1 | /* Compute x * y + z as ternary operation. |
2 | Copyright (C) 2010-2017 Free Software Foundation, Inc. |
3 | This file is part of the GNU C Library. |
4 | Contributed by Jakub Jelinek <jakub@redhat.com>, 2010. |
5 | |
6 | The GNU C Library is free software; you can redistribute it and/or |
7 | modify it under the terms of the GNU Lesser General Public |
8 | License as published by the Free Software Foundation; either |
9 | version 2.1 of the License, or (at your option) any later version. |
10 | |
11 | The GNU C Library is distributed in the hope that it will be useful, |
12 | but WITHOUT ANY WARRANTY; without even the implied warranty of |
13 | MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU |
14 | Lesser General Public License for more details. |
15 | |
16 | You should have received a copy of the GNU Lesser General Public |
17 | License along with the GNU C Library; if not, see |
18 | <http://www.gnu.org/licenses/>. */ |
19 | |
20 | #include <float.h> |
21 | #include <math.h> |
22 | #include <fenv.h> |
23 | #include <ieee754.h> |
24 | #include <math_private.h> |
25 | #include <tininess.h> |
26 | |
27 | /* This implementation uses rounding to odd to avoid problems with |
28 | double rounding. See a paper by Boldo and Melquiond: |
29 | http://www.lri.fr/~melquion/doc/08-tc.pdf */ |
30 | |
31 | double |
32 | __fma (double x, double y, double z) |
33 | { |
34 | union ieee754_double u, v, w; |
35 | int adjust = 0; |
36 | u.d = x; |
37 | v.d = y; |
38 | w.d = z; |
39 | if (__builtin_expect (u.ieee.exponent + v.ieee.exponent |
40 | >= 0x7ff + IEEE754_DOUBLE_BIAS - DBL_MANT_DIG, 0) |
41 | || __builtin_expect (u.ieee.exponent >= 0x7ff - DBL_MANT_DIG, 0) |
42 | || __builtin_expect (v.ieee.exponent >= 0x7ff - DBL_MANT_DIG, 0) |
43 | || __builtin_expect (w.ieee.exponent >= 0x7ff - DBL_MANT_DIG, 0) |
44 | || __builtin_expect (u.ieee.exponent + v.ieee.exponent |
45 | <= IEEE754_DOUBLE_BIAS + DBL_MANT_DIG, 0)) |
46 | { |
47 | /* If z is Inf, but x and y are finite, the result should be |
48 | z rather than NaN. */ |
49 | if (w.ieee.exponent == 0x7ff |
50 | && u.ieee.exponent != 0x7ff |
51 | && v.ieee.exponent != 0x7ff) |
52 | return (z + x) + y; |
53 | /* If z is zero and x are y are nonzero, compute the result |
54 | as x * y to avoid the wrong sign of a zero result if x * y |
55 | underflows to 0. */ |
56 | if (z == 0 && x != 0 && y != 0) |
57 | return x * y; |
58 | /* If x or y or z is Inf/NaN, or if x * y is zero, compute as |
59 | x * y + z. */ |
60 | if (u.ieee.exponent == 0x7ff |
61 | || v.ieee.exponent == 0x7ff |
62 | || w.ieee.exponent == 0x7ff |
63 | || x == 0 |
64 | || y == 0) |
65 | return x * y + z; |
66 | /* If fma will certainly overflow, compute as x * y. */ |
67 | if (u.ieee.exponent + v.ieee.exponent > 0x7ff + IEEE754_DOUBLE_BIAS) |
68 | return x * y; |
69 | /* If x * y is less than 1/4 of DBL_TRUE_MIN, neither the |
70 | result nor whether there is underflow depends on its exact |
71 | value, only on its sign. */ |
72 | if (u.ieee.exponent + v.ieee.exponent |
73 | < IEEE754_DOUBLE_BIAS - DBL_MANT_DIG - 2) |
74 | { |
75 | int neg = u.ieee.negative ^ v.ieee.negative; |
76 | double tiny = neg ? -0x1p-1074 : 0x1p-1074; |
77 | if (w.ieee.exponent >= 3) |
78 | return tiny + z; |
79 | /* Scaling up, adding TINY and scaling down produces the |
80 | correct result, because in round-to-nearest mode adding |
81 | TINY has no effect and in other modes double rounding is |
82 | harmless. But it may not produce required underflow |
83 | exceptions. */ |
84 | v.d = z * 0x1p54 + tiny; |
85 | if (TININESS_AFTER_ROUNDING |
