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_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  }