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.
5
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.
10
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.
15
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/>. */
19
20#include <float.h>
21#include <math.h>
22#include <fenv.h>
23#include <ieee754.h>
24#include <math-barriers.h>
25#include <math_private.h>
26#include <libm-alias-double.h>
27#include <tininess.h>
28
29/* This implementation uses rounding to odd to avoid problems with
30 double rounding. See a paper by Boldo and Melquiond:
31 http://www.lri.fr/~melquion/doc/08-tc.pdf */
32
33double
34__fma (double x, double y, double z)
35{
36 union ieee754_double u, v, w;
37 int adjust = 0;
38 u.d = x;
39 v.d = y;
40 w.d = z;
41 if (__builtin_expect (u.ieee.exponent + v.ieee.exponent
42 >= 0x7ff + IEEE754_DOUBLE_BIAS - DBL_MANT_DIG, 0)
43 || __builtin_expect (u.ieee.exponent >= 0x7ff - DBL_MANT_DIG, 0)
44 || __builtin_expect (v.ieee.exponent >= 0x7ff - DBL_MANT_DIG, 0)
45 || __builtin_expect (w.ieee.exponent >= 0x7ff - DBL_MANT_DIG, 0)
46 || __builtin_expect (u.ieee.exponent + v.ieee.exponent
47 <= IEEE754_DOUBLE_BIAS + DBL_MANT_DIG, 0))
48 {
49 /* If z is Inf, but x and y are finite, the result should be
50 z rather than NaN. */
51 if (w.ieee.exponent == 0x7ff
52 && u.ieee.exponent != 0x7ff
53 && v.ieee.exponent != 0x7ff)
54 return (z + x) + y;
55 /* If z is zero and x are y are nonzero, compute the result
56 as x * y to avoid the wrong sign of a zero result if x * y
57 underflows to 0. */
58 if (z == 0 && x != 0 && y != 0)
59 return x * y;
60 /* If x or y or z is Inf/NaN, or if x * y is zero, compute as
61 x * y + z. */
62 if (u.ieee.exponent == 0x7ff
63 || v.ieee.exponent == 0x7ff
64 || w.ieee.exponent == 0x7ff
65 || x == 0
66 || y == 0)
67 return x * y + z;
68 /* If fma will certainly overflow, compute as x * y. */
69 if (u.ieee.exponent + v.ieee.exponent > 0x7ff + IEEE754_DOUBLE_BIAS)
70 return x * y;
71 /* If x * y is less than 1/4 of DBL_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 < IEEE754_DOUBLE_BIAS - DBL_MANT_DIG - 2)
76 {
77 int neg = u.ieee.negative ^ v.ieee.negative;
78 double tiny = neg ? -0x1p-1074 : 0x1p-1074;
79 if (w.ieee.exponent >= 3)
80 return tiny + z;
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
85 exceptions. */
86 v.d = z * 0x1p54 + tiny;
87 if (TININESS_AFTER_ROUNDING
88 ? v.ieee.exponent < 55
89 : (w.ieee.exponent == 0
90 || (w.ieee.exponent == 1
91 && w.ieee.negative != neg
92 && w.ieee.mantissa1 == 0
93 && w.ieee.mantissa0 == 0)))
94 {
95 double force_underflow = x * y;
96 math_force_eval (force_underflow);
97 }
98 return v.d * 0x1p-54;
99 }
100 if (u.ieee.exponent + v.ieee.exponent
101 >= 0x7ff + IEEE754_DOUBLE_BIAS - DBL_MANT_DIG)
102 {
103 /* Compute 1p-53 times smaller result and multiply
104 at the end. */
105 if (u.ieee.exponent > v.ieee.exponent)
106 u.ieee.exponent -= DBL_MANT_DIG;
107 else
108 v.ieee.exponent -= DBL_MANT_DIG;
109 /* If x + y exponent is very large and z exponent is very small,
110 it doesn't matter if we don't adjust it. */
111 if (w.ieee.exponent > DBL_MANT_DIG)
112 w.ieee.exponent -= DBL_MANT_DIG;
113 adjust = 1;
114 }
115 else if (w.ieee.exponent >= 0x7ff - DBL_MANT_DIG)
116 {
117 /* Similarly.
