Subversion Repositories openrisc

/*                                                      log1pl.c

 *

 *      Relative error logarithm

 *      Natural logarithm of 1+x, 128-bit long double precision

 *

 *

 *

 * SYNOPSIS:

 *

 * long double x, y, log1pl();

 *

 * y = log1pl( 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, write to the Free Software

    Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307  USA */

#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 = 1.000000000000000000000000000000000000000E0Q, */

  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 = 1.000000000000000000000000000000000000000E0Q, */

  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;

static const __float128 zero = 0.0Q;

__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 >= 0x7fff0000)

    return xm1;

  /* log1p(+- 0) = +- 0.  */

  if (((hx & 0x7fffffff) == 0)

      && (u.words32.w1 | u.words32.w2 | u.words32.w3) == 0)

    return xm1;

  x = xm1 + 1.0Q;

  /* log1p(-1) = -inf */

  if (x <= 0.0Q)

    {

      if (x == 0.0Q)

        return (-1.0Q / (x - x));

      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 = 2.0Q * x - 1.0Q;    /*  2x - 1  */

      else

        x = xm1;

    }

  else

    {

      if (e != 0)

        x = x - 1.0Q;

      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);

}