]>
git.ipfire.org Git - thirdparty/glibc.git/blob - sysdeps/ieee754/ldbl-128/s_fmal.c
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 __fmal (long double x
, long double y
, long double z
)
34 union ieee854_long_double u
, v
, w
;
39 if (__builtin_expect (u
.ieee
.exponent
+ v
.ieee
.exponent
40 >= 0x7fff + IEEE854_LONG_DOUBLE_BIAS
42 || __builtin_expect (u
.ieee
.exponent
>= 0x7fff - LDBL_MANT_DIG
, 0)
43 || __builtin_expect (v
.ieee
.exponent
>= 0x7fff - LDBL_MANT_DIG
, 0)
44 || __builtin_expect (w
.ieee
.exponent
>= 0x7fff - LDBL_MANT_DIG
, 0)
45 || __builtin_expect (u
.ieee
.exponent
+ v
.ieee
.exponent
46 <= IEEE854_LONG_DOUBLE_BIAS
+ LDBL_MANT_DIG
, 0))
48 /* If z is Inf, but x and y are finite, the result should be
50 if (w
.ieee
.exponent
== 0x7fff
51 && u
.ieee
.exponent
!= 0x7fff
52 && v
.ieee
.exponent
!= 0x7fff)
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
== 0x7fff
62 || v
.ieee
.exponent
== 0x7fff
63 || w
.ieee
.exponent
== 0x7fff
67 /* If fma will certainly overflow, compute as x * y. */
68 if (u
.ieee
.exponent
+ v
.ieee
.exponent
69 > 0x7fff + IEEE854_LONG_DOUBLE_BIAS
)
71 /* If x * y is less than 1/4 of LDBL_TRUE_MIN, neither the
72 result nor whether there is underflow depends on its exact
73 value, only on its sign. */
74 if (u
.ieee
.exponent
+ v
.ieee
.exponent
75 < IEEE854_LONG_DOUBLE_BIAS
- LDBL_MANT_DIG
- 2)
77 int neg
= u
.ieee
.negative
^ v
.ieee
.negative
;
78 long double tiny
= neg
? -0x1p
-16494L : 0x1p
-16494L;
79 if (w
.ieee
.exponent
>= 3)
81 /* Scaling up, adding TINY and scaling down produces the
82 correct result, because in round-to-nearest mode adding
83 TINY has no effect and in other modes double rounding is
84 harmless. But it may not produce required underflow
86 v
.d
= z
* 0x1p
114L + tiny
;
87 if (TININESS_AFTER_ROUNDING
88 ? v
.ieee
.exponent
< 115
89 : (w
.ieee
.exponent
== 0
90 || (w
.ieee
.exponent
== 1
91 && w
.ieee
.negative
!= neg
92 && w
.ieee
.mantissa3
== 0
93 && w
.ieee
.mantissa2
== 0
94 && w
.ieee
.mantissa1
== 0
95 && w
.ieee
.mantissa0
== 0)))
97 long double force_underflow
= x
* y
;
98 math_force_eval (force_underflow
);
100 return v
.d
* 0x1p
-114L;
102 if (u
.ieee
.exponent
+ v
.ieee
.exponent
103 >= 0x7fff + IEEE854_LONG_DOUBLE_BIAS
- LDBL_MANT_DIG
)
105 /* Compute 1p-113 times smaller result and multiply
107 if (u
.ieee
.exponent
> v
.ieee
.exponent
)
108 u
.ieee
.exponent
-= LDBL_MANT_DIG
;
110 v
.ieee
.exponent
-= LDBL_MANT_DIG
;
111 /* If x + y exponent is very large and z exponent is very small,
112 it doesn't matter if we don't adjust it. */
113 if (w
.ieee
.exponent
> LDBL_MANT_DIG
)
114 w
.ieee
.exponent
-= LDBL_MANT_DIG
;
117 else if (w
.ieee
.exponent
>= 0x7fff - LDBL_MANT_DIG
)
120 If z exponent is very large and x and y exponents are
121 very small, adjust them up to avoid spurious underflows,
123 if (u
.ieee
.exponent
+ v
.ieee
.exponent
124 <= IEEE854_LONG_DOUBLE_BIAS
+ 2 * LDBL_MANT_DIG
)
126 if (u
.ieee
.exponent
> v
.ieee
.exponent
)
127 u
.ieee
.exponent
+= 2 * LDBL_MANT_DIG
+ 2;
129 v
.ieee
.exponent
+= 2 * LDBL_MANT_DIG
+ 2;
131 else if (u
.ieee
.exponent
> v
.ieee
.exponent
)
133 if (u
.ieee
.exponent
> LDBL_MANT_DIG
)
134 u
.ieee
.exponent
-= LDBL_MANT_DIG
;
136 else if (v
.ieee
.exponent
> LDBL_MANT_DIG
)
137 v
.ieee
.exponent
-= LDBL_MANT_DIG
;
138 w
.ieee
.exponent
-= LDBL_MANT_DIG
;
141 else if (u
.ieee
.exponent
>= 0x7fff - LDBL_MANT_DIG
)
143 u
.ieee
.exponent
-= LDBL_MANT_DIG
;
145 v
.ieee
.exponent
+= LDBL_MANT_DIG
;
149 else if (v
.ieee
.exponent
