kc3-lang/gnulib/lib/expm1l.c

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

  maint: run 'make update-copyright'
  

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• lib/expm1l.c

• /* Exponential function minus one.
     Copyright (C) 2011-2023 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>
  
  #if HAVE_SAME_LONG_DOUBLE_AS_DOUBLE
  
  long double
  expm1l (long double x)
  {
    return expm1 (x);
  }
  
  #else
  
  # include <float.h>
  
  /* A value slightly larger than log(2).  */
  #define LOG2_PLUS_EPSILON 0.6931471805599454L
  
  /* Best possible approximation of log(2) as a 'long double'.  */
  #define LOG2 0.693147180559945309417232121458176568075L
  
  /* Best possible approximation of 1/log(2) as a 'long double'.  */
  #define LOG2_INVERSE 1.44269504088896340735992468100189213743L
  
  /* Best possible approximation of log(2)/256 as a 'long double'.  */
  #define LOG2_BY_256 0.00270760617406228636491106297444600221904L
  
  /* Best possible approximation of 256/log(2) as a 'long double'.  */
  #define LOG2_BY_256_INVERSE 369.329930467574632284140718336484387181L
  
  /* The upper 32 bits of log(2)/256.  */
  #define LOG2_BY_256_HI_PART 0.0027076061733168899081647396087646484375L
  /* log(2)/256 - LOG2_HI_PART.  */
  #define LOG2_BY_256_LO_PART \
    0.000000000000745396456746323365681353781544922399845L
  
  long double
  expm1l (long double x)
  {
    if (isnanl (x))
      return x;
  
    if (x >= (long double) LDBL_MAX_EXP * LOG2_PLUS_EPSILON)
      /* x > LDBL_MAX_EXP * log(2)
         hence exp(x) > 2^LDBL_MAX_EXP, overflows to Infinity.  */
      return HUGE_VALL;
  
    if (x <= (long double) (- LDBL_MANT_DIG) * LOG2_PLUS_EPSILON)
      /* x < (- LDBL_MANT_DIG) * log(2)
         hence 0 < exp(x) < 2^-LDBL_MANT_DIG,
         hence -1 < exp(x)-1 < -1 + 2^-LDBL_MANT_DIG
         rounds to -1.  */
      return -1.0L;
  
    if (x <= - LOG2_PLUS_EPSILON)
      /* 0 < exp(x) < 1/2.
         Just compute exp(x), then subtract 1.  */
      return expl (x) - 1.0L;
  
    if (x == 0.0L)
      /* 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^13 term is < 0.0007^12 < 2^-120 <= 2^-LDBL_MANT_DIG, therefore we
         can truncate the series after the z^11 term.
  
       Given the usual bounds LDBL_MAX_EXP <= 16384, LDBL_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 LDBL_MANT_DIG bits, we have to use an approximation
       to log(2) that has 14+LDBL_MANT_DIG bits.  */
  
    {
      long double nm = roundl (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^-LDBL_MANT_DIG.  */
      long double y_tmp = x - nm * LOG2_BY_256_HI_PART;
      long double y = y_tmp - nm * LOG2_BY_256_LO_PART;
      long double z = 0.5L * y;
  
  /* Coefficients of the power series for tanh(z).  */
  #define TANH_COEFF_1   1.0L
  #define TANH_COEFF_3  -0.333333333333333333333333333333333333334L
  #define TANH_COEFF_5   0.133333333333333333333333333333333333334L
  #define TANH_COEFF_7  -0.053968253968253968253968253968253968254L
  #define TANH_COEFF_9   0.0218694885361552028218694885361552028218L
  #define TANH_COEFF_11 -0.00886323552990219656886323552990219656886L
  #define TANH_COEFF_13  0.00359212803657248101692546136990581435026L
  #define TANH_COEFF_15 -0.00145583438705131826824948518070211191904L
  
      long double z2 = z * z;
      long double tanh_z =
        (((((TANH_COEFF_11
             * z2 + TANH_COEFF_9)
            * z2 + TANH_COEFF_7)
           * z2 + TANH_COEFF_5)
          * z2 + TANH_COEFF_3)
         * z2 + TANH_COEFF_1)
        * z;
  
      long double exp_y_minus_1 = 2.0L * tanh_z / (1.0L - tanh_z);
  
      int n = (int) roundl (nm * (1.0L / 256.0L));
      int m = (int) nm - 256 * n;
  
