1 | /* Compute x * y + z as ternary operation. |
2 | Copyright (C) 2011-2022 Free Software Foundation, Inc. |
3 | This file is part of the GNU C Library. |
4 | |
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. |
9 | |
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. |
14 | |
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/>. */ |
18 | |
19 | #define NO_MATH_REDIRECT |
20 | #include <fenv.h> |
21 | #include <float.h> |
22 | #include <math.h> |
23 | #include <math-barriers.h> |
24 | #include <math_private.h> |
25 | #include <fenv_private.h> |
26 | #include <math-underflow.h> |
27 | #include <math_ldbl_opt.h> |
28 | #include <mul_split.h> |
29 | #include <stdlib.h> |
30 | |
31 | /* Calculate X + Y exactly and store the result in *HI + *LO. It is |
32 | given that |X| >= |Y| and the values are small enough that no |
33 | overflow occurs. */ |
34 | |
35 | static void |
36 | add_split (double *hi, double *lo, double x, double y) |
37 | { |
38 | /* Apply Dekker's algorithm. */ |
39 | *hi = x + y; |
40 | *lo = (x - *hi) + y; |
41 | } |
42 | |
43 | /* Value with extended range, used in intermediate computations. */ |
44 | typedef struct |
45 | { |
46 | /* Value in [0.5, 1), as from frexp, or 0. */ |
47 | double val; |
48 | /* Exponent of power of 2 it is multiplied by, or 0 for zero. */ |
49 | int exp; |
50 | } ext_val; |
51 | |
52 | /* Store D as an ext_val value. */ |
53 | |
54 | static void |
55 | store_ext_val (ext_val *v, double d) |
56 | { |
57 | v->val = __frexp (x: d, exponent: &v->exp); |
58 | } |
59 | |
60 | /* Store X * Y as ext_val values *V0 and *V1. */ |
61 | |
62 | static void |
63 | mul_ext_val (ext_val *v0, ext_val *v1, double x, double y) |
64 | { |
65 | int xexp, yexp; |
66 | x = __frexp (x: x, exponent: &xexp); |
67 | y = __frexp (x: y, exponent: &yexp); |
68 | double hi, lo; |
69 | mul_split (hi: &hi, lo: &lo, x, y); |
70 | store_ext_val (v: v0, d: hi); |
71 | if (hi != 0) |
72 | v0->exp += xexp + yexp; |
73 | store_ext_val (v: v1, d: lo); |
74 | if (lo != 0) |
75 | v1->exp += xexp + yexp; |
76 | } |
77 | |
78 | /* Compare absolute values of ext_val values pointed to by P and Q for |
79 | qsort. */ |
80 | |
81 | static int |
82 | compare (const void *p, const void *q) |
83 | { |
84 | const ext_val *pe = p; |
85 | const ext_val *qe = q; |
86 | if (pe->val == 0) |
87 | return qe->val == 0 ? 0 : -1; |
88 | else if (qe->val == 0) |
89 | return 1; |
90 | else if (pe->exp < qe->exp) |
91 | return -1; |
92 | else if (pe->exp > qe->exp) |
93 | return 1; |
94 | else |
95 | { |
96 | double pd = fabs (x: pe->val); |
97 | double qd = fabs (x: qe->val); |
98 | if (pd < qd) |
99 | return -1; |
100 | else if (pd == qd) |
101 | return 0; |
102 | else |
103 | return 1; |
104 | } |
105 | } |
106 | |
107 | /* Calculate *X + *Y exactly, storing the high part in *X (rounded to |
108 | nearest) and the low part in *Y. It is given that |X| >= |Y|. */ |
109 | |
110 | static void |
111 | add_split_ext (ext_val *x, ext_val *y) |
112 | { |
113 | int xexp = x->exp, yexp = y->exp; |
114 | if (y->val == 0 || xexp - yexp > 53) |
115 | return; |
116 | double hi = x->val; |
117 | double lo = __scalbn (x: y->val, n: yexp - xexp); |
118 | add_split (hi: &hi, lo: &lo, x: hi, y: lo); |
119 | store_ext_val (v: x, d: hi); |
120 | if (hi != 0) |
121 | x->exp += xexp; |
122 | store_ext_val (v: y, d: lo); |
123 | if (lo != 0) |
124 | y->exp += xexp; |
125 | } |
126 | |
127 | long double |
128 | __fmal (long double x, long double y, long double z) |
129 | { |
130 | double xhi, xlo, yhi, ylo, zhi, zlo; |
131 | int64_t hx, hy, hz; |
132 | int xexp, yexp, zexp; |
133 | double scale_val; |
134 | int scale_exp; |
135 | ldbl_unpack (x, &xhi, &xlo); |
136 | EXTRACT_WORDS64 (hx, xhi); |
137 | xexp = (hx & 0x7ff0000000000000LL) >> 52; |
138 | ldbl_unpack (y, &yhi, &ylo); |
139 | EXTRACT_WORDS64 (hy, yhi); |
140 | yexp = (hy & 0x7ff0000000000000LL) >> 52; |
141 | ldbl_unpack (z, &zhi, &zlo); |
142 | EXTRACT_WORDS64 (hz, zhi); |
143 | zexp = (hz & 0x7ff0000000000000LL) >> 52; |
144 | |
145 | /* If z is Inf or NaN, but x and y are finite, avoid any exceptions |
