1 /* Compute x * y + z as ternary operation.
2 Copyright (C) 2010-2015 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/>. */
24 #include <math_private.h>
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 */
32 __fma (double x
, double y
, double z
)
34 union ieee754_double u
, v
, w
;
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))
47 /* If z is Inf, but x and y are finite, the result should be
49 if (w
.ieee
.exponent
== 0x7ff
50 && u
.ieee
.exponent
!= 0x7ff
51 && v
.ieee
.exponent
!= 0x7ff)
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
56 if (z
== 0 && x
!= 0 && y
!= 0)
58 /* If x or y or z is Inf/NaN, or if x * y is zero, compute as
60 if (u
.ieee
.exponent
== 0x7ff
61 || v
.ieee
.exponent
== 0x7ff
62 || w
.ieee
.exponent
== 0x7ff
66 /* If fma will certainly overflow, compute as x * y. */
67 if (u
.ieee
.exponent
+ v
.ieee
.exponent
> 0x7ff + IEEE754_DOUBLE_BIAS
)
69 /* If x * y is less than 1/4 of DBL_DENORM_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)
75 int neg
= u
.ieee
.negative
^ v
.ieee
.negative
;
76 double tiny
= neg
? -0x1p
-1074 : 0x1p
-1074;
77 if (w
.ieee
.exponent
>= 3)
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
84 v
.d
= z
* 0x1p
54 + 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)))
93 volatile double force_underflow
= x
* y
;
94 (void) force_underflow
;
98 if (u
.ieee
.exponent
+ v
.ieee
.exponent
99 >= 0x7ff + IEEE754_DOUBLE_BIAS
- DBL_MANT_DIG
)
101 /* Compute 1p-53 times smaller result and multiply
103 if (u
.ieee
.exponent
> v
.ieee
.exponent
)
104 u
.ieee
.exponent
-= DBL_MANT_DIG
;
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
;
113 else if (w
.ieee
.exponent
>= 0x7ff - DBL_MANT_DIG
)
116 If z exponent is very large and x and y exponents are
117 very small, adjust them up to avoid spurious underflows,
119 if (u
.ieee
.exponent
+ v
.ieee
.exponent
120 <= IEEE754_DOUBLE_BIAS
+ DBL_MANT_DIG
)
122 if (u
.ieee
.exponent
> v
.ieee
.exponent
)
123 u
.ieee
.exponent
+= 2 * DBL_MANT_DIG
+ 2;
125 v
.ieee
.exponent
+= 2 * DBL_MANT_DIG
+ 2;
127 else if (u
.ieee
.exponent
> v
.ieee
.exponent
)
129 if (u
.ieee
.exponent
> DBL_MANT_DIG
)
130 u
.ieee
.exponent
-= DBL_MANT_DIG
;
132 else if (v
.ieee
.exponent
> DBL_MANT_DIG
)
133 v
.ieee
.exponent
-= DBL_MANT_DIG
;
134 w
.ieee
.exponent
-= DBL_MANT_DIG
;
137 else if (u
.ieee
.exponent
>= 0x7ff - DBL_MANT_DIG
)
139 u
.ieee
.exponent
-= DBL_MANT_DIG
;
141 v
.ieee
.exponent
+= DBL_MANT_DIG
;
145 else if (v
.ieee
.exponent
>= 0x7ff - DBL_MANT_DIG
)
147 v
.ieee
.exponent
-= DBL_MANT_DIG
;
149 u
.ieee
.exponent
+= DBL_MANT_DIG
;
153 else /* if (u.ieee.exponent + v.ieee.exponent
154 <= IEEE754_DOUBLE_BIAS + DBL_MANT_DIG) */
156 if (u
.ieee
.exponent
> v
.ieee
.exponent
)
157 u
.ieee
.exponent
+= 2 * DBL_MANT_DIG
+ 2;
159 v
.ieee
.exponent
+= 2 * DBL_MANT_DIG
+ 2;
160 if (w
.ieee
.exponent
<= 4 * DBL_MANT_DIG
+ 6)
163 w
.ieee
.exponent
+= 2 * DBL_MANT_DIG
+ 2;
168 /* Otherwise x * y should just affect inexact
176 /* Ensure correct sign of exact 0 + 0. */
