kc3-lang/gnulib/lib/expm1.c

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  Date : 2023-01-01T01:14:21
  

  maint: run 'make update-copyright'
  

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• lib/expm1.c

• /* Exponential function minus one.
     Copyright (C) 2012-2023 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>
  
  /* 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,
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          -0.133681908988844467561490046485836530346,
          -0.131333082363146875502898959063916619876,
          -0.128977887422421778270943284404535317759,
          -0.126616306900415529961291721709773157771,
          -0.1242483234840609219490048572320697039866,
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          -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,
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          -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,
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          0.0246677928971356451482890762708149276281,
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          0.096429081816376823386138295859248481766,
          0.099401802630221985463696968238829904039,
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          0.144972144431804219394413888222915895793,
          0.148076478840179006778799662697342680031,
          0.15118922995298270581775963520198253612,
          0.154310420590216039548221528724806960684,
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          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;
    }
  }
  

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