1 /* Compute x * y + z as ternary operation.
2 Copyright (C) 2010-2018 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>
25 #include <libm-alias-double.h>
28 /* This implementation uses rounding to odd to avoid problems with
29 double rounding. See a paper by Boldo and Melquiond:
30 http://www.lri.fr/~melquion/doc/08-tc.pdf */
33 __fma (double x
, double y
, double z
)
35 union ieee754_double u
, v
, w
;
40 if (__builtin_expect (u
.ieee
.exponent
+ v
.ieee
.exponent
41 >= 0x7ff + IEEE754_DOUBLE_BIAS
- DBL_MANT_DIG
, 0)
42 || __builtin_expect (u
.ieee
.exponent
>= 0x7ff - DBL_MANT_DIG
, 0)
43 || __builtin_expect (v
.ieee
.exponent
>= 0x7ff - DBL_MANT_DIG
, 0)
44 || __builtin_expect (w
.ieee
.exponent
>= 0x7ff - DBL_MANT_DIG
, 0)
45 || __builtin_expect (u
.ieee
.exponent
+ v
.ieee
.exponent
46 <= IEEE754_DOUBLE_BIAS
+ DBL_MANT_DIG
, 0))
48 /* If z is Inf, but x and y are finite, the result should be
50 if (w
.ieee
.exponent
== 0x7ff
51 && u
.ieee
.exponent
!= 0x7ff
52 && v
.ieee
.exponent
!= 0x7ff)
54 /* If z is zero and x are y are nonzero, compute the result
55 as x * y to avoid the wrong sign of a zero result if x * y
57 if (z
== 0 && x
!= 0 && y
!= 0)
59 /* If x or y or z is Inf/NaN, or if x * y is zero, compute as
61 if (u
.ieee
.exponent
== 0x7ff
62 || v
.ieee
.exponent
== 0x7ff
63 || w
.ieee
.exponent
== 0x7ff
67 /* If fma will certainly overflow, compute as x * y. */
68 if (u
.ieee
.exponent
+ v
.ieee
.exponent
> 0x7ff + IEEE754_DOUBLE_BIAS
)
70 /* If x * y is less than 1/4 of DBL_TRUE_MIN, neither the
71 result nor whether there is underflow depends on its exact
72 value, only on its sign. */
73 if (u
.ieee
.exponent
+ v
.ieee
.exponent
74 < IEEE754_DOUBLE_BIAS
- DBL_MANT_DIG
- 2)
76 int neg
= u
.ieee
.negative
^ v
.ieee
.negative
;
77 double tiny
= neg
? -0x1p
-1074 : 0x1p
-1074;
78 if (w
.ieee
.exponent
>= 3)
80 /* Scaling up, adding TINY and scaling down produces the
81 correct result, because in round-to-nearest mode adding
82 TINY has no effect and in other modes double rounding is
83 harmless. But it may not produce required underflow
85 v
.d
= z
* 0x1p
54 + tiny
;
86 if (TININESS_AFTER_ROUNDING
87 ? v
.ieee
.exponent
< 55
88 : (w
.ieee
.exponent
== 0
89 || (w
.ieee
.exponent
== 1
90 && w
.ieee
.negative
!= neg
91 && w
.ieee
.mantissa1
== 0
92 && w
.ieee
.mantissa0
== 0)))
94 double force_underflow
= x
* y
;
95 math_force_eval (force_underflow
);
99 if (u
.ieee
.exponent
+ v
.ieee
.exponent
100 >= 0x7ff + IEEE754_DOUBLE_BIAS
- DBL_MANT_DIG
)
102 /* Compute 1p-53 times smaller result and multiply
104 if (u
.ieee
.exponent
> v
.ieee
.exponent
)
105 u
.ieee
.exponent
-= DBL_MANT_DIG
;
107 v
.ieee
.exponent
-= DBL_MANT_DIG
;
108 /* If x + y exponent is very large and z exponent is very small,
109 it doesn't matter if we don't adjust it. */
110 if (w
.ieee
.exponent
> DBL_MANT_DIG
)
111 w
.ieee
.exponent
-= DBL_MANT_DIG
;
114 else if (w
.ieee
.exponent
>= 0x7ff - DBL_MANT_DIG
)
117 If z exponent is very large and x and y exponents are
118 very small, adjust them up to avoid spurious underflows,
120 if (u
.ieee
.exponent
+ v
.ieee
.exponent
121 <= IEEE754_DOUBLE_BIAS
+ 2 * DBL_MANT_DIG
)
123 if (u
.ieee
.exponent
> v
.ieee
.exponent
)
124 u
.ieee
.exponent
+= 2 * DBL_MANT_DIG
+ 2;
126 v
.ieee
.exponent
+= 2 * DBL_MANT_DIG
+ 2;
128 else if (u
.ieee
.exponent
> v
.ieee
.exponent
)
130 if (u
.ieee
.exponent
> DBL_MANT_DIG
)
131 u
.ieee
.exponent
-= DBL_MANT_DIG
;
133 else if (v
.ieee
.exponent
> DBL_MANT_DIG
)
134 v
.ieee
.exponent
-= DBL_MANT_DIG
;
135 w
.ieee
.exponent
-= DBL_MANT_DIG
;
138 else if (u
.ieee
.exponent
>= 0x7ff - DBL_MANT_DIG
)
140 u
.ieee
