1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
|
/* Compute x * y + z as ternary operation.
Copyright (C) 2010-2020 Free Software Foundation, Inc.
This file is part of the GNU C Library.
Contributed by Jakub Jelinek <jakub@redhat.com>, 2010.
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
<https://www.gnu.org/licenses/>. */
#include <float.h>
#include <math.h>
#include <fenv.h>
#include <ieee754.h>
#include <math-barriers.h>
#include <math_private.h>
#include <libm-alias-ldouble.h>
#include <tininess.h>
/* This implementation uses rounding to odd to avoid problems with
double rounding. See a paper by Boldo and Melquiond:
http://www.lri.fr/~melquion/doc/08-tc.pdf */
_Float128
__fmal (_Float128 x, _Float128 y, _Float128 z)
{
union ieee854_long_double u, v, w;
int adjust = 0;
u.d = x;
v.d = y;
w.d = z;
if (__builtin_expect (u.ieee.exponent + v.ieee.exponent
>= 0x7fff + IEEE854_LONG_DOUBLE_BIAS
- LDBL_MANT_DIG, 0)
|| __builtin_expect (u.ieee.exponent >= 0x7fff - LDBL_MANT_DIG, 0)
|| __builtin_expect (v.ieee.exponent >= 0x7fff - LDBL_MANT_DIG, 0)
|| __builtin_expect (w.ieee.exponent >= 0x7fff - LDBL_MANT_DIG, 0)
|| __builtin_expect (u.ieee.exponent + v.ieee.exponent
<= IEEE854_LONG_DOUBLE_BIAS + LDBL_MANT_DIG, 0))
{
/* If z is Inf, but x and y are finite, the result should be
z rather than NaN. */
if (w.ieee.exponent == 0x7fff
&& u.ieee.exponent != 0x7fff
&& v.ieee.exponent != 0x7fff)
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 (u.ieee.exponent == 0x7fff
|| v.ieee.exponent == 0x7fff
|| w.ieee.exponent == 0x7fff
|| x == 0
|| y == 0)
return x * y + z;
/* If fma will certainly overflow, compute as x * y. */
if (u.ieee.exponent + v.ieee.exponent
> 0x7fff + IEEE854_LONG_DOUBLE_BIAS)
return x * y;
/* If x * y is less than 1/4 of LDBL_TRUE_MIN, neither the
result nor whether there is underflow depends on its exact
value, only on its sign. */
if (u.ieee.exponent + v.ieee.exponent
< IEEE854_LONG_DOUBLE_BIAS - LDBL_MANT_DIG - 2)
{
int neg = u.ieee.negative ^ v.ieee.negative;
_Float128 tiny = neg ? L(-0x1p-16494) : L(0x1p-16494);
if (w.ieee.exponent >= 3)
return tiny + z;
/* Scaling up, adding TINY and scaling down produces the
correct result, because in round-to-nearest mode adding
TINY has no effect and in other modes double rounding is
harmless. But it may not produce required underflow
exceptions. */
v.d = z * L(0x1p114) + tiny;
if (TININESS_AFTER_ROUNDING
? v.ieee.exponent < 115
: (w.ieee.exponent == 0
|| (w.ieee.exponent == 1
&& w.ieee.negative != neg
&& w.ieee.mantissa3 == 0
&& w.ieee.mantissa2 == 0
&& w.ieee.mantissa1 == 0
&& w.ieee.mantissa0 == 0)))
{
_Float128 force_underflow = x * y;
math_force_eval (force_underflow);
}
return v.d * L(0x1p-114);
}
if (u.ieee.exponent + v.ieee.exponent
>= 0x7fff + IEEE854_LONG_DOUBLE_BIAS - LDBL_MANT_DIG)
{
/* Compute 1p-113 times smaller result and multiply
at the end. */
if (u.ieee.exponent > v.ieee.exponent)
u.ieee.exponent -= LDBL_MANT_DIG;
else
v.ieee.exponent -= LDBL_MANT_DIG;
/* If x + y exponent is very large and z exponent is very small,
it doesn't matter if we don't adjust it. */
if (w.ieee.exponent > LDBL_MANT_DIG)
w.ieee.exponent -= LDBL_MANT_DIG;
adjust = 1;
}
else if (w.ieee.exponent >= 0x7fff - LDBL_MANT_DIG)
{
/* Similarly.
