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.
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.
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.
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/>. */
20 #include "quadmath-imp.h"
25 # if defined HAVE_FEHOLDEXCEPT && defined HAVE_FESETROUND \
26 && defined HAVE_FEUPDATEENV && defined HAVE_FETESTEXCEPT \
27 && defined FE_TOWARDZERO && defined FE_INEXACT
32 /* This implementation uses rounding to odd to avoid problems with
33 double rounding. See a paper by Boldo and Melquiond:
34 http://www.lri.fr/~melquion/doc/08-tc.pdf */
37 fmaq (__float128 x
, __float128 y
, __float128 z
)
39 ieee854_float128 u
, v
, w
;
44 if (__builtin_expect (u
.ieee
.exponent
+ v
.ieee
.exponent
45 >= 0x7fff + IEEE854_FLOAT128_BIAS
47 || __builtin_expect (u
.ieee
.exponent
>= 0x7fff - FLT128_MANT_DIG
, 0)
48 || __builtin_expect (v
.ieee
.exponent
>= 0x7fff - FLT128_MANT_DIG
, 0)
49 || __builtin_expect (w
.ieee
.exponent
>= 0x7fff - FLT128_MANT_DIG
, 0)
50 || __builtin_expect (u
.ieee
.exponent
+ v
.ieee
.exponent
51 <= IEEE854_FLOAT128_BIAS
+ FLT128_MANT_DIG
, 0))
53 /* If z is Inf, but x and y are finite, the result should be
55 if (w
.ieee
.exponent
== 0x7fff
56 && u
.ieee
.exponent
!= 0x7fff
57 && v
.ieee
.exponent
!= 0x7fff)
59 /* If z is zero and x are y are nonzero, compute the result
60 as x * y to avoid the wrong sign of a zero result if x * y
62 if (z
== 0 && x
!= 0 && y
!= 0)
64 /* If x or y or z is Inf/NaN, or if x * y is zero, compute as
66 if (u
.ieee
.exponent
== 0x7fff
67 || v
.ieee
.exponent
== 0x7fff
68 || w
.ieee
.exponent
== 0x7fff
72 /* If fma will certainly overflow, compute as x * y. */
73 if (u
.ieee
.exponent
+ v
.ieee
.exponent
74 > 0x7fff + IEEE854_FLOAT128_BIAS
)
76 /* If x * y is less than 1/4 of FLT128_DENORM_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 < IEEE854_FLOAT128_BIAS
- FLT128_MANT_DIG
- 2)
82 int neg
= u
.ieee
.negative
^ v
.ieee
.negative
;
83 __float128 tiny
= neg
? -0x1p
-16494Q
: 0x1p
-16494Q
;
84 if (w
.ieee
.exponent
>= 3)
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
91 v
.value
= z
* 0x1p
114Q
+ tiny
;
92 if (TININESS_AFTER_ROUNDING
93 ? v
.ieee
.exponent
< 115
94 : (w
.ieee
.exponent
== 0
95 || (w
.ieee
.exponent
== 1
96 && w
.ieee
.negative
!= neg
97 && w
.ieee
.mant_low
== 0
98 && w
.ieee
.mant_high
== 0)))
100 __float128 force_underflow
= x
* y
;
101 math_force_eval (force_underflow
);
103 return v
.value
* 0x1p
-114Q
;
105 if (u
.ieee
.exponent
+ v
.ieee
.exponent
106 >= 0x7fff + IEEE854_FLOAT128_BIAS
- FLT128_MANT_DIG
)
108 /* Compute 1p-113 times smaller result and multiply
110 if (u
.ieee
.exponent
> v
.ieee
.exponent
)
111 u
.ieee
.exponent
-= FLT128_MANT_DIG
;
113 v
.ieee
.exponent
-= FLT128_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
> FLT128_MANT_DIG
)
117 w
.ieee
.exponent
-= FLT128_MANT_DIG
;
120 else if (w
.ieee
.exponent
>= 0x7fff - FLT128_MANT_DIG
)
123 If z exponent is very large and x and y exponents are
124 very small, adjust them up to avoid spurious underflows,
126 if (u
.ieee
.exponent
+ v
.ieee
.exponent
127 <= IEEE854_FLOAT128_BIAS
+ FLT128_MANT_DIG
)
129 if (u
.ieee
.exponent
> v
.ieee
.exponent
)
130 u
.ieee
.exponent
+= 2 * FLT128_MANT_DIG
+ 2;
132 v
.ieee
.exponent
+= 2 * FLT128_MANT_DIG
+ 2;
134 else if (u
.ieee
.exponent
> v
.ieee
.exponent
)
136 if (u
.ieee
.exponent
> FLT128_MANT_DIG
)
137 u
.ieee
.exponent
-= FLT128_MANT_DIG
;
139 else if (v
.ieee
.exponent
> FLT128_MANT_DIG
)
140 v
.ieee
.exponent
-= FLT128_MANT_DIG
;
141 w
.ieee
.exponent
-= FLT128_MANT_DIG
;
144 else if (u
.ieee
.exponent
>= 0x7fff - FLT128_MANT_DIG
)
146 u
.ieee
.exponent
-= FLT128_MANT_DIG
;
148 v
.ieee
.exponent
+= FLT128_MANT_DIG
;
152 else if (v
.ieee
.exponent
>= 0x7fff - FLT128_MANT_DIG
)
154 v
.ieee
.exponent
-= FLT128_MANT_DIG
;
156 u
.ieee
.exponent
+= FLT128_MANT_DIG
;
160 else /* if (u.ieee.exponent + v.ieee.exponent