86 | ? v.ieee.exponent < 55 |
87 | : (w.ieee.exponent == 0 |
88 | || (w.ieee.exponent == 1 |
89 | && w.ieee.negative != neg |
90 | && w.ieee.mantissa1 == 0 |
91 | && w.ieee.mantissa0 == 0))) |
92 | { |
93 | double force_underflow = x * y; |
94 | math_force_eval (force_underflow); |
95 | } |
96 | return v.d * 0x1p-54; |
97 | } |
98 | if (u.ieee.exponent + v.ieee.exponent |
99 | >= 0x7ff + IEEE754_DOUBLE_BIAS - DBL_MANT_DIG) |
100 | { |
101 | /* Compute 1p-53 times smaller result and multiply |
102 | at the end. */ |
103 | if (u.ieee.exponent > v.ieee.exponent) |
104 | u.ieee.exponent -= DBL_MANT_DIG; |
105 | else |
106 | v.ieee.exponent -= DBL_MANT_DIG; |
107 | /* If x + y exponent is very large and z exponent is very small, |
108 | it doesn't matter if we don't adjust it. */ |
109 | if (w.ieee.exponent > DBL_MANT_DIG) |
110 | w.ieee.exponent -= DBL_MANT_DIG; |
111 | adjust = 1; |
112 | } |
113 | else if (w.ieee.exponent >= 0x7ff - DBL_MANT_DIG) |
114 | { |
115 | /* Similarly. |
116 | If z exponent is very large and x and y exponents are |
117 | very small, adjust them up to avoid spurious underflows, |
118 | rather than down. */ |
119 | if (u.ieee.exponent + v.ieee.exponent |
120 | <= IEEE754_DOUBLE_BIAS + 2 * DBL_MANT_DIG) |
121 | { |
122 | if (u.ieee.exponent > v.ieee.exponent) |
123 | u.ieee.exponent += 2 * DBL_MANT_DIG + 2; |
124 | else |
125 | v.ieee.exponent += 2 * DBL_MANT_DIG + 2; |
126 | } |
127 | else if (u.ieee.exponent > v.ieee.exponent) |
128 | { |
129 | if (u.ieee.exponent > DBL_MANT_DIG) |
130 | u.ieee.exponent -= DBL_MANT_DIG; |
131 | } |
132 | else if (v.ieee.exponent > DBL_MANT_DIG) |
133 | v.ieee.exponent -= DBL_MANT_DIG; |
134 | w.ieee.exponent -= DBL_MANT_DIG; |
135 | adjust = 1; |
136 | } |
137 | else if (u.ieee.exponent >= 0x7ff - DBL_MANT_DIG) |
138 | { |
139 | u.ieee.exponent -= DBL_MANT_DIG; |
140 | if (v.ieee.exponent) |
141 | v.ieee.exponent += DBL_MANT_DIG; |
142 | else |
143 | v.d *= 0x1p53; |
144 | } |
145 | else if (v.ieee.exponent >= 0x7ff - DBL_MANT_DIG) |
146 | { |
147 | v.ieee.exponent -= DBL_MANT_DIG; |
148 | if (u.ieee.exponent) |
149 | u.ieee.exponent += DBL_MANT_DIG; |
150 | else |
151 | u.d *= 0x1p53; |
152 | } |
153 | else /* if (u.ieee.exponent + v.ieee.exponent |
154 | <= IEEE754_DOUBLE_BIAS + DBL_MANT_DIG) */ |
155 | { |
156 | if (u.ieee.exponent > v.ieee.exponent) |
157 | u.ieee.exponent += 2 * DBL_MANT_DIG + 2; |
158 | else |
159 | v.ieee.exponent += 2 * DBL_MANT_DIG + 2; |
160 | if (w.ieee.exponent <= 4 * DBL_MANT_DIG + 6) |
161 | { |
162 | if (w.ieee.exponent) |
163 | w.ieee.exponent += 2 * DBL_MANT_DIG + 2; |
164 | else |
165 | w.d *= 0x1p108; |
166 | adjust = -1; |
167 | } |
168 | /* Otherwise x * y should just affect inexact |
169 | and nothing else. */ |
170 | } |
171 | x = u.d; |
172 | y = v.d; |
173 | z = w.d; |
174 | } |
175 | |
176 | /* Ensure correct sign of exact 0 + 0. */ |
177 | if (__glibc_unlikely ((x == 0 || y == 0) && z == 0)) |
178 | { |
179 | x = math_opt_barrier (x); |
180 | return x * y + z; |
181 | } |
182 | |
183 | fenv_t env; |
184 | libc_feholdexcept_setround (&env, FE_TONEAREST); |
185 | |
186 | /* Multiplication m1 + m2 = x * y using Dekker's algorithm. */ |
187 | #define C ((1 << (DBL_MANT_DIG + 1) / 2) + 1) |
188 | double x1 = x * C; |
189 | double y1 = y * C; |
190 | double m1 = x * y; |
191 | x1 = (x - x1) + x1; |
192 | y1 = (y - y1) + y1; |
193 | double x2 = x - x1; |
194 | double y2 = y - y1; |
195 | double m2 = (((x1 * y1 - m1) + x1 * y2) + x2 * y1) + x2 * y2; |
196 | |
197 | /* Addition a1 + a2 = z + m1 using Knuth's algorithm. */ |
