| /* Compute x * y + z as ternary operation. |
| Copyright (C) 2011-2018 Free Software Foundation, Inc. |
| This file is part of the GNU C Library. |
| Contributed by David Flaherty <flaherty@linux.vnet.ibm.com>. |
| |
| 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 <fenv.h> |
| #include <float.h> |
| #include <math.h> |
| #include <math_private.h> |
| #include <math_ldbl_opt.h> |
| #include <mul_split.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 void |
| add_split (double *hi, double *lo, double x, double y) |
| { |
| /* Apply Dekker's algorithm. */ |
| *hi = x + y; |
| *lo = (x - *hi) + y; |
| } |
| |
| /* Value with extended range, used in intermediate computations. */ |
| typedef struct |
| { |
| /* Value in [0.5, 1), as from frexp, or 0. */ |
| double val; |
| /* Exponent of power of 2 it is multiplied by, or 0 for zero. */ |
| int exp; |
| } ext_val; |
| |
| /* Store D as an ext_val value. */ |
| |
| static void |
| store_ext_val (ext_val *v, double d) |
| { |
| v->val = __frexp (d, &v->exp); |
| } |
| |
| /* Store X * Y as ext_val values *V0 and *V1. */ |
| |
| static void |
| mul_ext_val (ext_val *v0, ext_val *v1, double x, double y) |
| { |
| int xexp, yexp; |
| x = __frexp (x, &xexp); |
| y = __frexp (y, &yexp); |
| double hi, lo; |
| mul_split (&hi, &lo, x, y); |
| store_ext_val (v0, hi); |
| if (hi != 0) |
| v0->exp += xexp + yexp; |
| store_ext_val (v1, lo); |
| if (lo != 0) |
| v1->exp += xexp + yexp; |
| } |
| |
| /* Compare absolute values of ext_val values pointed to by P and Q for |
| qsort. */ |
| |
| static int |
| compare (const void *p, const void *q) |
| { |
| const ext_val *pe = p; |
| const ext_val *qe = q; |
| if (pe->val == 0) |
| return qe->val == 0 ? 0 : -1; |
| else if (qe->val == 0) |
| return 1; |
| else if (pe->exp < qe->exp) |
| return -1; |
| else if (pe->exp > qe->exp) |
| return 1; |
| else |
| { |
| double pd = fabs (pe->val); |
| double qd = fabs (qe->val); |
| if (pd < qd) |
| return -1; |
| else if (pd == qd) |
| return 0; |
| else |
| return 1; |
| } |
| } |
| |
| /* Calculate *X + *Y exactly, storing the high part in *X (rounded to |
| nearest) and the low part in *Y. It is given that |X| >= |Y|. */ |
| |
| static void |
| add_split_ext (ext_val *x, ext_val *y) |
| { |
| int xexp = x->exp, yexp = y->exp; |
| if (y->val == 0 || xexp - yexp > 53) |
| return; |
| double hi = x->val; |
| double lo = __scalbn (y->val, yexp - xexp); |
| add_split (&hi, &lo, hi, lo); |
| store_ext_val (x, hi); |
| if (hi != 0) |
| x->exp += xexp; |
| store_ext_val (y, lo); |
| if (lo != 0) |
| y->exp += xexp; |
| } |
| |
| long double |
| __fmal (long double x, long double y, long double z) |
| { |
| double xhi, xlo, yhi, ylo, zhi, zlo; |
| int64_t hx, hy, hz; |
| int xexp, yexp, zexp; |
| double scale_val; |
| int scale_exp; |
| ldbl_unpack (x, &xhi, &xlo); |
| EXTRACT_WORDS64 (hx, xhi); |
| xexp = (hx & 0x7ff0000000000000LL) >> 52; |
| ldbl_unpack (y, &yhi, &ylo); |
| EXTRACT_WORDS64 (hy, yhi); |
| yexp = (hy & 0x7ff0000000000000LL) >> 52; |
| ldbl_unpack (z, &zhi, &zlo); |
| EXTRACT_WORDS64 (hz, zhi); |
| zexp = (hz & 0x7ff0000000000000LL) >> 52; |
| |
| /* If z is Inf or NaN, but x and y are finite, avoid any exceptions |
| from computing x * y. */ |
| if (zexp == 0x7ff && xexp != 0x7ff && yexp != 0x7ff) |
