1  /* Compute x^2 + y^2 - 1, without large cancellation error.
       2     Copyright (C) 2012-2023 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_split.h>
      23  #include <stdlib.h>
      24  
      25  /* Calculate X + Y exactly and store the result in *HI + *LO.  It is
      26     given that |X| >= |Y| and the values are small enough that no
      27     overflow occurs.  */
      28  
      29  static inline void
      30  add_split (double *hi, double *lo, double x, double y)
      31  {
      32    /* Apply Dekker's algorithm.  */
      33    *hi = x + y;
      34    *lo = (x - *hi) + y;
      35  }
      36  
      37  /* Compare absolute values of floating-point values pointed to by P
      38     and Q for qsort.  */
      39  
      40  static int
      41  compare (const void *p, const void *q)
      42  {
      43    double pd = fabs (*(const double *) p);
      44    double qd = fabs (*(const double *) q);
      45    if (pd < qd)
      46      return -1;
      47    else if (pd == qd)
      48      return 0;
      49    else
      50      return 1;
      51  }
      52  
      53  /* Return X^2 + Y^2 - 1, computed without large cancellation error.
      54     It is given that 1 > X >= Y >= epsilon / 2, and that X^2 + Y^2 >=
      55     0.5.  */
      56  
      57  long double
      58  __x2y2m1l (long double x, long double y)
      59  {
      60    double vals[13];
      61    SET_RESTORE_ROUND (FE_TONEAREST);
      62    union ibm_extended_long_double xu, yu;
      63    xu.ld = x;
      64    yu.ld = y;
      65    if (fabs (xu.d[1].d) < 0x1p-500)
      66      xu.d[1].d = 0.0;
      67    if (fabs (yu.d[1].d) < 0x1p-500)
      68      yu.d[1].d = 0.0;
      69    mul_split (&vals[1], &vals[0], xu.d[0].d, xu.d[0].d);
      70    mul_split (&vals[3], &vals[2], xu.d[0].d, xu.d[1].d);
      71    vals[2] *= 2.0;
      72    vals[3] *= 2.0;
      73    mul_split (&vals[5], &vals[4], xu.d[1].d, xu.d[1].d);
      74    mul_split (&vals[7], &vals[6], yu.d[0].d, yu.d[0].d);
      75    mul_split (&vals[9], &vals[8], yu.d[0].d, yu.d[1].d);
      76    vals[8] *= 2.0;
      77    vals[9] *= 2.0;
      78    mul_split (&vals[11], &vals[10], yu.d[1].d, yu.d[1].d);
      79    vals[12] = -1.0;
      80    qsort (vals, 13, sizeof (double), compare);
      81    /* Add up the values so that each element of VALS has absolute value
      82       at most equal to the last set bit of the next nonzero
      83       element.  */
      84    for (size_t i = 0; i <= 11; i++)
      85      {
      86        add_split (&vals[i + 1], &vals[i], vals[i + 1], vals[i]);
      87        qsort (vals + i + 1, 12 - i, sizeof (double), compare);
      88      }
      89    /* Now any error from this addition will be small.  */
      90    long double retval = (long double) vals[12];
      91    for (size_t i = 11; i != (size_t) -1; i--)
      92      retval += (long double) vals[i];
      93    return retval;
      94  }