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_private.h>
25#include <libm-alias-ldouble.h>
26#include <tininess.h>
27
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 */
31
32_Float128
33__fmal (_Float128 x, _Float128 y, _Float128 z)
34{
35 union ieee854_long_double u, v, w;
36 int adjust = 0;
37 u.d = x;
38 v.d = y;
39 w.d = z;
40 if (__builtin_expect (u.ieee.exponent + v.ieee.exponent
41 >= 0x7fff + IEEE854_LONG_DOUBLE_BIAS
42 - LDBL_MANT_DIG, 0)
43 || __builtin_expect (u.ieee.exponent >= 0x7fff - LDBL_MANT_DIG, 0)
44 || __builtin_expect (v.ieee.exponent >= 0x7fff - LDBL_MANT_DIG, 0)
45 || __builtin_expect (w.ieee.exponent >= 0x7fff - LDBL_MANT_DIG, 0)
46 || __builtin_expect (u.ieee.exponent + v.ieee.exponent
47 <= IEEE854_LONG_DOUBLE_BIAS + LDBL_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 == 0x7fff
52 && u.ieee.exponent != 0x7fff
53 && v.ieee.exponent != 0x7fff)
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 == 0x7fff
63 || v.ieee.exponent == 0x7fff
64 || w.ieee.exponent == 0x7fff
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
70 > 0x7fff + IEEE854_LONG_DOUBLE_BIAS)
71 return x * y;
72 /* If x * y is less than 1/4 of LDBL_TRUE_MIN, neither the
73 result nor whether there is underflow depends on its exact
74 value, only on its sign. */
75 if (u.ieee.exponent + v.ieee.exponent
76 < IEEE854_LONG_DOUBLE_BIAS - LDBL_MANT_DIG - 2)
77 {
78 int neg = u.ieee.negative ^ v.ieee.negative;
79 _Float128 tiny = neg ? L(-0x1p-16494) : L(0x1p-16494);
80 if (w.ieee.exponent >= 3)
81 return tiny + z;
82 /* Scaling up, adding TINY and scaling down produces the
83 correct result, because in round-to-nearest mode adding
84 TINY has no effect and in other modes double rounding is
85 harmless. But it may not produce required underflow
86 exceptions. */
87 v.d = z * L(0x1p114) + tiny;
88 if (TININESS_AFTER_ROUNDING
89 ? v.ieee.exponent < 115
90 : (w.ieee.exponent == 0
91 || (w.ieee.exponent == 1
92 && w.ieee.negative != neg
93 && w.ieee.mantissa3 == 0
94 && w.ieee.mantissa2 == 0
95 && w.ieee.mantissa1 == 0
96 && w.ieee.mantissa0 == 0)))
97 {
98 _Float128 force_underflow = x * y;
99 math_force_eval (force_underflow);
100 }
101 return v.d * L(0x1p-114);
102 }
103 if (u.ieee.exponent + v.ieee.exponent
104 >= 0x7fff + IEEE854_LONG_DOUBLE_BIAS - LDBL_MANT_DIG)
105 {
106 /* Compute 1p-113 times smaller result and multiply
107 at the end. */
108 if (u.ieee.exponent > v.ieee.exponent)
109 u.ieee.exponent -= LDBL_MANT_DIG;
110 else
111 v.ieee.exponent -= LDBL_MANT_DIG;
112 /* If x + y exponent is very large and z exponent is very small,
113 it doesn't matter if we don't adjust it. */
114 if (w.ieee.exponent > LDBL_MANT_DIG)
115 w.ieee.exponent -= LDBL_MANT_DIG;
116 adjust = 1;
117 }
118 else if (w.ieee.exponent >= 0x7fff - LDBL_MANT_DIG)
119 {
120 /* Similarly.
