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_split.h>
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
30 add_split (double *hi
, double *lo
, double x
, double y
)
32 /* Apply Dekker's algorithm. */
37 /* Compare absolute values of floating-point values pointed to by P
41 compare (const void *p
, const void *q
)
43 double pd
= fabs (*(const double *) p
);
44 double qd
= fabs (*(const double *) q
);
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 >=
58 __x2y2m1l (long double x
, long double y
)
61 SET_RESTORE_ROUND (FE_TONEAREST
);
62 union ibm_extended_long_double xu
, yu
;
65 if (fabs (xu
.d
[1].d
) < 0x1p
-500)
67 if (fabs (yu
.d
[1].d
) < 0x1p
-500)
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
);
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
);
78 mul_split (&vals
[11], &vals
[10], yu
.d
[1].d
, yu
.d
[1].d
);
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
84 for (size_t i
= 0; i
<= 11; i
++)
86 add_split (&vals
[i
+ 1], &vals
[i
], vals
[i
+ 1], vals
[i
]);
87 qsort (vals
+ i
+ 1, 12 - i
, sizeof (double), compare
);
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
];