/* Compute x^2 + y^2 - 1, without large cancellation error.

   Copyright (C) 2012-2018 Free Software Foundation, Inc.

   This file is part of the GNU C Library.

   The GNU C 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.

   The GNU C 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 the GNU C Library; if not, see

   <http://www.gnu.org/licenses/>.  */

#include <math.h>

#include <math_private.h>

#include <mul_split.h>

#include <stdlib.h>

/* Calculate X + Y exactly and store the result in *HI + *LO.  It is

   given that |X| >= |Y| and the values are small enough that no

   overflow occurs.  */

static inline void

add_split (double *hi, double *lo, double x, double y)

{

  /* Apply Dekker's algorithm.  */

  *hi = x + y;

  *lo = (x - *hi) + y;

}

/* Compare absolute values of floating-point values pointed to by P

   and Q for qsort.  */

static int

compare (const void *p, const void *q)

{

  double pd = fabs (*(const double *) p);

  double qd = fabs (*(const double *) q);

  if (pd < qd)

    return -1;

  else if (pd == qd)

    return 0;

  else

    return 1;

}

/* Return X^2 + Y^2 - 1, computed without large cancellation error.

   It is given that 1 > X >= Y >= epsilon / 2, and that X^2 + Y^2 >=

   0.5.  */

long double

__x2y2m1l (long double x, long double y)

{

  double vals[13];

  SET_RESTORE_ROUND (FE_TONEAREST);

  union ibm_extended_long_double xu, yu;

  xu.ld = x;

  yu.ld = y;

  if (fabs (xu.d[1].d) < 0x1p-500)

    xu.d[1].d = 0.0;

  if (fabs (yu.d[1].d) < 0x1p-500)

    yu.d[1].d = 0.0;

  mul_split (&vals[1], &vals[0], xu.d[0].d, xu.d[0].d);

  mul_split (&vals[3], &vals[2], xu.d[0].d, xu.d[1].d);

  vals[2] *= 2.0;

  vals[3] *= 2.0;

  mul_split (&vals[5], &vals[4], xu.d[1].d, xu.d[1].d);

  mul_split (&vals[7], &vals[6], yu.d[0].d, yu.d[0].d);

  mul_split (&vals[9], &vals[8], yu.d[0].d, yu.d[1].d);

  vals[8] *= 2.0;

  vals[9] *= 2.0;

  mul_split (&vals[11], &vals[10], yu.d[1].d, yu.d[1].d);

  vals[12] = -1.0;

  qsort (vals, 13, sizeof (double), compare);

  /* Add up the values so that each element of VALS has absolute value

     at most equal to the last set bit of the next nonzero

     element.  */

  for (size_t i = 0; i <= 11; i++)

    {

      add_split (&vals[i + 1], &vals[i], vals[i + 1], vals[i]);

      qsort (vals + i + 1, 12 - i, sizeof (double), compare);

    }

  /* Now any error from this addition will be small.  */

  long double retval = (long double) vals[12];

  for (size_t i = 11; i != (size_t) -1; i--)

    retval += (long double) vals[i];

  return retval;

}