118 If z exponent is very large and x and y exponents are
119 very small, adjust them up to avoid spurious underflows,
120 rather than down. */
121 if (u.ieee.exponent + v.ieee.exponent
122 <= IEEE754_DOUBLE_BIAS + 2 * DBL_MANT_DIG)
123 {
124 if (u.ieee.exponent > v.ieee.exponent)
125 u.ieee.exponent += 2 * DBL_MANT_DIG + 2;
126 else
127 v.ieee.exponent += 2 * DBL_MANT_DIG + 2;
128 }
129 else if (u.ieee.exponent > v.ieee.exponent)
130 {
131 if (u.ieee.exponent > DBL_MANT_DIG)
132 u.ieee.exponent -= DBL_MANT_DIG;
133 }
134 else if (v.ieee.exponent > DBL_MANT_DIG)
135 v.ieee.exponent -= DBL_MANT_DIG;
136 w.ieee.exponent -= DBL_MANT_DIG;
137 adjust = 1;
138 }
139 else if (u.ieee.exponent >= 0x7ff - DBL_MANT_DIG)
140 {
141 u.ieee.exponent -= DBL_MANT_DIG;
142 if (v.ieee.exponent)
143 v.ieee.exponent += DBL_MANT_DIG;
144 else
145 v.d *= 0x1p53;
146 }
147 else if (v.ieee.exponent >= 0x7ff - DBL_MANT_DIG)
148 {
149 v.ieee.exponent -= DBL_MANT_DIG;
150 if (u.ieee.exponent)
151 u.ieee.exponent += DBL_MANT_DIG;
152 else
153 u.d *= 0x1p53;
154 }
155 else /* if (u.ieee.exponent + v.ieee.exponent
156 <= IEEE754_DOUBLE_BIAS + DBL_MANT_DIG) */
157 {
158 if (u.ieee.exponent > v.ieee.exponent)
159 u.ieee.exponent += 2 * DBL_MANT_DIG + 2;
160 else
161 v.ieee.exponent += 2 * DBL_MANT_DIG + 2;
162 if (w.ieee.exponent <= 4 * DBL_MANT_DIG + 6)
163 {
164 if (w.ieee.exponent)
165 w.ieee.exponent += 2 * DBL_MANT_DIG + 2;
166 else
167 w.d *= 0x1p108;
168 adjust = -1;
169 }
170 /* Otherwise x * y should just affect inexact
171 and nothing else. */
172 }
173 x = u.d;
174 y = v.d;
175 z = w.d;
176 }
177
178 /* Ensure correct sign of exact 0 + 0. */
179 if (__glibc_unlikely ((x == 0 || y == 0) && z == 0))
180 {
181 x = math_opt_barrier (x);
182 return x * y + z;
183 }
184
185 fenv_t env;
186 libc_feholdexcept_setround (&env, FE_TONEAREST);
187
188 /* Multiplication m1 + m2 = x * y using Dekker's algorithm. */
189#define C ((1 << (DBL_MANT_DIG + 1) / 2) + 1)
190 double x1 = x * C;
191 double y1 = y * C;
192 double m1 = x * y;
193 x1 = (x - x1) + x1;
194 y1 = (y - y1) + y1;
195 double x2 = x - x1;
196 double y2 = y - y1;
197 double m2 = (((x1 * y1 - m1) + x1 * y2) + x2 * y1) + x2 * y2;
198
199 /* Addition a1 + a2 = z + m1 using Knuth's algorithm. */
200 double a1 = z + m1;
201 double t1 = a1 - z;
202 double t2 = a1 - t1;
203 t1 = m1 - t1;
204 t2 = z - t2;
205 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);
210
211 /* If the result is an exact zero, ensure it has the correct sign. */
212 if (a1 == 0 && m2 == 0)
213 {
214 libc_feupdateenv (&env);
215 /* Ensure that round-to-nearest value of z + m1 is not reused. */
216 z = math_opt_barrier (z);
217 return z + m1;
218 }
219
220 libc_fesetround (FE_TOWARDZERO);
221
222 /* Perform m2 + a2 addition with round to odd. */
223 u.d = a2 + m2;
224
225 if (__glibc_unlikely (adjust < 0))
226 {
227 if ((u.ieee.mantissa1 & 1) == 0)
228 u.ieee.mantissa1 |= libc_fetestexcept (FE_INEXACT) != 0;
229 v.d = a1 + u.d;
230 /* Ensure the addition is not scheduled after fetestexcept call. */
231 math_force_eval (v.d);
232 }
233
234 /* Reset rounding mode and test for inexact simultaneously. */
235 int j = libc_feupdateenv_test (&env, FE_INEXACT) != 0;
236
237 if (__glibc_likely (adjust == 0))
238 {
239 if ((u.ieee.mantissa1 & 1) == 0 && u.ieee.exponent != 0x7ff)
240 u.ieee.mantissa1 |= j;
241 /* Result is a1 + u.d. */
242 return a1 + u.d;
243 }
244 else if (__glibc_likely (adjust > 0))
245 {
246 if ((u.ieee.mantissa1 & 1) == 0 && u.ieee.exponent != 0x7ff)
247 u.ieee.mantissa1 |= j;
248 /* Result is a1 + u.d, scaled up. */
249 return (a1 + u.d) * 0x1p53;
250 }
251 else
252 {
253 /* If a1 + u.d is exact, the only rounding happens during
254 scaling down. */
255 if (j == 0)
256 return v.d * 0x1p-108;
257 /* If result rounded to zero is not subnormal, no double
258 rounding will occur. */
259 if (v.ieee.exponent > 108)
260 return (a1 + u.d) * 0x1p-108;
261 /* If v.d * 0x1p-108 with round to zero is a subnormal above
262 or equal to DBL_MIN / 2, then v.d * 0x1p-108 shifts mantissa
263 down just by 1 bit, which means v.ieee.mantissa1 |= j would
264 change the round bit, not sticky or guard bit.
265 v.d * 0x1p-108 never normalizes by shifting up,
266 so round bit plus sticky bit should be already enough
267 for proper rounding. */
268 if (v.ieee.exponent == 108)
269 {
270 /* If the exponent would be in the normal range when
271 rounding to normal precision with unbounded exponent
272 range, the exact result is known and spurious underflows
273 must be avoided on systems detecting tininess after
274 rounding. */
275 if (TININESS_AFTER_ROUNDING)
276 {
277 w.d = a1 + u.d;
278 if (w.ieee.exponent == 109)
279 return w.d * 0x1p-108;
280 }
281 /* v.ieee.mantissa1 & 2 is LSB bit of the result before rounding,
282 v.ieee.mantissa1 & 1 is the round bit and j is our sticky
283 bit. */
284 w.d = 0.0;
285 w.ieee.mantissa1 = ((v.ieee.mantissa1 & 3) << 1) | j;
286 w.ieee.negative = v.ieee.negative;
287 v.ieee.mantissa1 &= ~3U;
288 v.d *= 0x1p-108;
289 w.d *= 0x1p-2;
290 return v.d + w.d;
291 }
292 v.ieee.mantissa1 |= j;
293 return v.d * 0x1p-108;
294 }
295}
296#ifndef __fma
297libm_alias_double (__fma, fma)
298#endif
299