>= 0x7fff - LDBL_MANT_DIG
)
151 v
.ieee
.exponent
-= LDBL_MANT_DIG
;
153 u
.ieee
.exponent
+= LDBL_MANT_DIG
;
157 else /* if (u.ieee.exponent + v.ieee.exponent
158 <= IEEE854_LONG_DOUBLE_BIAS + LDBL_MANT_DIG) */
160 if (u
.ieee
.exponent
> v
.ieee
.exponent
)
161 u
.ieee
.exponent
+= 2 * LDBL_MANT_DIG
+ 2;
163 v
.ieee
.exponent
+= 2 * LDBL_MANT_DIG
+ 2;
164 if (w
.ieee
.exponent
<= 4 * LDBL_MANT_DIG
+ 6)
167 w
.ieee
.exponent
+= 2 * LDBL_MANT_DIG
+ 2;
172 /* Otherwise x * y should just affect inexact
180 /* Ensure correct sign of exact 0 + 0. */
181 if (__glibc_unlikely ((x
== 0 || y
== 0) && z
== 0))
186 fesetround (FE_TONEAREST
);
188 /* Multiplication m1 + m2 = x * y using Dekker's algorithm. */
189 #define C ((1LL << (LDBL_MANT_DIG + 1) / 2) + 1)
190 long double x1
= x
* C
;
191 long double y1
= y
* C
;
192 long double m1
= x
* y
;
195 long double x2
= x
- x1
;
196 long double y2
= y
- y1
;
197 long double m2
= (((x1
* y1
- m1
) + x1
* y2
) + x2
* y1
) + x2
* y2
;
199 /* Addition a1 + a2 = z + m1 using Knuth's algorithm. */
200 long double a1
= z
+ m1
;
201 long double t1
= a1
- z
;
202 long double t2
= a1
- t1
;
205 long double a2
= t1
+ t2
;
206 /* Ensure the arithmetic is not scheduled after feclearexcept call. */
207 math_force_eval (m2
);
208 math_force_eval (a2
);
209 feclearexcept (FE_INEXACT
);
211 /* If the result is an exact zero, ensure it has the correct sign. */
212 if (a1
== 0 && m2
== 0)
215 /* Ensure that round-to-nearest value of z + m1 is not reused. */
216 z
= math_opt_barrier (z
);
220 fesetround (FE_TOWARDZERO
);
221 /* Perform m2 + a2 addition with round to odd. */
224 if (__glibc_likely (adjust
== 0))
226 if ((u
.ieee
.mantissa3
& 1) == 0 && u
.ieee
.exponent
!= 0x7fff)
227 u
.ieee
.mantissa3
|= fetestexcept (FE_INEXACT
) != 0;
229 /* Result is a1 + u.d. */
232 else if (__glibc_likely (adjust
> 0))
234 if ((u
.ieee
.mantissa3
& 1) == 0 && u
.ieee
.exponent
!= 0x7fff)
235 u
.ieee
.mantissa3
|= fetestexcept (FE_INEXACT
) != 0;
237 /* Result is a1 + u.d, scaled up. */
238 return (a1
+ u
.d
) * 0x1p
113L;
242 if ((u
.ieee
.mantissa3
& 1) == 0)
243 u
.ieee
.mantissa3
|= fetestexcept (FE_INEXACT
) != 0;
245 /* Ensure the addition is not scheduled after fetestexcept call. */
246 math_force_eval (v
.d
);
247 int j
= fetestexcept (FE_INEXACT
) != 0;
249 /* Ensure the following computations are performed in default rounding
250 mode instead of just reusing the round to zero computation. */
251 asm volatile ("" : "=m" (u
) : "m" (u
));
252 /* If a1 + u.d is exact, the only rounding happens during
255 return v
.d
* 0x1p
-228L;
256 /* If result rounded to zero is not subnormal, no double
257 rounding will occur. */
258 if (v
.ieee
.exponent
> 228)
259 return (a1
+ u
.d
) * 0x1p
-228L;
260 /* If v.d * 0x1p-228L with round to zero is a subnormal above
261 or equal to LDBL_MIN / 2, then v.d * 0x1p-228L shifts mantissa
262 down just by 1 bit, which means v.ieee.mantissa3 |= j would
263 change the round bit, not sticky or guard bit.
264 v.d * 0x1p-228L 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
== 228)
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
== 229)
278 return w
.d
* 0x1p
-228L;
280 /* v.ieee.mantissa3 & 2 is LSB bit of the result before rounding,
281 v.ieee.mantissa3 & 1 is the round bit and j is our sticky
284 w
.ieee
.mantissa3
= ((v
.ieee
.mantissa3
& 3) << 1) | j
;
285 w
.ieee
.negative
= v
.ieee
.negative
;
286 v
.ieee
.mantissa3
&= ~3U;
291 v
.ieee
.mantissa3
|= j
;
292 return v
.d
* 0x1p
-228L;
295 weak_alias (__fmal
, fmal
)