1/* Compute x * y + z as ternary operation.
2 Copyright (C) 2010-2020 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 <https://www.gnu.org/licenses/>. */
19
20#include <float.h>
21#include <math.h>
22#include <fenv.h>
23#include <ieee754.h>
24#include <math-barriers.h>
25#include <fenv_private.h>
26#include <libm-alias-double.h>
27#include <tininess.h>
28#include <math-use-builtins.h>
29
30/* This implementation uses rounding to odd to avoid problems with
31 double rounding. See a paper by Boldo and Melquiond:
32 http://www.lri.fr/~melquion/doc/08-tc.pdf */
33
34double
35__fma (double x, double y, double z)
36{
37#if USE_FMA_BUILTIN
38 return __builtin_fma (x, y, z);
39#else
40 /* Use generic implementation. */
41 union ieee754_double u, v, w;
42 int adjust = 0;
43 u.d = x;
44 v.d = y;
45 w.d = z;
46 if (__builtin_expect (u.ieee.exponent + v.ieee.exponent
47 >= 0x7ff + IEEE754_DOUBLE_BIAS - DBL_MANT_DIG, 0)
48 || __builtin_expect (u.ieee.exponent >= 0x7ff - DBL_MANT_DIG, 0)
49 || __builtin_expect (v.ieee.exponent >= 0x7ff - DBL_MANT_DIG, 0)
50 || __builtin_expect (w.ieee.exponent >= 0x7ff - DBL_MANT_DIG, 0)
51 || __builtin_expect (u.ieee.exponent + v.ieee.exponent
52 <= IEEE754_DOUBLE_BIAS + DBL_MANT_DIG, 0))
53 {
54 /* If z is Inf, but x and y are finite, the result should be
55 z rather than NaN. */
56 if (w.ieee.exponent == 0x7ff
57 && u.ieee.exponent != 0x7ff
58 && v.ieee.exponent != 0x7ff)
59 return (z + x) + y;
60 /* If z is zero and x are y are nonzero, compute the result
61 as x * y to avoid the wrong sign of a zero result if x * y
62 underflows to 0. */
63 if (z == 0 && x != 0 && y != 0)
64 return x * y;
65 /* If x or y or z is Inf/NaN, or if x * y is zero, compute as
66 x * y + z. */
67 if (u.ieee.exponent == 0x7ff
68 || v.ieee.exponent == 0x7ff
69 || w.ieee.exponent == 0x7ff
70 || x == 0
71 || y == 0)
72 return x * y + z;
73 /* If fma will certainly overflow, compute as x * y. */
74 if (u.ieee.exponent + v.ieee.exponent > 0x7ff + IEEE754_DOUBLE_BIAS)
75 return x * y;
76 /* If x * y is less than 1/4 of DBL_TRUE_MIN, neither the
77 result nor whether there is underflow depends on its exact
78 value, only on its sign. */
79 if (u.ieee.exponent + v.ieee.exponent
80 < IEEE754_DOUBLE_BIAS - DBL_MANT_DIG - 2)
81 {
82 int neg = u.ieee.negative ^ v.ieee.negative;
83 double tiny = neg ? -0x1p-1074 : 0x1p-1074;
84 if (w.ieee.exponent >= 3)
85 return tiny + z;
86 /* Scaling up, adding TINY and scaling down produces the
87 correct result, because in round-to-nearest mode adding
88 TINY has no effect and in other modes double rounding is
89 harmless. But it may not produce required underflow
90 exceptions. */
91 v.d = z * 0x1p54 + tiny;
92 if (TININESS_AFTER_ROUNDING
93 ? v.ieee.exponent < 55
94 : (w.ieee.exponent == 0
95 || (w.ieee.exponent == 1
96 && w.ieee.negative != neg
97 && w.ieee.mantissa1 == 0
98 && w.ieee.mantissa0 == 0)))
99 {
100 double force_underflow = x * y;
101 math_force_eval (force_underflow);
102 }
103 return v.d * 0x1p-54;
104 }
105 if (u.ieee.exponent + v.ieee.exponent
106 >= 0x7ff + IEEE754_DOUBLE_BIAS - DBL_MANT_DIG)
107 {
108 /* Compute 1p-53 times smaller result and multiply
109 at the end. */
110 if (u.ieee.exponent > v.ieee.exponent)
111 u.ieee.exponent -= DBL_MANT_DIG;
112 else
113 v.ieee.exponent -= DBL_MANT_DIG;
114 /* If x + y exponent is very large and z exponent is very small,
115 it doesn't matter if we don't adjust it. */
116 if (w.ieee.exponent > DBL_MANT_DIG)
117 w.ieee.exponent -= DBL_MANT_DIG;
118 adjust = 1;
119 }
120 else if (w.ieee.exponent >= 0x7ff - DBL_MANT_DIG)
121 {
122 /* Similarly.
