| /* 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 (double *hi, double *lo, double x, 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 (double *hi, double *lo, double x, double y) |
| { |
| #ifdef __FP_FAST_FMA |
| /* Fast built-in fused multiply-add. */ |
| *hi = x * y; |
| *lo = __builtin_fma (x, y, -*hi); |
| #elif defined FP_FAST_FMA |
| /* Fast library fused multiply-add, compiler before GCC 4.6. */ |
| *hi = x * y; |
| *lo = __fma (x, y, -*hi); |
| #else |
| /* Apply Dekker's algorithm. */ |
| *hi = x * y; |
| # define C ((1 << (DBL_MANT_DIG + 1) / 2) + 1) |
| double x1 = x * C; |
| double y1 = y * C; |
| # undef C |
| x1 = (x - x1) + x1; |
| y1 = (y - y1) + y1; |
| double x2 = x - x1; |
| 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) |
| { |
| double pd = fabs (*(const double *) p); |
| double qd = fabs (*(const double *) q); |
| if (pd < qd) |
| return -1; |
| else if (pd == qd) |
| 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) |
| { |
| double vals[12]; |
| SET_RESTORE_ROUND (FE_TONEAREST); |
| union ibm_extended_long_double xu, yu; |
| xu.ld = x; |
| yu.ld = y; |
| if (fabs (xu.d[1].d) < 0x1p-500) |
| xu.d[1].d = 0.0; |
| if (fabs (yu.d[1].d) < 0x1p-500) |
| yu.d[1].d = 0.0; |
| mul_split (&vals[1], &vals[0], xu.d[0].d, xu.d[0].d); |
| mul_split (&vals[3], &vals[2], xu.d[0].d, xu.d[1].d); |
| vals[2] *= 2.0; |
| vals[3] *= 2.0; |
| mul_split (&vals[5], &vals[4], xu.d[1].d, xu.d[1].d); |
| mul_split (&vals[7], &vals[6], yu.d[0].d, yu.d[0].d); |
| mul_split (&vals[9], &vals[8], yu.d[0].d, yu.d[1].d); |
| vals[8] *= 2.0; |
| vals[9] *= 2.0; |
| mul_split (&vals[11], &vals[10], yu.d[1].d, yu.d[1].d); |
| if (xu.d[0].d >= 0.75) |
| vals[1] -= 1.0; |
| else |
| { |
| vals[1] -= 0.5; |
| vals[7] -= 0.5; |
| } |
| qsort (vals, 12, sizeof (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 <= 10; i++) |
| { |
| add_split (&vals[i + 1], &vals[i], vals[i + 1], vals[i]); |
| qsort (vals + i + 1, 11 - i, sizeof (double), compare); |
| } |
| /* Now any error from this addition will be small. */ |
| long double retval = (long double) vals[11]; |
| for (size_t i = 10; i != (size_t) -1; i--) |
| retval += (long double) vals[i]; |
| return retval; |
| } |
