| /* Compute x^2 + y^2 - 1, without large cancellation error. |
| Copyright (C) 2012-2014 Free Software Foundation, Inc. |
| This file is part of the GNU C Library. |
| |
| The GNU C Library is free software; you can redistribute it and/or |
| modify it under the terms of the GNU Lesser General Public |
| License as published by the Free Software Foundation; either |
| version 2.1 of the License, or (at your option) any later version. |
| |
| The GNU C Library is distributed in the hope that it will be useful, |
| but WITHOUT ANY WARRANTY; without even the implied warranty of |
| MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU |
| Lesser General Public License for more details. |
| |
| You should have received a copy of the GNU Lesser General Public |
| License along with the GNU C Library; if not, see |
| <http://www.gnu.org/licenses/>. */ |
| |
| #include <math.h> |
| #include <math_private.h> |
| #include <float.h> |
| #include <stdlib.h> |
| |
| /* Calculate X + Y exactly and store the result in *HI + *LO. It is |
| given that |X| >= |Y| and the values are small enough that no |
| overflow occurs. */ |
| |
| static inline void |
| add_split (long double *hi, long double *lo, long double x, long double y) |
| { |
| /* Apply Dekker's algorithm. */ |
| *hi = x + y; |
| *lo = (x - *hi) + y; |
| } |
| |
| /* Calculate X * Y exactly and store the result in *HI + *LO. It is |
| given that the values are small enough that no overflow occurs and |
| large enough (or zero) that no underflow occurs. */ |
| |
| static inline void |
| mul_split (long double *hi, long double *lo, long double x, long double y) |
| { |
| #ifdef __FP_FAST_FMAL |
| /* Fast built-in fused multiply-add. */ |
| *hi = x * y; |
| *lo = __builtin_fmal (x, y, -*hi); |
| #elif defined FP_FAST_FMAL |
| /* Fast library fused multiply-add, compiler before GCC 4.6. */ |
| *hi = x * y; |
| *lo = __fmal (x, y, -*hi); |
| #else |
| /* Apply Dekker's algorithm. */ |
| *hi = x * y; |
| # define C ((1LL << (LDBL_MANT_DIG + 1) / 2) + 1) |
| long double x1 = x * C; |
| long double y1 = y * C; |
| # undef C |
| x1 = (x - x1) + x1; |
| y1 = (y - y1) + y1; |
| long double x2 = x - x1; |
| long double y2 = y - y1; |
| *lo = (((x1 * y1 - *hi) + x1 * y2) + x2 * y1) + x2 * y2; |
| #endif |
| } |
| |
| /* Compare absolute values of floating-point values pointed to by P |
| and Q for qsort. */ |
| |
| static int |
| compare (const void *p, const void *q) |
| { |
| long double pld = fabsl (*(const long double *) p); |
| long double qld = fabsl (*(const long double *) q); |
| if (pld < qld) |
| return -1; |
| else if (pld == qld) |
| return 0; |
| else |
| return 1; |
| } |
| |
| /* Return X^2 + Y^2 - 1, computed without large cancellation error. |
| It is given that 1 > X >= Y >= epsilon / 2, and that either X >= |
| 0.75 or Y >= 0.5. */ |
| |
| long double |
| __x2y2m1l (long double x, long double y) |
| { |
| long double vals[4]; |
| SET_RESTORE_ROUNDL (FE_TONEAREST); |
| mul_split (&vals[1], &vals[0], x, x); |
| mul_split (&vals[3], &vals[2], y, y); |
| if (x >= 0.75L) |
| vals[1] -= 1.0L; |
| else |
| { |
| vals[1] -= 0.5L; |
| vals[3] -= 0.5L; |
| } |
| qsort (vals, 4, sizeof (long double), compare); |
| /* Add up the values so that each element of VALS has absolute value |
| at most equal to the last set bit of the next nonzero |
| element. */ |
| for (size_t i = 0; i <= 2; i++) |
| { |
| add_split (&vals[i + 1], &vals[i], vals[i + 1], vals[i]); |
| qsort (vals + i + 1, 3 - i, sizeof (long double), compare); |
| } |
| /* Now any error from this addition will be small. */ |
| return vals[3] + vals[2] + vals[1] + vals[0]; |
| } |