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-ldouble.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
33_Float128
34__fmal (_Float128 x, _Float128 y, _Float128 z)
35{
36 union ieee854_long_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 >= 0x7fff + IEEE854_LONG_DOUBLE_BIAS
43 - LDBL_MANT_DIG, 0)
44 || __builtin_expect (u.ieee.exponent >= 0x7fff - LDBL_MANT_DIG, 0)
45 || __builtin_expect (v.ieee.exponent >= 0x7fff - LDBL_MANT_DIG, 0)
46 || __builtin_expect (w.ieee.exponent >= 0x7fff - LDBL_MANT_DIG, 0)
47 || __builtin_expect (u.ieee.exponent + v.ieee.exponent
48 <= IEEE854_LONG_DOUBLE_BIAS + LDBL_MANT_DIG, 0))
49 {
50 /* If z is Inf, but x and y are finite, the result should be
51 z rather than NaN. */
52 if (w.ieee.exponent == 0x7fff
53 && u.ieee.exponent != 0x7fff
54 && v.ieee.exponent != 0x7fff)
55 return (z + x) + y;
56 /* If z is zero and x are y are nonzero, compute the result
57 as x * y to avoid the wrong sign of a zero result if x * y
58 underflows to 0. */
59 if (z == 0 && x != 0 && y != 0)
60 return x * y;
61 /* If x or y or z is Inf/NaN, or if x * y is zero, compute as
62 x * y + z. */
63 if (u.ieee.exponent == 0x7fff
64 || v.ieee.exponent == 0x7fff
65 || w.ieee.exponent == 0x7fff
66 || x == 0
67 || y == 0)
68 return x * y + z;
69 /* If fma will certainly overflow, compute as x * y. */
70 if (u.ieee.exponent + v.ieee.exponent
71 > 0x7fff + IEEE854_LONG_DOUBLE_BIAS)
72 return x * y;
73 /* If x * y is less than 1/4 of LDBL_TRUE_MIN, neither the
74 result nor whether there is underflow depends on its exact
75 value, only on its sign. */
76 if (u.ieee.exponent + v.ieee.exponent
77 < IEEE854_LONG_DOUBLE_BIAS - LDBL_MANT_DIG - 2)
78 {
79 int neg = u.ieee.negative ^ v.ieee.negative;
80 _Float128 tiny = neg ? L(-0x1p-16494) : L(0x1p-16494);
81 if (w.ieee.exponent >= 3)
82 return tiny + z;
83 /* Scaling up, adding TINY and scaling down produces the
84 correct result, because in round-to-nearest mode adding
85 TINY has no effect and in other modes double rounding is
86 harmless. But it may not produce required underflow
87 exceptions. */
88 v.d = z * L(0x1p114) + tiny;
89 if (TININESS_AFTER_ROUNDING
90 ? v.ieee.exponent < 115
91 : (w.ieee.exponent == 0
92 || (w.ieee.exponent == 1
93 && w.ieee.negative != neg
94 && w.ieee.mantissa3 == 0
95 && w.ieee.mantissa2 == 0
96 && w.ieee.mantissa1 == 0
97 && w.ieee.mantissa0 == 0)))
98 {
99 _Float128 force_underflow = x * y;
100 math_force_eval (force_underflow);
101 }
102 return v.d * L(0x1p-114);
103 }
104 if (u.ieee.exponent + v.ieee.exponent
105 >= 0x7fff + IEEE854_LONG_DOUBLE_BIAS - LDBL_MANT_DIG)
106 {
107 /* Compute 1p-113 times smaller result and multiply
108 at the end. */
109 if (u.ieee.exponent > v.ieee.exponent)
110 u.ieee.exponent -= LDBL_MANT_DIG;
111 else
112 v.ieee.exponent -= LDBL_MANT_DIG;
113 /* If x + y exponent is very large and z exponent is very small,
114 it doesn't matter if we don't adjust it. */
115 if (w.ieee.exponent > LDBL_MANT_DIG)
116 w.ieee.exponent -= LDBL_MANT_DIG;
117 adjust = 1;
118 }
119 else if (w.ieee.exponent >= 0x7fff - LDBL_MANT_DIG)
120 {
121 /* Similarly.