161 <= IEEE854_FLOAT128_BIAS + FLT128_MANT_DIG) */
163 if (u
.ieee
.exponent
> v
.ieee
.exponent
)
164 u
.ieee
.exponent
+= 2 * FLT128_MANT_DIG
+ 2;
166 v
.ieee
.exponent
+= 2 * FLT128_MANT_DIG
+ 2;
167 if (w
.ieee
.exponent
<= 4 * FLT128_MANT_DIG
+ 6)
170 w
.ieee
.exponent
+= 2 * FLT128_MANT_DIG
+ 2;
175 /* Otherwise x * y should just affect inexact
183 /* Ensure correct sign of exact 0 + 0. */
184 if (__builtin_expect ((x
== 0 || y
== 0) && z
== 0, 0))
186 x
= math_opt_barrier (x
);
193 fesetround (FE_TONEAREST
);
196 /* Multiplication m1 + m2 = x * y using Dekker's algorithm. */
197 #define C ((1LL << (FLT128_MANT_DIG + 1) / 2) + 1)
198 __float128 x1
= x
* C
;
199 __float128 y1
= y
* C
;
200 __float128 m1
= x
* y
;
203 __float128 x2
= x
- x1
;
204 __float128 y2
= y
- y1
;
205 __float128 m2
= (((x1
* y1
- m1
) + x1
* y2
) + x2
* y1
) + x2
* y2
;
207 /* Addition a1 + a2 = z + m1 using Knuth's algorithm. */
208 __float128 a1
= z
+ m1
;
209 __float128 t1
= a1
- z
;
210 __float128 t2
= a1
- t1
;
213 __float128 a2
= t1
+ t2
;
214 /* Ensure the arithmetic is not scheduled after feclearexcept call. */
215 math_force_eval (m2
);
216 math_force_eval (a2
);
218 feclearexcept (FE_INEXACT
);
221 /* If the result is an exact zero, ensure it has the correct sign. */
222 if (a1
== 0 && m2
== 0)
227 /* Ensure that round-to-nearest value of z + m1 is not reused. */
228 z
= math_opt_barrier (z
);
233 fesetround (FE_TOWARDZERO
);
235 /* Perform m2 + a2 addition with round to odd. */
238 if (__builtin_expect (adjust
== 0, 1))
241 if ((u
.ieee
.mant_low
& 1) == 0 && u
.ieee
.exponent
!= 0x7fff)
242 u
.ieee
.mant_low
|= fetestexcept (FE_INEXACT
) != 0;
245 /* Result is a1 + u.value. */
248 else if (__builtin_expect (adjust
> 0, 1))
251 if ((u
.ieee
.mant_low
& 1) == 0 && u
.ieee
.exponent
!= 0x7fff)
252 u
.ieee
.mant_low
|= fetestexcept (FE_INEXACT
) != 0;
255 /* Result is a1 + u.value, scaled up. */
256 return (a1
+ u
.value
) * 0x1p
113Q
;
261 if ((u
.ieee
.mant_low
& 1) == 0)
262 u
.ieee
.mant_low
|= fetestexcept (FE_INEXACT
) != 0;
264 v
.value
= a1
+ u
.value
;
265 /* Ensure the addition is not scheduled after fetestexcept call. */
266 asm volatile ("" : : "m" (v
.value
));
268 int j
= fetestexcept (FE_INEXACT
) != 0;
273 /* Ensure the following computations are performed in default rounding
274 mode instead of just reusing the round to zero computation. */
275 asm volatile ("" : "=m" (u
) : "m" (u
));
276 /* If a1 + u.value is exact, the only rounding happens during
279 return v
.value
* 0x1p
-228Q
;
280 /* If result rounded to zero is not subnormal, no double
281 rounding will occur. */
282 if (v
.ieee
.exponent
> 228)
283 return (a1
+ u
.value
) * 0x1p
-228Q
;
284 /* If v.value * 0x1p-228Q with round to zero is a subnormal above
285 or equal to FLT128_MIN / 2, then v.value * 0x1p-228Q shifts mantissa
286 down just by 1 bit, which means v.ieee.mant_low |= j would
287 change the round bit, not sticky or guard bit.
288 v.value * 0x1p-228Q never normalizes by shifting up,
289 so round bit plus sticky bit should be already enough
290 for proper rounding. */
291 if (v
.ieee
.exponent
== 228)
293 /* If the exponent would be in the normal range when
294 rounding to normal precision with unbounded exponent
295 range, the exact result is known and spurious underflows
296 must be avoided on systems detecting tininess after
298 if (TININESS_AFTER_ROUNDING
)
300 w
.value
= a1
+ u
.value
;
301 if (w
.ieee
.exponent
== 229)
302 return w
.value
* 0x1p
-228Q
;
304 /* v.ieee.mant_low & 2 is LSB bit of the result before rounding,
305 v.ieee.mant_low & 1 is the round bit and j is our sticky
308 w
.ieee
.mant_low
= ((v
.ieee
.mant_low
& 3) << 1) | j
;
309 w
.ieee
.negative
= v
.ieee
.negative
;
310 v
.ieee
.mant_low
&= ~3U;
311 v
.value
*= 0x1p
-228Q
;
313 return v
.value
+ w
.value
;
315 v
.ieee
.mant_low
|= j
;
316 return v
.value
* 0x1p
-228Q
;