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

   1 /* Compute x^2 + y^2 - 1, without large cancellation error.
   2    Copyright (C) 2012-2014 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 <float.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 /* Calculate X * Y exactly and store the result in *HI + *LO.  It is
  37    given that the values are small enough that no overflow occurs and
  38    large enough (or zero) that no underflow occurs.  */
  39
  40 static inline void
  41 mul_split (double *hi, double *lo, double x, double y)
  42 {
  43 #ifdef __FP_FAST_FMA
  44   /* Fast built-in fused multiply-add.  */
  45   *hi = x * y;
  46   *lo = __builtin_fma (x, y, -*hi);
  47 #elif defined FP_FAST_FMA
  48   /* Fast library fused multiply-add, compiler before GCC 4.6.  */
  49   *hi = x * y;
  50   *lo = __fma (x, y, -*hi);
  51 #else
  52   /* Apply Dekker's algorithm.  */
  53   *hi = x * y;
  54 # define C ((1 << (DBL_MANT_DIG + 1) / 2) + 1)
  55   double x1 = x * C;
  56   double y1 = y * C;
  57 # undef C
  58   x1 = (x - x1) + x1;
  59   y1 = (y - y1) + y1;
  60   double x2 = x - x1;
  61   double y2 = y - y1;
  62   *lo = (((x1 * y1 - *hi) + x1 * y2) + x2 * y1) + x2 * y2;
  63 #endif
  64 }
  65
  66 /* Compare absolute values of floating-point values pointed to by P
  67    and Q for qsort.  */
  68
  69 static int
  70 compare (const void *p, const void *q)
  71 {
  72   double pd = fabs (*(const double *) p);
  73   double qd = fabs (*(const double *) q);
  74   if (pd < qd)
  75     return -1;
  76   else if (pd == qd)
  77     return 0;
  78   else
  79     return 1;
  80 }
  81
  82 /* Return X^2 + Y^2 - 1, computed without large cancellation error.
  83    It is given that 1 > X >= Y >= epsilon / 2, and that either X >=
  84    0.75 or Y >= 0.5.  */
  85
  86 long double
  87 __x2y2m1l (long double x, long double y)
  88 {
  89   double vals[12];
  90   SET_RESTORE_ROUND (FE_TONEAREST);
  91   union ibm_extended_long_double xu, yu;
  92   xu.ld = x;
  93   yu.ld = y;
  94   if (fabs (xu.d[1].d) < 0x1p-500)
  95     xu.d[1].d = 0.0;
  96   if (fabs (yu.d[1].d) < 0x1p-500)
  97     yu.d[1].d = 0.0;
  98   mul_split (&vals[1], &vals[0], xu.d[0].d, xu.d[0].d);
  99   mul_split (&vals[3], &vals[2], xu.d[0].d, xu.d[1].d);
 100   vals[2] *= 2.0;
 101   vals[3] *= 2.0;
 102   mul_split (&vals[5], &vals[4], xu.d[1].d, xu.d[1].d);
 103   mul_split (&vals[7], &vals[6], yu.d[0].d, yu.d[0].d);
 104   mul_split (&vals[9], &vals[8], yu.d[0].d, yu.d[1].d);
 105   vals[8] *= 2.0;
 106   vals[9] *= 2.0;
 107   mul_split (&vals[11], &vals[10], yu.d[1].d, yu.d[1].d);
 108   if (xu.d[0].d >= 0.75)
 109     vals[1] -= 1.0;
 110   else
 111     {
 112       vals[1] -= 0.5;
 113       vals[7] -= 0.5;
 114     }
 115   qsort (vals, 12, sizeof (double), compare);
 116   /* Add up the values so that each element of VALS has absolute value
 117      at most equal to the last set bit of the next nonzero
 118      element.  */
 119   for (size_t i = 0; i <= 10; i++)
 120     {
 121       add_split (&vals[i + 1], &vals[i], vals[i + 1], vals[i]);
 122       qsort (vals + i + 1, 11 - i, sizeof (double), compare);
 123     }
 124   /* Now any error from this addition will be small.  */
 125   long double retval = (long double) vals[11];
 126   for (size_t i = 10; i != (size_t) -1; i--)
 127     retval += (long double) vals[i];
 128   return retval;
 129 }