122 If z exponent is very large and x and y exponents are
123 very small, adjust them up to avoid spurious underflows,
124 rather than down. */
125 if (u.ieee.exponent + v.ieee.exponent
126 <= IEEE854_LONG_DOUBLE_BIAS + 2 * LDBL_MANT_DIG)
127 {
128 if (u.ieee.exponent > v.ieee.exponent)
129 u.ieee.exponent += 2 * LDBL_MANT_DIG + 2;
130 else
131 v.ieee.exponent += 2 * LDBL_MANT_DIG + 2;
132 }
133 else if (u.ieee.exponent > v.ieee.exponent)
134 {
135 if (u.ieee.exponent > LDBL_MANT_DIG)
136 u.ieee.exponent -= LDBL_MANT_DIG;
137 }
138 else if (v.ieee.exponent > LDBL_MANT_DIG)
139 v.ieee.exponent -= LDBL_MANT_DIG;
140 w.ieee.exponent -= LDBL_MANT_DIG;
141 adjust = 1;
142 }
143 else if (u.ieee.exponent >= 0x7fff - LDBL_MANT_DIG)
144 {
145 u.ieee.exponent -= LDBL_MANT_DIG;
146 if (v.ieee.exponent)
147 v.ieee.exponent += LDBL_MANT_DIG;
148 else
149 v.d *= L(0x1p113);
150 }
151 else if (v.ieee.exponent >= 0x7fff - LDBL_MANT_DIG)
152 {
153 v.ieee.exponent -= LDBL_MANT_DIG;
154 if (u.ieee.exponent)
155 u.ieee.exponent += LDBL_MANT_DIG;
156 else
157 u.d *= L(0x1p113);
158 }
159 else /* if (u.ieee.exponent + v.ieee.exponent
160 <= IEEE854_LONG_DOUBLE_BIAS + LDBL_MANT_DIG) */
161 {
162 if (u.ieee.exponent > v.ieee.exponent)
163 u.ieee.exponent += 2 * LDBL_MANT_DIG + 2;
164 else
165 v.ieee.exponent += 2 * LDBL_MANT_DIG + 2;
166 if (w.ieee.exponent <= 4 * LDBL_MANT_DIG + 6)
167 {
168 if (w.ieee.exponent)
169 w.ieee.exponent += 2 * LDBL_MANT_DIG + 2;
170 else
171 w.d *= L(0x1p228);
172 adjust = -1;
173 }
174 /* Otherwise x * y should just affect inexact
175 and nothing else. */
176 }
177 x = u.d;
178 y = v.d;
179 z = w.d;
180 }
181
182 /* Ensure correct sign of exact 0 + 0. */
183 if (__glibc_unlikely ((x == 0 || y == 0) && z == 0))
184 {
185 x = math_opt_barrier (x);
186 return x * y + z;
187 }
188
189 fenv_t env;
190 feholdexcept (&env);
191 fesetround (FE_TONEAREST);
192
193 /* Multiplication m1 + m2 = x * y using Dekker's algorithm. */
194#define C ((1LL << (LDBL_MANT_DIG + 1) / 2) + 1)
195 _Float128 x1 = x * C;
196 _Float128 y1 = y * C;
197 _Float128 m1 = x * y;
198 x1 = (x - x1) + x1;
199 y1 = (y - y1) + y1;
200 _Float128 x2 = x - x1;
201 _Float128 y2 = y - y1;
202 _Float128 m2 = (((x1 * y1 - m1) + x1 * y2) + x2 * y1) + x2 * y2;
203
204 /* Addition a1 + a2 = z + m1 using Knuth's algorithm. */
205 _Float128 a1 = z + m1;
206 _Float128 t1 = a1 - z;
207 _Float128 t2 = a1 - t1;
208 t1 = m1 - t1;
209 t2 = z - t2;
210 _Float128 a2 = t1 + t2;
211 /* Ensure the arithmetic is not scheduled after feclearexcept call. */
212 math_force_eval (m2);
213 math_force_eval (a2);
214 feclearexcept (FE_INEXACT);
215
216 /* If the result is an exact zero, ensure it has the correct sign. */
217 if (a1 == 0 && m2 == 0)
218 {
219 feupdateenv (&env);
