sysdeps/ieee754/ldbl-128ibm/x2y2m1l.c

   1 /* Compute x^2 + y^2 - 1, without large cancellation error.
   2    Copyright (C) 2012-2016 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 <math.h>
  20 #include <math_private.h>
  21 #include <mul_split.h>
  22 #include <stdlib.h>
  23
  24 /* Calculate X + Y exactly and store the result in *HI + *LO.  It is
  25    given that |X| >= |Y| and the values are small enough that no
  26    overflow occurs.  */
  27
  28 static inline void
  29 add_split (double *hi, double *lo, double x, double y)
  30 {
  31   /* Apply Dekker's algorithm.  */
  32   *hi = x + y;
  33   *lo = (x - *hi) + y;
  34 }
  35
  36 /* Compare absolute values of floating-point values pointed to by P
  37    and Q for qsort.  */
  38
  39 static int
  40 compare (const void *p, const void *q)
  41 {
  42   double pd = fabs (*(const double *) p);
  43   double qd = fabs (*(const double *) q);
  44   if (pd < qd)
  45     return -1;
  46   else if (pd == qd)
  47     return 0;
  48   else
  49     return 1;
  50 }
  51
  52 /* Return X^2 + Y^2 - 1, computed without large cancellation error.
  53    It is given that 1 > X >= Y >= epsilon / 2, and that X^2 + Y^2 >=
  54    0.5.  */
  55
  56 long double
  57 __x2y2m1l (long double x, long double y)
  58 {
  59   double vals[13];
  60   SET_RESTORE_ROUND (FE_TONEAREST);
  61   union ibm_extended_long_double xu, yu;
  62   xu.ld = x;
  63   yu.ld = y;
  64   if (fabs (xu.d[1].d) < 0x1p-500)
  65     xu.d[1].d = 0.0;
  66   if (fabs (yu.d[1].d) < 0x1p-500)
  67     yu.d[1].d = 0.0;
  68   mul_split (&vals[1], &vals[0], xu.d[0].d, xu.d[0].d);
  69   mul_split (&vals[3], &vals[2], xu.d[0].d, xu.d[1].d);
  70   vals[2] *= 2.0;
  71   vals[3] *= 2.0;
  72   mul_split (&vals[5], &vals[4], xu.d[1].d, xu.d[1].d);
  73   mul_split (&vals[7], &vals[6], yu.d[0].d, yu.d[0].d);
  74   mul_split (&vals[9], &vals[8], yu.d[0].d, yu.d[1].d);
  75   vals[8] *= 2.0;
  76   vals[9] *= 2.0;
  77   mul_split (&vals[11], &vals[10], yu.d[1].d, yu.d[1].d);
  78   vals[12] = -1.0;
  79   qsort (vals, 13, sizeof (double), compare);
  80   /* Add up the values so that each element of VALS has absolute value
  81      at most equal to the last set bit of the next nonzero
  82      element.  */
  83   for (size_t i = 0; i <= 11; i++)
  84     {
  85       add_split (&vals[i + 1], &vals[i], vals[i + 1], vals[i]);
  86       qsort (vals + i + 1, 12 - i, sizeof (double), compare);
  87     }
  88   /* Now any error from this addition will be small.  */
  89   long double retval = (long double) vals[12];
  90   for (size_t i = 11; i != (size_t) -1; i--)
  91     retval += (long double) vals[i];
  92   return retval;
  93 }