123 If z exponent is very large and x and y exponents are
124 very small, adjust them up to avoid spurious underflows,
125 rather than down. */
126 if (u.ieee.exponent + v.ieee.exponent
127 <= IEEE754_DOUBLE_BIAS + 2 * DBL_MANT_DIG)
128 {
129 if (u.ieee.exponent > v.ieee.exponent)
130 u.ieee.exponent += 2 * DBL_MANT_DIG + 2;
131 else
132 v.ieee.exponent += 2 * DBL_MANT_DIG + 2;
133 }
134 else if (u.ieee.exponent > v.ieee.exponent)
135 {
136 if (u.ieee.exponent > DBL_MANT_DIG)
137 u.ieee.exponent -= DBL_MANT_DIG;
138 }
139 else if (v.ieee.exponent > DBL_MANT_DIG)
140 v.ieee.exponent -= DBL_MANT_DIG;
141 w.ieee.exponent -= DBL_MANT_DIG;
142 adjust = 1;
143 }
144 else if (u.ieee.exponent >= 0x7ff - DBL_MANT_DIG)
145 {
146 u.ieee.exponent -= DBL_MANT_DIG;
147 if (v.ieee.exponent)
148 v.ieee.exponent += DBL_MANT_DIG;
149 else
150 v.d *= 0x1p53;
151 }
152 else if (v.ieee.exponent >= 0x7ff - DBL_MANT_DIG)
153 {
154 v.ieee.exponent -= DBL_MANT_DIG;
155 if (u.ieee.exponent)
156 u.ieee.exponent += DBL_MANT_DIG;
157 else
158 u.d *= 0x1p53;
159 }
160 else /* if (u.ieee.exponent + v.ieee.exponent
161 <= IEEE754_DOUBLE_BIAS + DBL_MANT_DIG) */
162 {
163 if (u.ieee.exponent > v.ieee.exponent)
164 u.ieee.exponent += 2 * DBL_MANT_DIG + 2;
165 else
166 v.ieee.exponent += 2 * DBL_MANT_DIG + 2;
167 if (w.ieee.exponent <= 4 * DBL_MANT_DIG + 6)
168 {
169 if (w.ieee.exponent)
170 w.ieee.exponent += 2 * DBL_MANT_DIG + 2;
171 else
172 w.d *= 0x1p108;
173 adjust = -1;
174 }
175 /* Otherwise x * y should just affect inexact
176 and nothing else. */
177 }
178 x = u.d;
179 y = v.d;
180 z = w.d;
181 }
182
183 /* Ensure correct sign of exact 0 + 0. */
184 if (__glibc_unlikely ((x == 0 || y == 0) && z == 0))
185 {
186 x = math_opt_barrier (x);
187 return x * y + z;
188 }
189
190 fenv_t env;
191 libc_feholdexcept_setround (&env, FE_TONEAREST);
192
193 /* Multiplication m1 + m2 = x * y using Dekker's algorithm. */
194#define C ((1 << (DBL_MANT_DIG + 1) / 2) + 1)
195 double x1 = x * C;
196 double y1 = y * C;
197 double m1 = x * y;
198 x1 = (x - x1) + x1;
199 y1 = (y - y1) + y1;
200 double x2 = x - x1;
201 double y2 = y - y1;
202 double m2 = (((x1 * y1 - m1) + x1 * y2) + x2 * y1) + x2 * y2;
203
204 /* Addition a1 + a2 = z + m1 using Knuth's algorithm. */
205 double a1 = z + m1;
206 double t1 = a1 - z;
207 double t2 = a1 - t1;
208 t1 = m1 - t1;
209 t2 = z - t2;
210 double a2 = t1 + t2;
211 /* Ensure the arithmetic is not scheduled after feclearexcept call. */
212 math_force_eval (m2);
213 math_force_eval (a2);
214 feclearexcept (FE_INEXACT);
215
216 /* If the result is an exact zero, ensure it has the correct sign. */
