1  /* Compute x^2 + y^2 - 1, without large cancellation error.
       2     Copyright (C) 2012-2018 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  
      21  /* Calculate X + Y exactly and store the result in *HI + *LO.  It is
      22     given that |X| >= |Y| and the values are small enough that no
      23     overflow occurs.  */
      24  
      25  static inline void
      26  add_split (__float128 *hi, __float128 *lo, __float128 x, __float128 y)
      27  {
      28    /* Apply Dekker's algorithm.  */
      29    *hi = x + y;
      30    *lo = (x - *hi) + y;
      31  }
      32  
      33  /* Compare absolute values of floating-point values pointed to by P
      34     and Q for qsort.  */
      35  
      36  static int
      37  compare (const void *p, const void *q)
      38  {
      39    __float128 pld = fabsq (*(const __float128 *) p);
      40    __float128 qld = fabsq (*(const __float128 *) q);
      41    if (pld < qld)
      42      return -1;
      43    else if (pld == qld)
      44      return 0;
      45    else
      46      return 1;
      47  }
      48  
      49  /* Return X^2 + Y^2 - 1, computed without large cancellation error.
      50     It is given that 1 > X >= Y >= epsilon / 2, and that X^2 + Y^2 >=
      51     0.5.  */
      52  
      53  __float128
      54  __quadmath_x2y2m1q (__float128 x, __float128 y)
      55  {
      56    __float128 vals[5];
      57    SET_RESTORE_ROUNDF128 (FE_TONEAREST);
      58    mul_splitq (&vals[1], &vals[0], x, x);
      59    mul_splitq (&vals[3], &vals[2], y, y);
      60    vals[4] = -1;
      61    qsort (vals, 5, sizeof (__float128), compare);
      62    /* Add up the values so that each element of VALS has absolute value
      63       at most equal to the last set bit of the next nonzero
      64       element.  */
      65    for (size_t i = 0; i <= 3; i++)
      66      {
      67        add_split (&vals[i + 1], &vals[i], vals[i + 1], vals[i]);
      68        qsort (vals + i + 1, 4 - i, sizeof (__float128), compare);
      69      }
      70    /* Now any error from this addition will be small.  */
      71    return vals[4] + vals[3] + vals[2] + vals[1] + vals[0];
      72  }