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/*                                                      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, write to the Free Software

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

 */

#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

/* 1.000000000000000000000000000000000000000E0Q, */

};

/* 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

/* 1.000000000000000000000000000000000000000E0Q, */

};

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

  if (hx < 0)

    return (x - x) / (x - x);

  if (hx >= 0x7fff000000000000LL)

    return (x + x);

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

    }

  else

    {

      x = x - 1.0Q;

    }

  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);

}