1 /* Compute x^2 + y^2 - 1, without large cancellation error.
2 Copyright (C) 2012-2015 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 <http://www.gnu.org/licenses/>. */
20 #include <math_private.h>
24 /* Calculate X + Y exactly and store the result in *HI + *LO. It is
25 given that |X| >= |Y| and the values are small enough that no
29 add_split (double *hi
, double *lo
, double x
, double y
)
31 /* Apply Dekker's algorithm. */
36 /* Calculate X * Y exactly and store the result in *HI + *LO. It is
37 given that the values are small enough that no overflow occurs and
38 large enough (or zero) that no underflow occurs. */
41 mul_split (double *hi
, double *lo
, double x
, double y
)
44 /* Fast built-in fused multiply-add. */
46 *lo
= __builtin_fma (x
, y
, -*hi
);
47 #elif defined FP_FAST_FMA
48 /* Fast library fused multiply-add, compiler before GCC 4.6. */
50 *lo
= __fma (x
, y
, -*hi
);
52 /* Apply Dekker's algorithm. */
54 # define C ((1 << (DBL_MANT_DIG + 1) / 2) + 1)
62 *lo
= (((x1
* y1
- *hi
) + x1
* y2
) + x2
* y1
) + x2
* y2
;
66 /* Compare absolute values of floating-point values pointed to by P
70 compare (const void *p
, const void *q
)
72 double pd
= fabs (*(const double *) p
);
73 double qd
= fabs (*(const double *) q
);
82 /* Return X^2 + Y^2 - 1, computed without large cancellation error.
83 It is given that 1 > X >= Y >= epsilon / 2, and that either X >=
87 __x2y2m1l (long double x
, long double y
)
90 SET_RESTORE_ROUND (FE_TONEAREST
);
91 union ibm_extended_long_double xu
, yu
;
94 if (fabs (xu
.d
[1].d
) < 0x1p
-500)
96 if (fabs (yu
.d
[1].d
) < 0x1p
-500)
98 mul_split (&vals
[1], &vals
[0], xu
.d
[0].d
, xu
.d
[0].d
);
99 mul_split (&vals
[3], &vals
[2], xu
.d
[0].d
, xu
.d
[1].d
);
102 mul_split (&vals
[5], &vals
[4], xu
.d
[1].d
, xu
.d
[1].d
);
103 mul_split (&vals
[7], &vals
[6], yu
.d
[0].d
, yu
.d
[0].d
);
104 mul_split (&vals
[9], &vals
[8], yu
.d
[0].d
, yu
.d
[1].d
);
107 mul_split (&vals
[11], &vals
[10], yu
.d
[1].d
, yu
.d
[1].d
);
108 if (xu
.d
[0].d
>= 0.75)
115 qsort (vals
, 12, sizeof (double), compare
);
116 /* Add up the values so that each element of VALS has absolute value
117 at most equal to the last set bit of the next nonzero
119 for (size_t i
= 0; i
<= 10; i
++)
121 add_split (&vals
[i
+ 1], &vals
[i
], vals
[i
+ 1], vals
[i
]);
122 qsort (vals
+ i
+ 1, 11 - i
, sizeof (double), compare
);
124 /* Now any error from this addition will be small. */
125 long double retval
= (long double) vals
[11];
126 for (size_t i
= 10; i
!= (size_t) -1; i
--)
127 retval
+= (long double) vals
[i
];