177 if (__glibc_unlikely ((x
== 0 || y
== 0) && z
== 0))
181 libc_feholdexcept_setround (&env
, FE_TONEAREST
);
183 /* Multiplication m1 + m2 = x * y using Dekker's algorithm. */
184 #define C ((1 << (DBL_MANT_DIG + 1) / 2) + 1)
192 double m2
= (((x1
* y1
- m1
) + x1
* y2
) + x2
* y1
) + x2
* y2
;
194 /* Addition a1 + a2 = z + m1 using Knuth's algorithm. */
201 /* Ensure the arithmetic is not scheduled after feclearexcept call. */
202 math_force_eval (m2
);
203 math_force_eval (a2
);
204 feclearexcept (FE_INEXACT
);
206 /* If the result is an exact zero, ensure it has the correct sign. */
207 if (a1
== 0 && m2
== 0)
209 libc_feupdateenv (&env
);
210 /* Ensure that round-to-nearest value of z + m1 is not reused. */
211 z
= math_opt_barrier (z
);
215 libc_fesetround (FE_TOWARDZERO
);
217 /* Perform m2 + a2 addition with round to odd. */
220 if (__glibc_unlikely (adjust
< 0))
222 if ((u
.ieee
.mantissa1
& 1) == 0)
223 u
.ieee
.mantissa1
|= libc_fetestexcept (FE_INEXACT
) != 0;
225 /* Ensure the addition is not scheduled after fetestexcept call. */
226 math_force_eval (v
.d
);
229 /* Reset rounding mode and test for inexact simultaneously. */
230 int j
= libc_feupdateenv_test (&env
, FE_INEXACT
) != 0;
232 if (__glibc_likely (adjust
== 0))
234 if ((u
.ieee
.mantissa1
& 1) == 0 && u
.ieee
.exponent
!= 0x7ff)
235 u
.ieee
.mantissa1
|= j
;
236 /* Result is a1 + u.d. */
239 else if (__glibc_likely (adjust
> 0))
241 if ((u
.ieee
.mantissa1
& 1) == 0 && u
.ieee
.exponent
!= 0x7ff)
242 u
.ieee
.mantissa1
|= j
;
243 /* Result is a1 + u.d, scaled up. */
244 return (a1
+ u
.d
) * 0x1p
53;
248 /* If a1 + u.d is exact, the only rounding happens during
251 return v
.d
* 0x1p
-108;
252 /* If result rounded to zero is not subnormal, no double
253 rounding will occur. */
254 if (v
.ieee
.exponent
> 108)
255 return (a1
+ u
.d
) * 0x1p
-108;
256 /* If v.d * 0x1p-108 with round to zero is a subnormal above
257 or equal to DBL_MIN / 2, then v.d * 0x1p-108 shifts mantissa
258 down just by 1 bit, which means v.ieee.mantissa1 |= j would
259 change the round bit, not sticky or guard bit.
260 v.d * 0x1p-108 never normalizes by shifting up,
261 so round bit plus sticky bit should be already enough
262 for proper rounding. */
263 if (v
.ieee
.exponent
== 108)
265 /* If the exponent would be in the normal range when
266 rounding to normal precision with unbounded exponent
267 range, the exact result is known and spurious underflows
268 must be avoided on systems detecting tininess after
270 if (TININESS_AFTER_ROUNDING
)
273 if (w
.ieee
.exponent
== 109)
274 return w
.d
* 0x1p
-108;
276 /* v.ieee.mantissa1 & 2 is LSB bit of the result before rounding,
277 v.ieee.mantissa1 & 1 is the round bit and j is our sticky
280 w
.ieee
.mantissa1
= ((v
.ieee
.mantissa1
& 3) << 1) | j
;
281 w
.ieee
.negative
= v
.ieee
.negative
;
282 v
.ieee
.mantissa1
&= ~3U;
287 v
.ieee
.mantissa1
|= j
;
288 return v
.d
* 0x1p
-108;
292 weak_alias (__fma
, fma
)
295 #ifdef NO_LONG_DOUBLE
296 strong_alias (__fma
, __fmal
)
297 weak_alias (__fmal
, fmal
)