kc3-lang/libquadmath/math/log1pq.c

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

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• math/log1pq.c

• /*							log1pq.c
   *
   *      Relative error logarithm
   *	Natural logarithm of 1+x, 128-bit long double precision
   *
   *
   *
   * SYNOPSIS:
   *
   * long double x, y, log1pq();
   *
   * y = log1pq( x );
   *
   *
   *
   * DESCRIPTION:
   *
   * Returns the base e (2.718...) logarithm of 1+x.
   *
   * The argument 1+x is separated into its exponent and fractional
   * parts.  If the exponent is between -1 and +1, the 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(w-1)/(w+1),
   *
   *     log(w) = z + z^3 P(z)/Q(z).
   *
   *
   *
   * ACCURACY:
   *
   *                      Relative error:
   * arithmetic   domain     # trials      peak         rms
   *    IEEE      -1, 8       100000      1.9e-34     4.3e-35
   */
  
  /* Copyright 2001 by Stephen L. Moshier
  
      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 log(1+x) = x - x^2 / 2 + x^3 P(x)/Q(x)
   * 1/sqrt(2) <= 1+x < sqrt(2)
   * Theoretical peak relative error = 5.3e-37,
   * relative peak error spread = 2.3e-14
   */
  static const __float128
    P12 = 1.538612243596254322971797716843006400388E-6Q,
    P11 = 4.998469661968096229986658302195402690910E-1Q,
    P10 = 2.321125933898420063925789532045674660756E1Q,
    P9 = 4.114517881637811823002128927449878962058E2Q,
    P8 = 3.824952356185897735160588078446136783779E3Q,
    P7 = 2.128857716871515081352991964243375186031E4Q,
    P6 = 7.594356839258970405033155585486712125861E4Q,
    P5 = 1.797628303815655343403735250238293741397E5Q,
    P4 = 2.854829159639697837788887080758954924001E5Q,
    P3 = 3.007007295140399532324943111654767187848E5Q,
    P2 = 2.014652742082537582487669938141683759923E5Q,
    P1 = 7.771154681358524243729929227226708890930E4Q,
    P0 = 1.313572404063446165910279910527789794488E4Q,
    /* Q12 = 1.000000000000000000000000000000000000000E0L, */
    Q11 = 4.839208193348159620282142911143429644326E1Q,
    Q10 = 9.104928120962988414618126155557301584078E2Q,
    Q9 = 9.147150349299596453976674231612674085381E3Q,
    Q8 = 5.605842085972455027590989944010492125825E4Q,
    Q7 = 2.248234257620569139969141618556349415120E5Q,
    Q6 = 6.132189329546557743179177159925690841200E5Q,
    Q5 = 1.158019977462989115839826904108208787040E6Q,
    Q4 = 1.514882452993549494932585972882995548426E6Q,
    Q3 = 1.347518538384329112529391120390701166528E6Q,
    Q2 = 7.777690340007566932935753241556479363645E5Q,
    Q1 = 2.626900195321832660448791748036714883242E5Q,
    Q0 = 3.940717212190338497730839731583397586124E4Q;
  
  /* 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
    R5 = -8.828896441624934385266096344596648080902E-1Q,
    R4 = 8.057002716646055371965756206836056074715E1Q,
    R3 = -2.024301798136027039250415126250455056397E3Q,
    R2 = 2.048819892795278657810231591630928516206E4Q,
    R1 = -8.977257995689735303686582344659576526998E4Q,
    R0 = 1.418134209872192732479751274970992665513E5Q,
    /* S6 = 1.000000000000000000000000000000000000000E0L, */
    S5 = -1.186359407982897997337150403816839480438E2Q,
    S4 = 3.998526750980007367835804959888064681098E3Q,
    S3 = -5.748542087379434595104154610899551484314E4Q,
    S2 = 4.001557694070773974936904547424676279307E5Q,
    S1 = -1.332535117259762928288745111081235577029E6Q,
    S0 = 1.701761051846631278975701529965589676574E6Q;
  
  /* C1 + C2 = ln 2 */
  static const __float128 C1 = 6.93145751953125E-1Q;
  static const __float128 C2 = 1.428606820309417232121458176568075500134E-6Q;
  
  static const __float128 sqrth = 0.7071067811865475244008443621048490392848Q;
  /* ln (2^16384 * (1 - 2^-113)) */
  static const __float128 zero = 0;
  
  __float128
  log1pq (__float128 xm1)
  {
    __float128 x, y, z, r, s;
    ieee854_float128 u;
    int32_t hx;
    int e;
  
    /* Test for NaN or infinity input. */
    u.value = xm1;
    hx = u.words32.w0;
    if ((hx & 0x7fffffff) >= 0x7fff0000)
      return xm1 + fabsq (xm1);
  
    /* log1p(+- 0) = +- 0.  */
    if (((hx & 0x7fffffff) == 0)
        && (u.words32.w1 | u.words32.w2 | u.words32.w3) == 0)
      return xm1;
  
    if ((hx & 0x7fffffff) < 0x3f8e0000)
      {
        math_check_force_underflow (xm1);
        if ((int) xm1 == 0)
  	return xm1;
      }
  
    if (xm1 >= 0x1p113Q)
      x = xm1;
    else
      x = xm1 + 1;
  
    /* log1p(-1) = -inf */
    if (x <= 0)
      {
        if (x == 0)
  	return (-1 / zero);  /* log1p(-1) = -inf */
        else
  	return (zero / (x - x));
      }
  
    /* Separate mantissa from exponent.  */
  
    /* Use frexp used so that denormal numbers will be handled properly.  */
    x = frexpq (x, &e);
  
    /* Logarithm using log(x) = z + z^3 P(z^2)/Q(z^2),
       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;
        r = ((((R5 * z
  	      + R4) * z
  	     + R3) * z
  	    + R2) * z
  	   + R1) * z
  	+ R0;
        s = (((((z
  	       + S5) * z
  	      + S4) * z
  	     + S3) * z
  	    + S2) * z
  	   + S1) * z
  	+ S0;
        z = x * (z * r / s);
        z = z + e * C2;
        z = z + x;
        z = z + e * C1;
        return (z);
      }
  
  
    /* Logarithm using log(1+x) = x - .5x^2 + x^3 P(x)/Q(x). */
  
    if (x < sqrth)
      {
        e -= 1;
        if (e != 0)
  	x = 2 * x - 1;	/*  2x - 1  */
        else
  	x = xm1;
      }
    else
      {
        if (e != 0)
  	x = x - 1;
        else
  	x = xm1;
      }
    z = x * x;
    r = (((((((((((P12 * x
  		 + P11) * x
  		+ P10) * x
  	       + P9) * x
  	      + P8) * x
  	     + P7) * x
  	    + P6) * x
  	   + P5) * x
  	  + P4) * x
  	 + P3) * x
  	+ P2) * x
         + P1) * x
      + P0;
    s = (((((((((((x
  		 + Q11) * x
  		+ Q10) * x
  	       + Q9) * x
  	      + Q8) * x
  	     + Q7) * x
  	    + Q6) * x
  	   + Q5) * x
  	  + Q4) * x
  	 + Q3) * x
  	+ Q2) * x
         + Q1) * x
      + Q0;
    y = x * (z * r / s);
    y = y + e * C2;
    z = y - 0.5Q * z;
    z = z + x;
    z = z + e * C1;
    return (z);
  }
  

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