146 | from computing x * y. */ |
147 | if (zexp == 0x7ff && xexp != 0x7ff && yexp != 0x7ff) |
148 | return (z + x) + y; |
149 | |
150 | /* If z is zero and x are y are nonzero, compute the result as x * y |
151 | to avoid the wrong sign of a zero result if x * y underflows to |
152 | 0. */ |
153 | if (z == 0 && x != 0 && y != 0) |
154 | return x * y; |
155 | |
156 | /* If x or y or z is Inf/NaN, or if x * y is zero, compute as x * y |
157 | + z. */ |
158 | if (xexp == 0x7ff || yexp == 0x7ff || zexp == 0x7ff |
159 | || x == 0 || y == 0) |
160 | return (x * y) + z; |
161 | |
162 | { |
163 | SET_RESTORE_ROUND (FE_TONEAREST); |
164 | |
165 | ext_val vals[10]; |
166 | store_ext_val (v: &vals[0], d: zhi); |
167 | store_ext_val (v: &vals[1], d: zlo); |
168 | mul_ext_val (v0: &vals[2], v1: &vals[3], x: xhi, y: yhi); |
169 | mul_ext_val (v0: &vals[4], v1: &vals[5], x: xhi, y: ylo); |
170 | mul_ext_val (v0: &vals[6], v1: &vals[7], x: xlo, y: yhi); |
171 | mul_ext_val (v0: &vals[8], v1: &vals[9], x: xlo, y: ylo); |
172 | qsort (base: vals, nmemb: 10, size: sizeof (ext_val), compar: compare); |
173 | /* Add up the values so that each element of VALS has absolute |
174 | value at most equal to the last set bit of the next nonzero |
175 | element. */ |
176 | for (size_t i = 0; i <= 8; i++) |
177 | { |
178 | add_split_ext (x: &vals[i + 1], y: &vals[i]); |
179 | qsort (base: vals + i + 1, nmemb: 9 - i, size: sizeof (ext_val), compar: compare); |
180 | } |
181 | /* Add up the values in the other direction, so that each element |
182 | of VALS has absolute value less than 5ulp of the next |
183 | value. */ |
184 | size_t dstpos = 9; |
185 | for (size_t i = 1; i <= 9; i++) |
186 | { |
187 | if (vals[dstpos].val == 0) |
188 | { |
189 | vals[dstpos] = vals[9 - i]; |
190 | vals[9 - i].val = 0; |
191 | vals[9 - i].exp = 0; |
192 | } |
193 | else |
194 | { |
195 | add_split_ext (x: &vals[dstpos], y: &vals[9 - i]); |
196 | if (vals[9 - i].val != 0) |
197 | { |
198 | if (9 - i < dstpos - 1) |
199 | { |
200 | vals[dstpos - 1] = vals[9 - i]; |
201 | vals[9 - i].val = 0; |
202 | vals[9 - i].exp = 0; |
203 | } |
204 | dstpos--; |
205 | } |
206 | } |
207 | } |
208 | /* If the result is an exact zero, it results from adding two |
209 | values with opposite signs; recompute in the original rounding |
210 | mode. */ |
211 | if (vals[9].val == 0) |
212 | goto zero_out; |
213 | /* Adding the top three values will now give a result as accurate |
214 | as the underlying long double arithmetic. */ |
215 | add_split_ext (x: &vals[9], y: &vals[8]); |
216 | if (compare (p: &vals[8], q: &vals[7]) < 0) |
217 | { |
218 | ext_val tmp = vals[7]; |
219 | vals[7] = vals[8]; |
220 | vals[8] = tmp; |
221 | } |
222 | add_split_ext (x: &vals[8], y: &vals[7]); |
223 | add_split_ext (x: &vals[9], y: &vals[8]); |
224 | if (vals[9].exp > DBL_MAX_EXP || vals[9].exp < DBL_MIN_EXP) |
225 | { |
226 | /* Overflow or underflow, with the result depending on the |
227 | original rounding mode, but not on the low part computed |
228 | here. */ |
229 | scale_val = vals[9].val; |
230 | scale_exp = vals[9].exp; |
231 | goto scale_out; |
232 | } |
233 | double hi = __scalbn (x: vals[9].val, n: vals[9].exp); |
234 | double lo = __scalbn (x: vals[8].val, n: vals[8].exp); |
235 | /* It is possible that the low part became subnormal and was |
236 | rounded so that the result is no longer canonical. */ |
237 | ldbl_canonicalize (&hi, &lo); |
238 | long double ret = ldbl_pack (hi, lo); |
239 | math_check_force_underflow (ret); |
240 | return ret; |
241 | } |
242 | |
243 | scale_out: |
244 | scale_val = math_opt_barrier (scale_val); |
245 | scale_val = __scalbn (x: scale_val, n: scale_exp); |
246 | if (fabs (x: scale_val) == DBL_MAX) |
247 | return copysignl (LDBL_MAX, y: scale_val); |
248 | math_check_force_underflow (scale_val); |
249 | return scale_val; |
250 | |
251 | zero_out:; |
252 | double zero = 0.0; |
253 | zero = math_opt_barrier (zero); |
254 | return zero - zero; |
255 | } |
256 | #if IS_IN (libm) |
257 | long_double_symbol (libm, __fmal, fmal); |
258 | #else |
259 | long_double_symbol (libc, __fmal, fmal); |
260 | #endif |
261 | |