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
25static inline void
26add_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
36static int
37compare (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