/* Exponential function minus one.
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 . */
#include
/* Specification. */
#include
#include
/* A value slightly larger than log(2). */
#define LOG2_PLUS_EPSILON 0.6931471805599454
/* 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
/* The upper 32 bits of log(2)/256. */
#define LOG2_BY_256_HI_PART 0.0027076061733168899081647396087646484375
/* log(2)/256 - LOG2_HI_PART. */
#define LOG2_BY_256_LO_PART \
0.000000000000745396456746323365681353781544922399845
double
expm1 (double x)
{
if (isnand (x))
return x;
if (x >= (double) DBL_MAX_EXP * LOG2_PLUS_EPSILON)
/* x > DBL_MAX_EXP * log(2)
hence exp(x) > 2^DBL_MAX_EXP, overflows to Infinity. */
return HUGE_VAL;
if (x <= (double) (- DBL_MANT_DIG) * LOG2_PLUS_EPSILON)
/* x < (- DBL_MANT_DIG) * log(2)
hence 0 < exp(x) < 2^-DBL_MANT_DIG,
hence -1 < exp(x)-1 < -1 + 2^-DBL_MANT_DIG
rounds to -1. */
return -1.0;
if (x <= - LOG2_PLUS_EPSILON)
/* 0 < exp(x) < 1/2.
Just compute exp(x), then subtract 1. */
return exp (x) - 1.0;
if (x == 0.0)
/* Return a zero with the same sign as x. */
return x;
/* Decompose x into
x = n * log(2) + m * log(2)/256 + y
where
n is an integer, n >= -1,
m is an integer, -128 <= m <= 128,
y is a number, |y| <= log(2)/512 + epsilon = 0.00135...
Then
exp(x) = 2^n * exp(m * log(2)/256) * exp(y)
Compute each factor minus one, then combine them through the
formula (1+a)*(1+b) = 1 + (a+b*(1+a)),
that is (1+a)*(1+b) - 1 = a + b*(1+a).
The first factor is an ldexpl() call.
The second factor is a table lookup.
The third factor minus one is computed
- either as sinh(y) + sinh(y)^2 / (cosh(y) + 1)
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 = 2 * 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.
Given the usual bounds DBL_MAX_EXP <= 16384, DBL_MANT_DIG <= 120, we
can estimate x: -84 <= x <= 11357.
This means, when dividing x by log(2), where we want x mod log(2)
to be precise to DBL_MANT_DIG bits, we have to use an approximation
to log(2) that has 14+DBL_MANT_DIG bits. */
{
double nm = round (x * LOG2_BY_256_INVERSE); /* = 256 * n + m */
/* n has at most 15 bits, nm therefore has at most 23 bits, therefore
n * LOG2_HI_PART is computed exactly, and n * LOG2_LO_PART is computed
with an absolute error < 2^15 * 2e-10 * 2^-DBL_MANT_DIG. */
double y_tmp = x - nm * LOG2_BY_256_HI_PART;
double y = y_tmp - nm * LOG2_BY_256_LO_PART;
double z = 0.5L * y;
/* 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_minus_1 = 2.0 * tanh_z / (1.0 - tanh_z);
int n = (int) round (nm * (1.0 / 256.0));
int m = (int) nm - 256 * n;
/* expm1_table[i] = exp((i - 128) * log(2)/256) - 1.
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))) 1) a))) */
static const double expm1_table[257] =
{
-0.292893218813452475599155637895150960716,
-0.290976057839792401079436677742323809165,
-0.289053698915417220095325702647879950038,
-0.287126127947252846596498423285616993819,
-0.285193330804014994382467110862430046956,
-0.283255293316105578740250215722626632811,
-0.281312001275508837198386957752147486471,
-0.279363440435687168635744042695052413926,
-0.277409596511476689981496879264164547161,
-0.275450455178982509740597294512888729286,
-0.273486002075473717576963754157712706214,
-0.271516222799278089184548475181393238264,
-0.269541102909676505674348554844689233423,
-0.267560627926797086703335317887720824384,
-0.265574783331509036569177486867109287348,
-0.263583554565316202492529493866889713058,
-0.261586927030250344306546259812975038038,
-0.259584886088764114771170054844048746036,
-0.257577417063623749727613604135596844722,
-0.255564505237801467306336402685726757248,
-0.253546135854367575399678234256663229163,
-0.251522294116382286608175138287279137577,
-0.2494929651867872398674385184702356751864,
-0.247458134188296727960327722100283867508,
-0.24541778620328863011699022448340323429,
-0.243371906273695048903181511842366886387,
-0.24132047940089265059510885341281062657,
-0.239263490545592708236869372901757573532,
-0.237200924627730846574373155241529522695,
-0.23513276652635648805745654063657412692,
-0.233059001079521999099699248246140670544,
-0.230979613084171535783261520405692115669,
-0.228894587296029588193854068954632579346,
-0.226803908429489222568744221853864674729,
-0.224707561157500020438486294646580877171,
-0.222605530111455713940842831198332609562,
-0.2204977998810815164831359552625710592544,
-0.218384355014321147927034632426122058645,
-0.2162651800172235534675441445217774245016,
-0.214140259353829315375718509234297186439,
-0.212009577446056756772364919909047495547,
-0.209873118673587736597751517992039478005,
-0.2077308673737531349400659265343210916196,
-0.205582807841418027883101951185666435317,
-0.2034289243288665510313756784404656320656,
-0.201269201045686450868589852895683430425,