217 if (a1 == 0 && m2 == 0)
218 {
219 libc_feupdateenv (&env);
220 /* Ensure that round-to-nearest value of z + m1 is not reused. */
221 z = math_opt_barrier (z);
222 return z + m1;
223 }
224
225 libc_fesetround (FE_TOWARDZERO);
226
227 /* Perform m2 + a2 addition with round to odd. */
228 u.d = a2 + m2;
229
230 if (__glibc_unlikely (adjust < 0))
231 {
232 if ((u.ieee.mantissa1 & 1) == 0)
233 u.ieee.mantissa1 |= libc_fetestexcept (FE_INEXACT) != 0;
234 v.d = a1 + u.d;
235 /* Ensure the addition is not scheduled after fetestexcept call. */
236 math_force_eval (v.d);
237 }
238
239 /* Reset rounding mode and test for inexact simultaneously. */
240 int j = libc_feupdateenv_test (&env, FE_INEXACT) != 0;
241
242 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. */
247 return a1 + u.d;
248 }
249 else if (__glibc_likely (adjust > 0))
250 {
251 if ((u.ieee.mantissa1 & 1) == 0 && u.ieee.exponent != 0x7ff)
252 u.ieee.mantissa1 |= j;
253 /* Result is a1 + u.d, scaled up. */
254 return (a1 + u.d) * 0x1p53;
255 }
256 else
257 {
258 /* If a1 + u.d is exact, the only rounding happens during
259 scaling down. */
260 if (j == 0)
261 return v.d * 0x1p-108;
262 /* If result rounded to zero is not subnormal, no double
263 rounding will occur. */
264 if (v.ieee.exponent > 108)
265 return (a1 + u.d) * 0x1p-108;
266 /* If v.d * 0x1p-108 with round to zero is a subnormal above
267 or equal to DBL_MIN / 2, then v.d * 0x1p-108 shifts mantissa
268 down just by 1 bit, which means v.ieee.mantissa1 |= j would
269 change the round bit, not sticky or guard bit.
270 v.d * 0x1p-108 never normalizes by shifting up,
271 so round bit plus sticky bit should be already enough
272 for proper rounding. */
273 if (v.ieee.exponent == 108)
274 {
275 /* If the exponent would be in the normal range when
276 rounding to normal precision with unbounded exponent
277 range, the exact result is known and spurious underflows
278 must be avoided on systems detecting tininess after
279 rounding. */
280 if (TININESS_AFTER_ROUNDING)
281 {
282 w.d = a1 + u.d;
283 if (w.ieee.exponent == 109)
284 return w.d * 0x1p-108;
285 }
286 /* v.ieee.mantissa1 & 2 is LSB bit of the result before rounding,
287 v.ieee.mantissa1 & 1 is the round bit and j is our sticky
288 bit. */
289 w.d = 0.0;
290 w.ieee.mantissa1 = ((v.ieee.mantissa1 & 3) << 1) | j;
291 w.ieee.negative = v.ieee.negative;
292 v.ieee.mantissa1 &= ~3U;
293 v.d *= 0x1p-108;
294 w.d *= 0x1p-2;
295 return v.d + w.d;
296 }
297 v.ieee.mantissa1 |= j;
298 return v.d * 0x1p-108;
299 }
300#endif /* ! USE_FMA_BUILTIN */
301}
302#ifndef __fma
303libm_alias_double (__fma, fma)
304#endif
305