| /* Exponential function minus one. |
| Copyright (C) 2012-2020 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 <https://www.gnu.org/licenses/>. */ |
| |
| #include <config.h> |
| |
| /* Specification. */ |
| #include <math.h> |
| |
| #include <float.h> |
| |
| /* 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; |
| } |
| } |