If z exponent is very large and x and y exponents are
very small, adjust them up to avoid spurious underflows,
rather than down. */
if (u.ieee.exponent + v.ieee.exponent
<= IEEE854_LONG_DOUBLE_BIAS + 2 * LDBL_MANT_DIG)
{
if (u.ieee.exponent > v.ieee.exponent)
u.ieee.exponent += 2 * LDBL_MANT_DIG + 2;
else
v.ieee.exponent += 2 * LDBL_MANT_DIG + 2;
}
else if (u.ieee.exponent > v.ieee.exponent)
{
if (u.ieee.exponent > LDBL_MANT_DIG)
u.ieee.exponent -= LDBL_MANT_DIG;
}
else if (v.ieee.exponent > LDBL_MANT_DIG)
v.ieee.exponent -= LDBL_MANT_DIG;
w.ieee.exponent -= LDBL_MANT_DIG;
adjust = 1;
}
else if (u.ieee.exponent >= 0x7fff - LDBL_MANT_DIG)
{
u.ieee.exponent -= LDBL_MANT_DIG;
if (v.ieee.exponent)
v.ieee.exponent += LDBL_MANT_DIG;
else
v.d *= L(0x1p113);
}
else if (v.ieee.exponent >= 0x7fff - LDBL_MANT_DIG)
{
v.ieee.exponent -= LDBL_MANT_DIG;
if (u.ieee.exponent)
u.ieee.exponent += LDBL_MANT_DIG;
else
u.d *= L(0x1p113);
}
else /* if (u.ieee.exponent + v.ieee.exponent
<= IEEE854_LONG_DOUBLE_BIAS + LDBL_MANT_DIG) */
{
if (u.ieee.exponent > v.ieee.exponent)
u.ieee.exponent += 2 * LDBL_MANT_DIG + 2;
else
v.ieee.exponent += 2 * LDBL_MANT_DIG + 2;
if (w.ieee.exponent <= 4 * LDBL_MANT_DIG + 6)
{
if (w.ieee.exponent)
w.ieee.exponent += 2 * LDBL_MANT_DIG + 2;
else
w.d *= L(0x1p228);
adjust = -1;
}
/* Otherwise x * y should just affect inexact
and nothing else. */
}
x = u.d;
y = v.d;
z = w.d;
}
/* Ensure correct sign of exact 0 + 0. */
if (__glibc_unlikely ((x == 0 || y == 0) && z == 0))
{
x = math_opt_barrier (x);
return x * y + z;
}
fenv_t env;
feholdexcept (&env);
fesetround (FE_TONEAREST);
/* Multiplication m1 + m2 = x * y using Dekker's algorithm. */
#define C ((1LL << (LDBL_MANT_DIG + 1) / 2) + 1)
_Float128 x1 = x * C;
_Float128 y1 = y * C;
_Float128 m1 = x * y;
x1 = (x - x1) + x1;
y1 = (y - y1) + y1;
_Float128 x2 = x - x1;
_Float128 y2 = y - y1;
_Float128 m2 = (((x1 * y1 - m1) + x1 * y2) + x2 * y1) + x2 * y2;
/* Addition a1 + a2 = z + m1 using Knuth's algorithm. */
_Float128 a1 = z + m1;
_Float128 t1 = a1 - z;
_Float128 t2 = a1 - t1;
t1 = m1 - t1;
t2 = z - t2;
_Float128 a2 = t1 + t2;
/* Ensure the arithmetic is not scheduled after feclearexcept call. */
math_force_eval (m2);
math_force_eval (a2);
feclearexcept (FE_INEXACT);
/* If the result is an exact zero, ensure it has the correct sign. */
if (a1 == 0 && m2 == 0)
{
feupdateenv (&env);
/* Ensure that round-to-nearest value of z + m1 is not reused. */
z = math_opt_barrier (z);
return z + m1;
}
fesetround (FE_TOWARDZERO);
/* Perform m2 + a2 addition with round to odd. */
u.d = a2 + m2;
if (__glibc_likely (adjust == 0))
{
if ((u.ieee.mantissa3 & 1) == 0 && u.ieee.exponent != 0x7fff)
u.ieee.mantissa3 |= fetestexcept (FE_INEXACT) != 0;
feupdateenv (&env);
/* Result is a1 + u.d. */
return a1 + u.d;
}
else if (__glibc_likely (adjust > 0))
{
if ((u.ieee.mantissa3 & 1) == 0 && u.ieee.exponent != 0x7fff)
u.ieee.mantissa3 |= fetestexcept (FE_INEXACT) != 0;
feupdateenv (&env);
/* Result is a1 + u.d, scaled up. */
return (a1 + u.d) * L(0x1p113);
}
else
{
if ((u.ieee.mantissa3 & 1) == 0)
u.ieee.mantissa3 |= fetestexcept (FE_INEXACT) != 0;
v.d = a1 + u.d;
/* Ensure the addition is not scheduled after fetestexcept call. */
math_force_eval (v.d);
int j = fetestexcept (FE_INEXACT) != 0;
feupdateenv (&env);
/* Ensure the following computations are performed in default rounding
mode instead of just reusing the round to zero computation. */
asm volatile ("" : "=m" (u) : "m" (u));
/* If a1 + u.d is exact, the only rounding happens during
scaling down. */
if (j == 0)
return v.d * L(0x1p-228);
/* If result rounded to zero is not subnormal, no double
rounding will occur. */
if (v.ieee.exponent > 228)
return (a1 + u.d) * L(0x1p-228);
/* If v.d * 0x1p-228L with round to zero is a subnormal above
or equal to LDBL_MIN / 2, then v.d * 0x1p-228L shifts mantissa
down just by 1 bit, which means v.ieee.mantissa3 |= j would
change the round bit, not sticky or guard bit.
v.d * 0x1p-228L never normalizes by shifting up,
so round bit plus sticky bit should be already enough
for proper rounding. */
if (v.ieee.exponent == 228)
{
/* If the exponent would be in the normal range when
rounding to normal precision with unbounded exponent
range, the exact result is known and spurious underflows
must be avoided on systems detecting tininess after
rounding. */
if (TININESS_AFTER_ROUNDING)
{
w.d = a1 + u.d;
if (w.ieee.exponent == 229)
return w.d * L(0x1p-228);
}
/* v.ieee.mantissa3 & 2 is LSB bit of the result before rounding,
v.ieee.mantissa3 & 1 is the round bit and j is our sticky
bit. */
w.d = 0;
w.ieee.mantissa3 = ((v.ieee.mantissa3 & 3) << 1) | j;
w.ieee.negative = v.ieee.negative;
v.ieee.mantissa3 &= ~3U;
v.d *= L(0x1p-228);
w.d *= L(0x1p-2);
return v.d + w.d;
}
v.ieee.mantissa3 |= j;
return v.d * L(0x1p-228);
}
}
libm_alias_ldouble (__fma, fma)
|
