1 /* Compute x^2 + y^2 - 1, without large cancellation error.
2 Copyright (C) 2012-2024 Free Software Foundation, Inc.
3 This file is part of the GNU C Library.
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.
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.
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/>. */
20 #include <math_private.h>
21 #include <fenv_private.h>
22 #include <mul_splitl.h>
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
31 add_split (_Float128
*hi
, _Float128
*lo
, _Float128 x
, _Float128 y
)
33 /* Apply Dekker's algorithm. */
38 /* Compare absolute values of floating-point values pointed to by P
42 compare (const void *p
, const void *q
)
44 _Float128 pld
= fabsl (*(const _Float128
*) p
);
45 _Float128 qld
= fabsl (*(const _Float128
*) q
);
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 >=
59 __x2y2m1l (_Float128 x
, _Float128 y
)
62 SET_RESTORE_ROUNDL (FE_TONEAREST
);
63 mul_splitl (&vals
[1], &vals
[0], x
, x
);
64 mul_splitl (&vals
[3], &vals
[2], y
, y
);
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
70 for (size_t i
= 0; i
<= 3; i
++)
72 add_split (&vals
[i
+ 1], &vals
[i
], vals
[i
+ 1], vals
[i
]);
73 qsort (vals
+ i
+ 1, 4 - i
, sizeof (_Float128
), compare
);
75 /* Now any error from this addition will be small. */
76 return vals
[4] + vals
[3] + vals
[2] + vals
[1] + vals
[0];