License as published by
+ the Free Software Foundation; either version 3 of the License, or
+ (at your option) any later version.
+
+ This program 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 General Public License for more details.
+
+ You should have received a copy of the GNU General Public License
+ along with this program. If not, see <http://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 error 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,
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+ 1.157440073633751029613085766293796821106,
+ 1.16057821202749874636945947257609098625,
+ 1.16372485877757751381357359909218531234,
+ 1.166880036952481570555516298414089287834,
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+ 1.17321608016363724753480435451324538889,
+ 1.176396991650281276284645728483848641054,
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+ 1.19243138258315122214272755814543101148,
+ 1.195664392039827374583837049865451975705,
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+ 1.202156731452703142096396957497765876003,
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+ 1.208684323626581577354792255889216998484,
+ 1.21196139927680119446816891773249304545,
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+ 1.24522483017525793277520496748615267417,
+ 1.24860097718920473662176609730249554519,
+ 1.25198627786631627006020603178920359732,
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+ 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,
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+ 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);
+ }
+}