220 /* Ensure that round-to-nearest value of z + m1 is not reused. */
221 z = math_opt_barrier (z);
222 return z + m1;
223 }
224
225 fesetround (FE_TOWARDZERO);
226 /* Perform m2 + a2 addition with round to odd. */
227 u.d = a2 + m2;
228
229 if (__glibc_likely (adjust == 0))
230 {
231 if ((u.ieee.mantissa3 & 1) == 0 && u.ieee.exponent != 0x7fff)
232 u.ieee.mantissa3 |= fetestexcept (FE_INEXACT) != 0;
233 feupdateenv (&env);
234 /* Result is a1 + u.d. */
235 return a1 + u.d;
236 }
237 else if (__glibc_likely (adjust > 0))
238 {
239 if ((u.ieee.mantissa3 & 1) == 0 && u.ieee.exponent != 0x7fff)
240 u.ieee.mantissa3 |= fetestexcept (FE_INEXACT) != 0;
241 feupdateenv (&env);
242 /* Result is a1 + u.d, scaled up. */
243 return (a1 + u.d) * L(0x1p113);
244 }
245 else
246 {
247 if ((u.ieee.mantissa3 & 1) == 0)
248 u.ieee.mantissa3 |= fetestexcept (FE_INEXACT) != 0;
249 v.d = a1 + u.d;
250 /* Ensure the addition is not scheduled after fetestexcept call. */
251 math_force_eval (v.d);
252 int j = fetestexcept (FE_INEXACT) != 0;
253 feupdateenv (&env);
254 /* Ensure the following computations are performed in default rounding
255 mode instead of just reusing the round to zero computation. */
256 asm volatile ("" : "=m" (u) : "m" (u));
257 /* If a1 + u.d is exact, the only rounding happens during
258 scaling down. */
259 if (j == 0)
260 return v.d * L(0x1p-228);
261 /* If result rounded to zero is not subnormal, no double
262 rounding will occur. */
263 if (v.ieee.exponent > 228)
264 return (a1 + u.d) * L(0x1p-228);
265 /* If v.d * 0x1p-228L with round to zero is a subnormal above
266 or equal to LDBL_MIN / 2, then v.d * 0x1p-228L shifts mantissa
267 down just by 1 bit, which means v.ieee.mantissa3 |= j would
268 change the round bit, not sticky or guard bit.
269 v.d * 0x1p-228L never normalizes by shifting up,
270 so round bit plus sticky bit should be already enough
271 for proper rounding. */
272 if (v.ieee.exponent == 228)
273 {
274 /* If the exponent would be in the normal range when
275 rounding to normal precision with unbounded exponent
276 range, the exact result is known and spurious underflows
277 must be avoided on systems detecting tininess after
278 rounding. */
279 if (TININESS_AFTER_ROUNDING)
280 {
281 w.d = a1 + u.d;
282 if (w.ieee.exponent == 229)
283 return w.d * L(0x1p-228);
284 }
285 /* v.ieee.mantissa3 & 2 is LSB bit of the result before rounding,
286 v.ieee.mantissa3 & 1 is the round bit and j is our sticky
287 bit. */
288 w.d = 0;
289 w.ieee.mantissa3 = ((v.ieee.mantissa3 & 3) << 1) | j;
290 w.ieee.negative = v.ieee.negative;
291 v.ieee.mantissa3 &= ~3U;
292 v.d *= L(0x1p-228);
293 w.d *= L(0x1p-2);
294 return v.d + w.d;
295 }
296 v.ieee.mantissa3 |= j;
297 return v.d * L(0x1p-228);
298 }
299}
300libm_alias_ldouble (__fma, fma)
301