198 | double a1 = z + m1; |
199 | double t1 = a1 - z; |
200 | double t2 = a1 - t1; |
201 | t1 = m1 - t1; |
202 | t2 = z - t2; |
203 | double a2 = t1 + t2; |
204 | /* Ensure the arithmetic is not scheduled after feclearexcept call. */ |
205 | math_force_eval (m2); |
206 | math_force_eval (a2); |
207 | feclearexcept (FE_INEXACT); |
208 | |
209 | /* If the result is an exact zero, ensure it has the correct sign. */ |
210 | if (a1 == 0 && m2 == 0) |
211 | { |
212 | libc_feupdateenv (&env); |
213 | /* Ensure that round-to-nearest value of z + m1 is not reused. */ |
214 | z = math_opt_barrier (z); |
215 | return z + m1; |
216 | } |
217 | |
218 | libc_fesetround (FE_TOWARDZERO); |
219 | |
220 | /* Perform m2 + a2 addition with round to odd. */ |
221 | u.d = a2 + m2; |
222 | |
223 | if (__glibc_unlikely (adjust < 0)) |
224 | { |
225 | if ((u.ieee.mantissa1 & 1) == 0) |
226 | u.ieee.mantissa1 |= libc_fetestexcept (FE_INEXACT) != 0; |
227 | v.d = a1 + u.d; |
228 | /* Ensure the addition is not scheduled after fetestexcept call. */ |
229 | math_force_eval (v.d); |
230 | } |
231 | |
232 | /* Reset rounding mode and test for inexact simultaneously. */ |
233 | int j = libc_feupdateenv_test (&env, FE_INEXACT) != 0; |
234 | |
235 | if (__glibc_likely (adjust == 0)) |
236 | { |
237 | if ((u.ieee.mantissa1 & 1) == 0 && u.ieee.exponent != 0x7ff) |
238 | u.ieee.mantissa1 |= j; |
239 | /* Result is a1 + u.d. */ |
240 | return a1 + u.d; |
241 | } |
242 | else if (__glibc_likely (adjust > 0)) |
243 | { |
244 | if ((u.ieee.mantissa1 & 1) == 0 && u.ieee.exponent != 0x7ff) |
245 | u.ieee.mantissa1 |= j; |
246 | /* Result is a1 + u.d, scaled up. */ |
247 | return (a1 + u.d) * 0x1p53; |
248 | } |
249 | else |
250 | { |
251 | /* If a1 + u.d is exact, the only rounding happens during |
252 | scaling down. */ |
253 | if (j == 0) |
254 | return v.d * 0x1p-108; |
255 | /* If result rounded to zero is not subnormal, no double |
256 | rounding will occur. */ |
257 | if (v.ieee.exponent > 108) |
258 | return (a1 + u.d) * 0x1p-108; |
259 | /* If v.d * 0x1p-108 with round to zero is a subnormal above |
260 | or equal to DBL_MIN / 2, then v.d * 0x1p-108 shifts mantissa |
261 | down just by 1 bit, which means v.ieee.mantissa1 |= j would |
262 | change the round bit, not sticky or guard bit. |
263 | v.d * 0x1p-108 never normalizes by shifting up, |
264 | so round bit plus sticky bit should be already enough |
265 | for proper rounding. */ |
266 | if (v.ieee.exponent == 108) |
267 | { |
268 | /* If the exponent would be in the normal range when |
269 | rounding to normal precision with unbounded exponent |
270 | range, the exact result is known and spurious underflows |
271 | must be avoided on systems detecting tininess after |
272 | rounding. */ |
273 | if (TININESS_AFTER_ROUNDING) |
274 | { |
275 | w.d = a1 + u.d; |
276 | if (w.ieee.exponent == 109) |
277 | return w.d * 0x1p-108; |
278 | } |
279 | /* v.ieee.mantissa1 & 2 is LSB bit of the result before rounding, |
280 | v.ieee.mantissa1 & 1 is the round bit and j is our sticky |
281 | bit. */ |
282 | w.d = 0.0; |
283 | w.ieee.mantissa1 = ((v.ieee.mantissa1 & 3) << 1) | j; |
284 | w.ieee.negative = v.ieee.negative; |
285 | v.ieee.mantissa1 &= ~3U; |
286 | v.d *= 0x1p-108; |
287 | w.d *= 0x1p-2; |
288 | return v.d + w.d; |
289 | } |
290 | v.ieee.mantissa1 |= j; |
291 | return v.d * 0x1p-108; |
292 | } |
293 | } |
294 | #ifndef __fma |
295 | weak_alias (__fma, fma) |
296 | #endif |
297 | |
298 | #ifdef NO_LONG_DOUBLE |
299 | strong_alias (__fma, __fmal) |
300 | weak_alias (__fmal, fmal) |
301 | #endif |
302 | |