libquadmath/math/x2y2m1q.c

/* Compute x^2 + y^2 - 1, without large cancellation error.
   Copyright (C) 2012 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 "quadmath-imp.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 (__float128 *hi, __float128 *lo, __float128 x, __float128 y)
{
  /* Apply Dekker's algorithm.  */
  *hi = x + y;
  *lo = (x - *hi) + y;
}

/* Calculate X * Y exactly and store the result in *HI + *LO.  It is
   given that the values are small enough that no overflow occurs and
   large enough (or zero) that no underflow occurs.  */

static inline void
mul_split (__float128 *hi, __float128 *lo, __float128 x, __float128 y)
{
  /* Fast built-in fused multiply-add.  */
  *hi = x * y;
  *lo = fmaq (x, y, -*hi);
}

/* Compare absolute values of floating-point values pointed to by P
   and Q for qsort.  */

static int
compare (const void *p, const void *q)
{
  __float128 pld = fabsq (*(const __float128 *) p);
  __float128 qld = fabsq (*(const __float128 *) q);
  if (pld < qld)
    return -1;
  else if (pld == qld)
    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 either X >=
   0.75 or Y >= 0.5.  */

__float128
__quadmath_x2y2m1q (__float128 x, __float128 y)
{
  __float128 vals[4];
  size_t i;

  /* FIXME:  SET_RESTORE_ROUNDL (FE_TONEAREST);  */
  mul_split (&vals[1], &vals[0], x, x);
  mul_split (&vals[3], &vals[2], y, y);
  if (x >= 0.75Q)
    vals[1] -= 1.0Q;
  else
    {
      vals[1] -= 0.5Q;
      vals[3] -= 0.5Q;
    }
  qsort (vals, 4, sizeof (__float128), 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 (i = 0; i <= 2; i++)
    {
      add_split (&vals[i + 1], &vals[i], vals[i + 1], vals[i]);
      qsort (vals + i + 1, 3 - i, sizeof (__float128), compare);
    }
  /* Now any error from this addition will be small.  */
  return vals[3] + vals[2] + vals[1] + vals[0];
}