libquadmath/math/x2y2m1q.c

   1 /* Compute x^2 + y^2 - 1, without large cancellation error.
   2    Copyright (C) 2012-2018 Free Software Foundation, Inc.
   3    This file is part of the GNU C Library.
   4
   5    The GNU C Library is free software; you can redistribute it and/or
   6    modify it under the terms of the GNU Lesser General Public
   7    License as published by the Free Software Foundation; either
   8    version 2.1 of the License, or (at your option) any later version.
   9
  10    The GNU C Library is distributed in the hope that it will be useful,
  11    but WITHOUT ANY WARRANTY; without even the implied warranty of
  12    MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
  13    Lesser General Public License for more details.
  14
  15    You should have received a copy of the GNU Lesser General Public
  16    License along with the GNU C Library; if not, see
  17    <http://www.gnu.org/licenses/>.  */
  18
  19 #include "quadmath-imp.h"
  20
  21 /* Calculate X + Y exactly and store the result in *HI + *LO.  It is
  22    given that |X| >= |Y| and the values are small enough that no
  23    overflow occurs.  */
  24
  25 static inline void
  26 add_split (__float128 *hi, __float128 *lo, __float128 x, __float128 y)
  27 {
  28   /* Apply Dekker's algorithm.  */
  29   *hi = x + y;
  30   *lo = (x - *hi) + y;
  31 }
  32
  33 /* Compare absolute values of floating-point values pointed to by P
  34    and Q for qsort.  */
  35
  36 static int
  37 compare (const void *p, const void *q)
  38 {
  39   __float128 pld = fabsq (*(const __float128 *) p);
  40   __float128 qld = fabsq (*(const __float128 *) q);
  41   if (pld < qld)
  42     return -1;
  43   else if (pld == qld)
  44     return 0;
  45   else
  46     return 1;
  47 }
  48
  49 /* Return X^2 + Y^2 - 1, computed without large cancellation error.
  50    It is given that 1 > X >= Y >= epsilon / 2, and that X^2 + Y^2 >=
  51    0.5.  */
  52
  53 __float128
  54 __quadmath_x2y2m1q (__float128 x, __float128 y)
  55 {
  56   __float128 vals[5];
  57   SET_RESTORE_ROUNDF128 (FE_TONEAREST);
  58   mul_splitq (&vals[1], &vals[0], x, x);
  59   mul_splitq (&vals[3], &vals[2], y, y);
  60   vals[4] = -1;
  61   qsort (vals, 5, sizeof (__float128), compare);
  62   /* Add up the values so that each element of VALS has absolute value
  63      at most equal to the last set bit of the next nonzero
  64      element.  */
  65   for (size_t i = 0; i <= 3; i++)
  66     {
  67       add_split (&vals[i + 1], &vals[i], vals[i + 1], vals[i]);
  68       qsort (vals + i + 1, 4 - i, sizeof (__float128), compare);
  69     }
  70   /* Now any error from this addition will be small.  */
  71   return vals[4] + vals[3] + vals[2] + vals[1] + vals[0];
  72 }