kc3-lang/libquadmath/math/log2q.c

Branch - branch -master

• Show log

  Commit

• Hash : 4f659ab9
  Author : Thomas de Grivel
  Date : 2025-11-02T19:33:45
  

  initial commit
  

Download

• math/log2q.c

• /*                                                      log2l.c
   *      Base 2 logarithm, 128-bit long double precision
   *
   *
   *
   * SYNOPSIS:
   *
   * long double x, y, log2l();
   *
   * y = log2l( x );
   *
   *
   *
   * DESCRIPTION:
   *
   * Returns the base 2 logarithm of x.
   *
   * The argument is separated into its exponent and fractional
   * parts.  If the exponent is between -1 and +1, the (natural)
   * logarithm of the fraction is approximated by
   *
   *     log(1+x) = x - 0.5 x^2 + x^3 P(x)/Q(x).
   *
   * Otherwise, setting  z = 2(x-1)/x+1),
   *
   *     log(x) = z + z^3 P(z)/Q(z).
   *
   *
   *
   * ACCURACY:
   *
   *                      Relative error:
   * arithmetic   domain     # trials      peak         rms
   *    IEEE      0.5, 2.0     100,000    2.6e-34     4.9e-35
   *    IEEE     exp(+-10000)  100,000    9.6e-35     4.0e-35
   *
   * In the tests over the interval exp(+-10000), the logarithms
   * of the random arguments were uniformly distributed over
   * [-10000, +10000].
   *
   */
  
  /*
     Cephes Math Library Release 2.2:  January, 1991
     Copyright 1984, 1991 by Stephen L. Moshier
     Adapted for glibc November, 2001
  
      This library 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 2.1 of the License, or (at your option) any later version.
  
      This library 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 library; if not, see <http://www.gnu.org/licenses/>.
   */
  
  #include "quadmath-imp.h"
  
  /* Coefficients for ln(1+x) = x - x**2/2 + x**3 P(x)/Q(x)
   * 1/sqrt(2) <= x < sqrt(2)
   * Theoretical peak relative error = 5.3e-37,
   * relative peak error spread = 2.3e-14
   */
  static const __float128 P[13] =
  {
    1.313572404063446165910279910527789794488E4Q,
    7.771154681358524243729929227226708890930E4Q,
    2.014652742082537582487669938141683759923E5Q,
    3.007007295140399532324943111654767187848E5Q,
    2.854829159639697837788887080758954924001E5Q,
    1.797628303815655343403735250238293741397E5Q,
    7.594356839258970405033155585486712125861E4Q,
    2.128857716871515081352991964243375186031E4Q,
    3.824952356185897735160588078446136783779E3Q,
    4.114517881637811823002128927449878962058E2Q,
    2.321125933898420063925789532045674660756E1Q,
    4.998469661968096229986658302195402690910E-1Q,
    1.538612243596254322971797716843006400388E-6Q
  };
  static const __float128 Q[12] =
  {
    3.940717212190338497730839731583397586124E4Q,
    2.626900195321832660448791748036714883242E5Q,
    7.777690340007566932935753241556479363645E5Q,
    1.347518538384329112529391120390701166528E6Q,
    1.514882452993549494932585972882995548426E6Q,
    1.158019977462989115839826904108208787040E6Q,
    6.132189329546557743179177159925690841200E5Q,
    2.248234257620569139969141618556349415120E5Q,
    5.605842085972455027590989944010492125825E4Q,
    9.147150349299596453976674231612674085381E3Q,
    9.104928120962988414618126155557301584078E2Q,
    4.839208193348159620282142911143429644326E1Q
  /* 1.000000000000000000000000000000000000000E0L, */
  };
  
