libquadmath/math/x2y2m1q.c

   1 /* Compute x^2 + y^2 - 1, without large cancellation error.
   2    Copyright (C) 2012 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 #include <stdlib.h>
  21
  22 /* Calculate X + Y exactly and store the result in *HI + *LO.  It is
  23    given that |X| >= |Y| and the values are small enough that no
  24    overflow occurs.  */
  25
  26 static inline void
  27 add_split (__float128 *hi, __float128 *lo, __float128 x, __float128 y)
  28 {
  29   /* Apply Dekker's algorithm.  */
  30   *hi = x + y;
  31   *lo = (x - *hi) + y;
  32 }
  33
  34 /* Calculate X * Y exactly and store the result in *HI + *LO.  It is
  35    given that the values are small enough that no overflow occurs and
  36    large enough (or zero) that no underflow occurs.  */
  37
  38 static inline void
  39 mul_split (__float128 *hi, __float128 *lo, __float128 x, __float128 y)
  40 {
  41   /* Fast built-in fused multiply-add.  */
  42   *hi = x * y;
  43   *lo = fmaq (x, y, -*hi);
  44 }
  45
  46 /* Compare absolute values of floating-point values pointed to by P
  47    and Q for qsort.  */
  48
  49 static int
  50 compare (const void *p, const void *q)
  51 {
  52   __float128 pld = fabsq (*(const __float128 *) p);
  53   __float128 qld = fabsq (*(const __float128 *) q);
  54   if (pld < qld)
  55     return -1;
  56   else if (pld == qld)
  57     return 0;
  58   else
  59     return 1;
  60 }
  61
  62 /* Return X^2 + Y^2 - 1, computed without large cancellation error.
  63    It is given that 1 > X >= Y >= epsilon / 2, and that either X >=
  64    0.75 or Y >= 0.5.  */
  65
  66 __float128
  67 __quadmath_x2y2m1q (__float128 x, __float128 y)
  68 {
  69   __float128 vals[4];
  70   size_t i;
  71
  72   /* FIXME:  SET_RESTORE_ROUNDL (FE_TONEAREST);  */
  73   mul_split (&vals[1], &vals[0], x, x);
  74   mul_split (&vals[3], &vals[2], y, y);
  75   if (x >= 0.75Q)
  76     vals[1] -= 1.0Q;
  77   else
  78     {
  79       vals[1] -= 0.5Q;
  80       vals[3] -= 0.5Q;
  81     }
  82   qsort (vals, 4, sizeof (__float128), compare);
  83   /* Add up the values so that each element of VALS has absolute value
  84      at most equal to the last set bit of the next nonzero
  85      element.  */
  86   for (i = 0; i <= 2; i++)
  87     {
  88       add_split (&vals[i + 1], &vals[i], vals[i + 1], vals[i]);
  89       qsort (vals + i + 1, 3 - i, sizeof (__float128), compare);
  90     }
  91   /* Now any error from this addition will be small.  */
  92   return vals[3] + vals[2] + vals[1] + vals[0];
  93 }