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1 /* Compute x^2 + y^2 - 1, without large cancellation error.
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2 Copyright (C) 2012 Free Software Foundation, Inc.
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3 This file is part of the GNU C Library.
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4
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5 The GNU C Library is free software; you can redistribute it and/or
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6 modify it under the terms of the GNU Lesser General Public
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7 License as published by the Free Software Foundation; either
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8 version 2.1 of the License, or (at your option) any later version.
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9
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10 The GNU C Library is distributed in the hope that it will be useful,
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11 but WITHOUT ANY WARRANTY; without even the implied warranty of
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12 MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
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13 Lesser General Public License for more details.
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14
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15 You should have received a copy of the GNU Lesser General Public
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16 License along with the GNU C Library; if not, see
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17 <http://www.gnu.org/licenses/>. */
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18
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19 #include "quadmath-imp.h"
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20 #include <stdlib.h>
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21
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22 /* Calculate X + Y exactly and store the result in *HI + *LO. It is
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23 given that |X| >= |Y| and the values are small enough that no
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24 overflow occurs. */
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25
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26 static inline void
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27 add_split (__float128 *hi, __float128 *lo, __float128 x, __float128 y)
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28 {
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29 /* Apply Dekker's algorithm. */
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30 *hi = x + y;
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31 *lo = (x - *hi) + y;
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32 }
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33
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34 /* Calculate X * Y exactly and store the result in *HI + *LO. It is
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35 given that the values are small enough that no overflow occurs and
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36 large enough (or zero) that no underflow occurs. */
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37
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38 static inline void
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39 mul_split (__float128 *hi, __float128 *lo, __float128 x, __float128 y)
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40 {
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41 /* Fast built-in fused multiply-add. */
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42 *hi = x * y;
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43 *lo = fmaq (x, y, -*hi);
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44 }
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45
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46 /* Compare absolute values of floating-point values pointed to by P
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47 and Q for qsort. */
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48
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49 static int
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50 compare (const void *p, const void *q)
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51 {
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52 __float128 pld = fabsq (*(const __float128 *) p);
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53 __float128 qld = fabsq (*(const __float128 *) q);
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54 if (pld < qld)
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55 return -1;
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56 else if (pld == qld)
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57 return 0;
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58 else
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59 return 1;
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60 }
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61
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62 /* Return X^2 + Y^2 - 1, computed without large cancellation error.
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63 It is given that 1 > X >= Y >= epsilon / 2, and that either X >=
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64 0.75 or Y >= 0.5. */
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65
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66 __float128
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67 __quadmath_x2y2m1q (__float128 x, __float128 y)
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68 {
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69 __float128 vals[4];
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70 size_t i;
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71
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72 /* FIXME: SET_RESTORE_ROUNDL (FE_TONEAREST); */
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73 mul_split (&vals[1], &vals[0], x, x);
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74 mul_split (&vals[3], &vals[2], y, y);
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75 if (x >= 0.75Q)
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76 vals[1] -= 1.0Q;
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77 else
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78 {
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79 vals[1] -= 0.5Q;
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80 vals[3] -= 0.5Q;
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81 }
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82 qsort (vals, 4, sizeof (__float128), compare);
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83 /* Add up the values so that each element of VALS has absolute value
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84 at most equal to the last set bit of the next nonzero
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85 element. */
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86 for (i = 0; i <= 2; i++)
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87 {
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88 add_split (&vals[i + 1], &vals[i], vals[i + 1], vals[i]);
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89 qsort (vals + i + 1, 3 - i, sizeof (__float128), compare);
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90 }
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91 /* Now any error from this addition will be small. */
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92 return vals[3] + vals[2] + vals[1] + vals[0];
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93 }
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