-0.199103622158653323103076879204523186316,
-0.196932171791614537151556053482436428417,
-0.19475483402537284591023966632129970827,
-0.192571592897569679960015418424270885733,
-0.190382432402568125350119133273631796029,
-0.188187336491335584102392022226559177731,
-0.185986289071326116575890738992992661386,
-0.183779274006362464829286135533230759947,
-0.181566275116517756116147982921992768975,
-0.17934727617799688564586793151548689933,
-0.1771222609230175777406216376370887771665,
-0.1748912130396911245164132617275148983224,
-0.1726541161719028012138814282020908791644,
-0.170410953919191957302175212789218768074,
-0.168161709836631782476831771511804777363,
-0.165906367434708746670203829291463807099,
-0.1636449101792017131905953879307692887046,
-0.161377321491060724103867675441291294819,
-0.15910358474628545696887452376678510496,
-0.15682368327580335203567701228614769857,
-0.154537600365347409013071332406381692911,
-0.152245319255333652509541396360635796882,
-0.149946823140738265249318713251248832456,
-0.147642095170974388162796469615281683674,
-0.145331118449768586448102562484668501975,
-0.143013876035036980698187522160833990549,
-0.140690350938761042185327811771843747742,
-0.138360526126863051392482883127641270248,
-0.136024384519081218878475585385633792948,
-0.133681908988844467561490046485836530346,
-0.131333082363146875502898959063916619876,
-0.128977887422421778270943284404535317759,
-0.126616306900415529961291721709773157771,
-0.1242483234840609219490048572320697039866,
-0.121873919813350258443919690312343389353,
-0.1194930784812080879189542126763637438278,
-0.11710578203336358947830887503073906297,
-0.1147120129682226132300120925687579825894,
-0.1123117537367393737247203999003383961205,
-0.1099049867422877955201404475637647649574,
-0.1074916943405325099278897180135900838485,
-0.1050718588392995019970556101123417014993,
-0.102645462498446406786148378936109092823,
-0.1002124875297324539725723033374854302454,
-0.097772916096688059846161368344495155786,
-0.0953267303144840657307406742107731280055,
-0.092873912249800621875082699818829828767,
-0.0904144439206957158520284361718212536293,
-0.0879483072964733445019372468353990225585,
-0.0854754842975513284540160873038416459095,
-0.0829959567953287682564584052058555719614,
-0.080509706612053141143695628825336081184,
-0.078016715520687037466429613329061550362,
-0.075516965244774535807472733052603963221,
-0.073010437458307215803773464831151680239,
-0.070497113785589807692349282254427317595,
-0.067976975801105477595185454402763710658,
-0.0654500050293807475554878955602008567352,
-0.06291618294485004933500052502277673278,
-0.0603754909717199109794126487955155117284,
-0.0578279104838327751561896480162548451191,
-0.055273422804530448266460732621318468453,
-0.0527120092065171793298906732865376926237,
-0.0501436509117223676387482401930039000769,
-0.0475683290911628981746625337821392744829,
-0.044986024864805103778829470427200864833,
-0.0423967193014263530636943648520845560749,
-0.0398003934184762630513928111129293882558,
-0.0371970281819375355214808849088086316225,
-0.0345866045061864160477270517354652168038,
-0.0319691032538527747009720477166542375817,
-0.0293445052356798073922893825624102948152,
-0.0267127912103833568278979766786970786276,
-0.0240739418845108520444897665995250062307,
-0.0214279379122998654908388741865642544049,
-0.018774759895536286618755114942929674984,
-0.016114388383412110943633198761985316073,
-0.01344680387238284353202993186779328685225,
-0.0107719868060245158708750409344163322253,
-0.00808991757489031507008688867384418356197,
-0.00540057651636682434752231377783368554176,
-0.00270394391452987374234008615207739887604,
0.0,
0.00271127505020248543074558845036204047301,
0.0054299011128028213513839559347998147001,
0.00815589811841751578309489081720103927357,
0.0108892860517004600204097905618605243881,
0.01363008495148943884025892906393992959584,
0.0163783149109530379404931137862940627635,
0.0191339960777379496848780958207928793998,
0.0218971486541166782344801347832994397821,
0.0246677928971356451482890762708149276281,
0.0274459491187636965388611939222137814994,
0.0302316376860410128717079024539045670944,
0.0330248790212284225001082839704609180866,
0.0358256936019571200299832090180813718441,
0.0386341019613787906124366979546397325796,
0.0414501246883161412645460790118931264803,
0.0442737824274138403219664787399290087847,
0.0471050958792898661299072502271122405627,
0.049944085800687266082038126515907909062,
0.0527907730046263271198912029807463031904,
0.05564517836055715880834132515293865216,
0.0585073227945126901057721096837166450754,
0.0613772272892620809505676780038837262945,
0.0642549128844645497886112570015802206798,
0.0671404006768236181695211209928091626068,
