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      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 }
     73