| return (z + x) + y; |
| |
| /* If z is zero and x are y are nonzero, compute the result as x * y |
| to avoid the wrong sign of a zero result if x * y underflows to |
| 0. */ |
| if (z == 0 && x != 0 && y != 0) |
| return x * y; |
| |
| /* If x or y or z is Inf/NaN, or if x * y is zero, compute as x * y |
| + z. */ |
| if (xexp == 0x7ff || yexp == 0x7ff || zexp == 0x7ff |
| || x == 0 || y == 0) |
| return (x * y) + z; |
| |
| { |
| SET_RESTORE_ROUND (FE_TONEAREST); |
| |
| ext_val vals[10]; |
| store_ext_val (&vals[0], zhi); |
| store_ext_val (&vals[1], zlo); |
| mul_ext_val (&vals[2], &vals[3], xhi, yhi); |
| mul_ext_val (&vals[4], &vals[5], xhi, ylo); |
| mul_ext_val (&vals[6], &vals[7], xlo, yhi); |
| mul_ext_val (&vals[8], &vals[9], xlo, ylo); |
| qsort (vals, 10, sizeof (ext_val), 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 <= 8; i++) |
| { |
| add_split_ext (&vals[i + 1], &vals[i]); |
| qsort (vals + i + 1, 9 - i, sizeof (ext_val), compare); |
| } |
| /* Add up the values in the other direction, so that each element |
| of VALS has absolute value less than 5ulp of the next |
| value. */ |
| size_t dstpos = 9; |
| for (size_t i = 1; i <= 9; i++) |
| { |
| if (vals[dstpos].val == 0) |
| { |
| vals[dstpos] = vals[9 - i]; |
| vals[9 - i].val = 0; |
| vals[9 - i].exp = 0; |
| } |
| else |
| { |
| add_split_ext (&vals[dstpos], &vals[9 - i]); |
| if (vals[9 - i].val != 0) |
| { |
| if (9 - i < dstpos - 1) |
| { |
| vals[dstpos - 1] = vals[9 - i]; |
| vals[9 - i].val = 0; |
| vals[9 - i].exp = 0; |
| } |
| dstpos--; |
| } |
| } |
| } |
| /* If the result is an exact zero, it results from adding two |
| values with opposite signs; recompute in the original rounding |
| mode. */ |
| if (vals[9].val == 0) |
| goto zero_out; |
| /* Adding the top three values will now give a result as accurate |
| as the underlying long double arithmetic. */ |
| add_split_ext (&vals[9], &vals[8]); |
| if (compare (&vals[8], &vals[7]) < 0) |
| { |
| ext_val tmp = vals[7]; |
| vals[7] = vals[8]; |
| vals[8] = tmp; |
| } |
| add_split_ext (&vals[8], &vals[7]); |
| add_split_ext (&vals[9], &vals[8]); |
| if (vals[9].exp > DBL_MAX_EXP || vals[9].exp < DBL_MIN_EXP) |
| { |
| /* Overflow or underflow, with the result depending on the |
| original rounding mode, but not on the low part computed |
| here. */ |
| scale_val = vals[9].val; |
| scale_exp = vals[9].exp; |
| goto scale_out; |
| } |
| double hi = __scalbn (vals[9].val, vals[9].exp); |
| double lo = __scalbn (vals[8].val, vals[8].exp); |
| /* It is possible that the low part became subnormal and was |
| rounded so that the result is no longer canonical. */ |
| ldbl_canonicalize (&hi, &lo); |
| long double ret = ldbl_pack (hi, lo); |
| math_check_force_underflow (ret); |
| return ret; |
| } |
| |
| scale_out: |
| scale_val = math_opt_barrier (scale_val); |
| scale_val = __scalbn (scale_val, scale_exp); |
| if (fabs (scale_val) == DBL_MAX) |
| return __copysignl (LDBL_MAX, scale_val); |
| math_check_force_underflow (scale_val); |
| return scale_val; |
| |
| zero_out:; |
| double zero = 0.0; |
| zero = math_opt_barrier (zero); |
| return zero - zero; |
| } |
| #if IS_IN (libm) |
| long_double_symbol (libm, __fmal, fmal); |
| #else |
| long_double_symbol (libc, __fmal, fmal); |
| #endif |