0.070033711820241773542411936757623568504,
0.0729348675259755513850354508738275853402,
0.0758438890627910378032286484760570740623,
0.0787607977571197937406800374384829584908,
0.081685614993215201942115594422531125645,
0.0846183622133092378161051719066143416095,
0.0875590609177696653467978309440397078697,
0.090507732665257659207010655760707978993,
0.0934643990728858542282201462504471620805,
0.096429081816376823386138295859248481766,
0.099401802630221985463696968238829904039,
0.1023825833078409435564142094256468575113,
0.1053714457017412555882746962569503110404,
0.1083684117236786380094236494266198501387,
0.111373503344817603850149254228916637444,
0.1143867425958925363088129569196030678004,
0.1174081515673691990545799630857802666544,
0.120437752409606684429003879866313012766,
0.1234755673330198007337297397753214319548,
0.1265216186082418997947986437870347776336,
0.12957592856628814599726498884024982591,
0.1326385195987192279870737236776230843835,
0.135709414157805514240390330676117013429,
0.1387886347566916537038302838415112547204,
0.14187620396956162271229760828788093894,
0.144972144431804219394413888222915895793,
0.148076478840179006778799662697342680031,
0.15118922995298270581775963520198253612,
0.154310420590216039548221528724806960684,
0.157440073633751029613085766293796821108,
0.160578212027498746369459472576090986253,
0.163724858777577513813573599092185312343,
0.166880036952481570555516298414089287832,
0.1700437696832501880802590357927385730016,
0.1732160801636372475348043545132453888896,
0.176396991650281276284645728483848641053,
0.1795865274628759454861005667694405189764,
0.182784710984341029924457204693850757963,
0.185991565660993831371265649534215563735,
0.189207115002721066717499970560475915293,
0.192431382583151222142727558145431011481,
0.1956643920398273745838370498654519757025,
0.1989061670743804817703025579763002069494,
0.202156731452703142096396957497765876,
0.205416109005123825604211432558411335666,
0.208684323626581577354792255889216998483,
0.211961399276801194468168917732493045449,
0.2152473599804688781165202513387984576236,
0.218542229827408361758207148117394510722,
0.221846032972757516903891841911570785834,
0.225158793637145437709464594384845353705,
0.2284805361068700056940089577927818403626,
0.231811284734075935884556653212794816605,
0.235151063936933305692912507415415760296,
0.238499898199816567833368865859612431546,
0.241857812073484048593677468726595605511,
0.245224830175257932775204967486152674173,
0.248600977189204736621766097302495545187,
0.251986277866316270060206031789203597321,
0.255380757024691089579390657442301194598,
0.258784439549716443077860441815162618762,
0.262197350394250708014010258518416459672,
0.265619514578806324196273999873453036297,
0.269050957191733222554419081032338004715,
0.272491703389402751236692044184602176772,
0.27594177839639210038120243475928938891,
0.279401207505669226913587970027852545961,
0.282870016078778280726669781021514051111,
0.286348229546025533601482208069738348358,
0.289835873406665812232747295491552189677,
0.293332973229089436725559789048704304684,
0.296839554651009665933754117792451159835,
0.300355643379650651014140567070917791291,
0.303881265191935898574523648951997368331,
0.30741644593467724479715157747196172848,
0.310961211524764341922991786330755849366,
0.314515587949354658485983613383997794966,
0.318079601266063994690185647066116617661,
0.321653277603157514326511812330609226158,
0.325236643159741294629537095498721674113,
0.32882972420595439547865089632866510792,
0.33243254708316144935164337949073577407,
0.336045138204145773442627904371869759286,
0.339667524053303005360030669724352576023,
0.343299731186835263824217146181630875424,
0.346941786232945835788173713229537282073,
0.350593715892034391408522196060133960038,
0.354255546936892728298014740140702804344,
0.357927306212901046494536695671766697444,
0.361609020638224755585535938831941474643,
0.365300717204011815430698360337542855432,
0.369002422974590611929601132982192832168,
0.372714165087668369284997857144717215791,
0.376435970754530100216322805518686960261,
0.380167867260238095581945274358283464698,
0.383909881963831954872659527265192818003,
0.387662042298529159042861017950775988895,
0.391424375771926187149835529566243446678,
0.395196909966200178275574599249220994717,
0.398979672538311140209528136715194969206,
0.402772691220204706374713524333378817108,
0.40657599381901544248361973255451684411,
0.410389608217270704414375128268675481146,
0.414213562373095048801688724209698078569
};
double t = expm1_table[128 + m];
/* (1+t) * (1+exp_y_minus_1) - 1 = t + (1+t)*exp_y_minus_1 */
double p_minus_1 = t + (1.0 + t) * exp_y_minus_1;
double s = ldexp (1.0, n) - 1.0;
/* (1+s) * (1+p_minus_1) - 1 = s + (1+s)*p_minus_1 */
return s + (1.0 + s) * p_minus_1;
}
}