sysdeps/ieee754/ldbl-128/x2y2m1l.c

   1 /* Compute x^2 + y^2 - 1, without large cancellation error.
   2    Copyright (C) 2012-2024 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    <https://www.gnu.org/licenses/>.  */
  18
  19 #include <math.h>
  20 #include <math_private.h>
  21 #include <fenv_private.h>
  22 #include <mul_splitl.h>
  23 #include <stdlib.h>
  24
  25
  26 /* Calculate X + Y exactly and store the result in *HI + *LO.  It is
  27    given that |X| >= |Y| and the values are small enough that no
  28    overflow occurs.  */
  29
  30 static inline void
  31 add_split (_Float128 *hi, _Float128 *lo, _Float128 x, _Float128 y)
  32 {
  33   /* Apply Dekker's algorithm.  */
  34   *hi = x + y;
  35   *lo = (x - *hi) + y;
  36 }
  37
  38 /* Compare absolute values of floating-point values pointed to by P
  39    and Q for qsort.  */
  40
  41 static int
  42 compare (const void *p, const void *q)
  43 {
  44   _Float128 pld = fabsl (*(const _Float128 *) p);
  45   _Float128 qld = fabsl (*(const _Float128 *) q);
  46   if (pld < qld)
  47     return -1;
  48   else if (pld == qld)
  49     return 0;
  50   else
  51     return 1;
  52 }
  53
  54 /* Return X^2 + Y^2 - 1, computed without large cancellation error.
  55    It is given that 1 > X >= Y >= epsilon / 2, and that X^2 + Y^2 >=
  56    0.5.  */
  57
  58 _Float128
  59 __x2y2m1l (_Float128 x, _Float128 y)
  60 {
  61   _Float128 vals[5];
  62   SET_RESTORE_ROUNDL (FE_TONEAREST);
  63   mul_splitl (&vals[1], &vals[0], x, x);
  64   mul_splitl (&vals[3], &vals[2], y, y);
  65   vals[4] = -1;
  66   qsort (vals, 5, sizeof (_Float128), compare);
  67   /* Add up the values so that each element of VALS has absolute value
  68      at most equal to the last set bit of the next nonzero
  69      element.  */
  70   for (size_t i = 0; i <= 3; i++)
  71     {
  72       add_split (&vals[i + 1], &vals[i], vals[i + 1], vals[i]);
  73       qsort (vals + i + 1, 4 - i, sizeof (_Float128), compare);
  74     }
  75   /* Now any error from this addition will be small.  */
  76   return vals[4] + vals[3] + vals[2] + vals[1] + vals[0];
  77 }