      /* expm1l_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 long double expm1l_table[257] =
        {
          -0.292893218813452475599155637895150960716L,
          -0.290976057839792401079436677742323809165L,
          -0.289053698915417220095325702647879950038L,
          -0.287126127947252846596498423285616993819L,
          -0.285193330804014994382467110862430046956L,
          -0.283255293316105578740250215722626632811L,
          -0.281312001275508837198386957752147486471L,
          -0.279363440435687168635744042695052413926L,
          -0.277409596511476689981496879264164547161L,
          -0.275450455178982509740597294512888729286L,
          -0.273486002075473717576963754157712706214L,
          -0.271516222799278089184548475181393238264L,
          -0.269541102909676505674348554844689233423L,
          -0.267560627926797086703335317887720824384L,
          -0.265574783331509036569177486867109287348L,
          -0.263583554565316202492529493866889713058L,
          -0.261586927030250344306546259812975038038L,
          -0.259584886088764114771170054844048746036L,
          -0.257577417063623749727613604135596844722L,
          -0.255564505237801467306336402685726757248L,
          -0.253546135854367575399678234256663229163L,
          -0.251522294116382286608175138287279137577L,
          -0.2494929651867872398674385184702356751864L,
          -0.247458134188296727960327722100283867508L,
          -0.24541778620328863011699022448340323429L,
          -0.243371906273695048903181511842366886387L,
          -0.24132047940089265059510885341281062657L,
          -0.239263490545592708236869372901757573532L,
          -0.237200924627730846574373155241529522695L,
          -0.23513276652635648805745654063657412692L,
          -0.233059001079521999099699248246140670544L,
          -0.230979613084171535783261520405692115669L,
          -0.228894587296029588193854068954632579346L,
          -0.226803908429489222568744221853864674729L,
          -0.224707561157500020438486294646580877171L,
          -0.222605530111455713940842831198332609562L,
          -0.2204977998810815164831359552625710592544L,
          -0.218384355014321147927034632426122058645L,
          -0.2162651800172235534675441445217774245016L,
          -0.214140259353829315375718509234297186439L,
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          -0.188187336491335584102392022226559177731L,
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          -0.1194930784812080879189542126763637438278L,
          -0.11710578203336358947830887503073906297L,
          -0.1147120129682226132300120925687579825894L,
          -0.1123117537367393737247203999003383961205L,
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          -0.00540057651636682434752231377783368554176L,
          -0.00270394391452987374234008615207739887604L,
          0.0L,
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          0.318079601266063994690185647066116617661L,
          0.321653277603157514326511812330609226158L,
          0.325236643159741294629537095498721674113L,
          0.32882972420595439547865089632866510792L,
          0.33243254708316144935164337949073577407L,
          0.336045138204145773442627904371869759286L,
          0.339667524053303005360030669724352576023L,
          0.343299731186835263824217146181630875424L,
          0.346941786232945835788173713229537282073L,
          0.350593715892034391408522196060133960038L,
          0.354255546936892728298014740140702804344L,
          0.357927306212901046494536695671766697444L,
          0.361609020638224755585535938831941474643L,
          0.365300717204011815430698360337542855432L,
          0.369002422974590611929601132982192832168L,
          0.372714165087668369284997857144717215791L,
          0.376435970754530100216322805518686960261L,
          0.380167867260238095581945274358283464698L,
          0.383909881963831954872659527265192818003L,
          0.387662042298529159042861017950775988895L,
          0.391424375771926187149835529566243446678L,
          0.395196909966200178275574599249220994717L,
          0.398979672538311140209528136715194969206L,
          0.402772691220204706374713524333378817108L,
          0.40657599381901544248361973255451684411L,
          0.410389608217270704414375128268675481146L,
          0.414213562373095048801688724209698078569L
        };
  
      long double t = expm1l_table[128 + m];
  
      /* (1+t) * (1+exp_y_minus_1) - 1 = t + (1+t)*exp_y_minus_1 */
      long double p_minus_1 = t + (1.0L + t) * exp_y_minus_1;
  
      long double s = ldexpl (1.0L, n) - 1.0L;
  
      /* (1+s) * (1+p_minus_1) - 1 = s + (1+s)*p_minus_1 */
      return s + (1.0L + s) * p_minus_1;
    }
  }
  
  #endif
  

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