.exponent
-= DBL_MANT_DIG
;
142 v
.ieee
.exponent
+= DBL_MANT_DIG
;
146 else if (v
.ieee
.exponent
>= 0x7ff - DBL_MANT_DIG
)
148 v
.ieee
.exponent
-= DBL_MANT_DIG
;
150 u
.ieee
.exponent
+= DBL_MANT_DIG
;
154 else /* if (u.ieee.exponent + v.ieee.exponent
155 <= IEEE754_DOUBLE_BIAS + DBL_MANT_DIG) */
157 if (u
.ieee
.exponent
> v
.ieee
.exponent
)
158 u
.ieee
.exponent
+= 2 * DBL_MANT_DIG
+ 2;
160 v
.ieee
.exponent
+= 2 * DBL_MANT_DIG
+ 2;
161 if (w
.ieee
.exponent
<= 4 * DBL_MANT_DIG
+ 6)
164 w
.ieee
.exponent
+= 2 * DBL_MANT_DIG
+ 2;
169 /* Otherwise x * y should just affect inexact
177 /* Ensure correct sign of exact 0 + 0. */
178 if (__glibc_unlikely ((x
== 0 || y
== 0) && z
== 0))
180 x
= math_opt_barrier (x
);
185 libc_feholdexcept_setround (&env
, FE_TONEAREST
);
187 /* Multiplication m1 + m2 = x * y using Dekker's algorithm. */
188 #define C ((1 << (DBL_MANT_DIG + 1) / 2) + 1)
196 double m2
= (((x1
* y1
- m1
) + x1
* y2
) + x2
* y1
) + x2
* y2
;
198 /* Addition a1 + a2 = z + m1 using Knuth's algorithm. */
205 /* Ensure the arithmetic is not scheduled after feclearexcept call. */
206 math_force_eval (m2
);
207 math_force_eval (a2
);
208 feclearexcept (FE_INEXACT
);
210 /* If the result is an exact zero, ensure it has the correct sign. */
211 if (a1
== 0 && m2
== 0)
213 libc_feupdateenv (&env
);
214 /* Ensure that round-to-nearest value of z + m1 is not reused. */
215 z
= math_opt_barrier (z
);
219 libc_fesetround (FE_TOWARDZERO
);
221 /* Perform m2 + a2 addition with round to odd. */
224 if (__glibc_unlikely (adjust
< 0))
226 if ((u
.ieee
.mantissa1
& 1) == 0)
227 u
.ieee
.mantissa1
|= libc_fetestexcept (FE_INEXACT
) != 0;
229 /* Ensure the addition is not scheduled after fetestexcept call. */
230 math_force_eval (v
.d
);
233 /* Reset rounding mode and test for inexact simultaneously. */
234 int j
= libc_feupdateenv_test (&env
, FE_INEXACT
) != 0;
236 if (__glibc_likely (adjust
== 0))
238 if ((u
.ieee
.mantissa1
& 1) == 0 && u
.ieee
.exponent
!= 0x7ff)
239 u
.ieee
.mantissa1
|= j
;
240 /* Result is a1 + u.d. */
243 else if (__glibc_likely (adjust
> 0))
245 if ((u
.ieee
.mantissa1
& 1) == 0 && u
.ieee
.exponent
!= 0x7ff)
246 u
.ieee
.mantissa1
|= j
;
247 /* Result is a1 + u.d, scaled up. */
248 return (a1
+ u
.d
) * 0x1p
53;
252 /* If a1 + u.d is exact, the only rounding happens during
255 return v
.d
* 0x1p
-108;
256 /* If result rounded to zero is not subnormal, no double
257 rounding will occur. */
258 if (v
.ieee
.exponent
> 108)
259 return (a1
+ u
.d
) * 0x1p
-108;
260 /* If v.d * 0x1p-108 with round to zero is a subnormal above
261 or equal to DBL_MIN / 2, then v.d * 0x1p-108 shifts mantissa
262 down just by 1 bit, which means v.ieee.mantissa1 |= j would
263 change the round bit, not sticky or guard bit.
264 v.d * 0x1p-108 never normalizes by shifting up,
265 so round bit plus sticky bit should be already enough
266 for proper rounding. */
267 if (v
.ieee
.exponent
== 108)
269 /* If the exponent would be in the normal range when
270 rounding to normal precision with unbounded exponent
271 range, the exact result is known and spurious underflows
272 must be avoided on systems detecting tininess after
274 if (TININESS_AFTER_ROUNDING
)
277 if (w
.ieee
.exponent
== 109)
278 return w
.d
* 0x1p
-108;
280 /* v.ieee.mantissa1 & 2 is LSB bit of the result before rounding,
281 v.ieee.mantissa1 & 1 is the round bit and j is our sticky
284 w
.ieee
.mantissa1
= ((v
.ieee
.mantissa1
& 3) << 1) | j
;
285 w
.ieee
.negative
= v
.ieee
.negative
;
286 v
.ieee
.mantissa1
&= ~3U;
291 v
.ieee
.mantissa1
|= j
;
292 return v
.d
* 0x1p
-108;
296 libm_alias_double (__fma
, fma
)