  /* Coefficients for log(x) = z + z^3 P(z^2)/Q(z^2),
   * where z = 2(x-1)/(x+1)
   * 1/sqrt(2) <= x < sqrt(2)
   * Theoretical peak relative error = 1.1e-35,
   * relative peak error spread 1.1e-9
   */
  static const __float128 R[6] =
  {
    1.418134209872192732479751274970992665513E5Q,
   -8.977257995689735303686582344659576526998E4Q,
    2.048819892795278657810231591630928516206E4Q,
   -2.024301798136027039250415126250455056397E3Q,
    8.057002716646055371965756206836056074715E1Q,
   -8.828896441624934385266096344596648080902E-1Q
  };
  static const __float128 S[6] =
  {
    1.701761051846631278975701529965589676574E6Q,
   -1.332535117259762928288745111081235577029E6Q,
    4.001557694070773974936904547424676279307E5Q,
   -5.748542087379434595104154610899551484314E4Q,
    3.998526750980007367835804959888064681098E3Q,
   -1.186359407982897997337150403816839480438E2Q
  /* 1.000000000000000000000000000000000000000E0L, */
  };
  
  static const __float128
  /* log2(e) - 1 */
  LOG2EA = 4.4269504088896340735992468100189213742664595E-1Q,
  /* sqrt(2)/2 */
  SQRTH = 7.071067811865475244008443621048490392848359E-1Q;
  
  
  /* Evaluate P[n] x^n  +  P[n-1] x^(n-1)  +  ...  +  P[0] */
  
  static __float128
  neval (__float128 x, const __float128 *p, int n)
  {
    __float128 y;
  
    p += n;
    y = *p--;
    do
      {
        y = y * x + *p--;
      }
    while (--n > 0);
    return y;
  }
  
  
  /* Evaluate x^n+1  +  P[n] x^(n)  +  P[n-1] x^(n-1)  +  ...  +  P[0] */
  
  static __float128
  deval (__float128 x, const __float128 *p, int n)
  {
    __float128 y;
  
    p += n;
    y = x + *p--;
    do
      {
        y = y * x + *p--;
      }
    while (--n > 0);
    return y;
  }
  
  
  
  __float128
  log2q (__float128 x)
  {
    __float128 z;
    __float128 y;
    int e;
    int64_t hx, lx;
  
  /* Test for domain */
    GET_FLT128_WORDS64 (hx, lx, x);
    if (((hx & 0x7fffffffffffffffLL) | lx) == 0)
      return (-1 / fabsq (x));		/* log2l(+-0)=-inf  */
    if (hx < 0)
      return (x - x) / (x - x);
    if (hx >= 0x7fff000000000000LL)
      return (x + x);
  
    if (x == 1)
      return 0;
  
  /* separate mantissa from exponent */
  
  /* Note, frexp is used so that denormal numbers
   * will be handled properly.
   */
    x = frexpq (x, &e);
  
  
  /* logarithm using log(x) = z + z**3 P(z)/Q(z),
   * where z = 2(x-1)/x+1)
   */
    if ((e > 2) || (e < -2))
      {
        if (x < SQRTH)
  	{			/* 2( 2x-1 )/( 2x+1 ) */
  	  e -= 1;
  	  z = x - 0.5Q;
  	  y = 0.5Q * z + 0.5Q;
  	}
        else
  	{			/*  2 (x-1)/(x+1)   */
  	  z = x - 0.5Q;
  	  z -= 0.5Q;
  	  y = 0.5Q * x + 0.5Q;
  	}
        x = z / y;
        z = x * x;
        y = x * (z * neval (z, R, 5) / deval (z, S, 5));
        goto done;
      }
  
  
  /* logarithm using log(1+x) = x - .5x**2 + x**3 P(x)/Q(x) */
  
    if (x < SQRTH)
      {
        e -= 1;
        x = 2.0 * x - 1;	/*  2x - 1  */
      }
    else
      {
        x = x - 1;
      }
    z = x * x;
    y = x * (z * neval (x, P, 12) / deval (x, Q, 11));
    y = y - 0.5 * z;
  
  done:
  
  /* Multiply log of fraction by log2(e)
   * and base 2 exponent by 1
   */
    z = y * LOG2EA;
    z += x * LOG2EA;
    z += y;
    z += x;
    z += e;
    return (z);
  }
  

Download