121 If z exponent is very large and x and y exponents are
122 very small, adjust them up to avoid spurious underflows,
123 rather than down. */
124 if (u.ieee.exponent + v.ieee.exponent
125 <= IEEE854_LONG_DOUBLE_BIAS + 2 * LDBL_MANT_DIG)
126 {
127 if (u.ieee.exponent > v.ieee.exponent)
128 u.ieee.exponent += 2 * LDBL_MANT_DIG + 2;
129 else
130 v.ieee.exponent += 2 * LDBL_MANT_DIG + 2;
131 }
132 else if (u.ieee.exponent > v.ieee.exponent)
133 {
134 if (u.ieee.exponent > LDBL_MANT_DIG)
135 u.ieee.exponent -= LDBL_MANT_DIG;
136 }
137 else if (v.ieee.exponent > LDBL_MANT_DIG)
138 v.ieee.exponent -= LDBL_MANT_DIG;
139 w.ieee.exponent -= LDBL_MANT_DIG;
140 adjust = 1;
141 }
142 else if (u.ieee.exponent >= 0x7fff - LDBL_MANT_DIG)
143 {
144 u.ieee.exponent -= LDBL_MANT_DIG;
145 if (v.ieee.exponent)
146 v.ieee.exponent += LDBL_MANT_DIG;
147 else
148 v.d *= L(0x1p113);
149 }
150 else if (v.ieee.exponent >= 0x7fff - LDBL_MANT_DIG)
151 {
152 v.ieee.exponent -= LDBL_MANT_DIG;
153 if (u.ieee.exponent)
154 u.ieee.exponent += LDBL_MANT_DIG;
155 else
156 u.d *= L(0x1p113);
157 }
158 else /* if (u.ieee.exponent + v.ieee.exponent
159 <= IEEE854_LONG_DOUBLE_BIAS + LDBL_MANT_DIG) */
160 {
161 if (u.ieee.exponent > v.ieee.exponent)
162 u.ieee.exponent += 2 * LDBL_MANT_DIG + 2;
163 else
164 v.ieee.exponent += 2 * LDBL_MANT_DIG + 2;
165 if (w.ieee.exponent <= 4 * LDBL_MANT_DIG + 6)
166 {
167 if (w.ieee.exponent)
168 w.ieee.exponent += 2 * LDBL_MANT_DIG + 2;
169 else
170 w.d *= L(0x1p228);
171 adjust = -1;
172 }
173 /* Otherwise x * y should just affect inexact
174 and nothing else. */
175 }
176 x = u.d;
177 y = v.d;
178 z = w.d;
179 }
180
181 /* Ensure correct sign of exact 0 + 0. */
182 if (__glibc_unlikely ((x == 0 || y == 0) && z == 0))
183 {
184 x = math_opt_barrier (x);
185 return x * y + z;
186 }
187
188 fenv_t env;
189 feholdexcept (&env);
190 fesetround (FE_TONEAREST);
191
192 /* Multiplication m1 + m2 = x * y using Dekker's algorithm. */
193#define C ((1LL << (LDBL_MANT_DIG + 1) / 2) + 1)
194 _Float128 x1 = x * C;
195 _Float128 y1 = y * C;
196 _Float128 m1 = x * y;
197 x1 = (x - x1) + x1;
198 y1 = (y - y1) + y1;
199 _Float128 x2 = x - x1;
200 _Float128 y2 = y - y1;
201 _Float128 m2 = (((x1 * y1 - m1) + x1 * y2) + x2 * y1) + x2 * y2;
202
203 /* Addition a1 + a2 = z + m1 using Knuth's algorithm. */
204 _Float128 a1 = z + m1;
205 _Float128 t1 = a1 - z;
206 _Float128 t2 = a1 - t1;
207 t1 = m1 - t1;
208 t2 = z - t2;
209 _Float128 a2 = t1 + t2;
210 /* Ensure the arithmetic is not scheduled after feclearexcept call. */
211 math_force_eval (m2);
212 math_force_eval (a2);
213 feclearexcept (FE_INEXACT);
214
215 /* If the result is an exact zero, ensure it has the correct sign. */
216 if (a1 == 0 && m2 == 0)
217 {
218 feupdateenv (&env);
219 /* Ensure that round-to-nearest value of z + m1 is not reused. */
220 z = math_opt_barrier (z);
221 return z + m1;
222 }
223
224 fesetround (FE_TOWARDZERO);
225 /* Perform m2 + a2 addition with round to odd. */
226 u.d = a2 + m2;
227
228 if (__glibc_likely (adjust == 0))
229 {
230 if ((u.ieee.mantissa3 & 1) == 0 && u.ieee.exponent != 0x7fff)
231 u.ieee.mantissa3 |= fetestexcept (FE_INEXACT) != 0;
232 feupdateenv (&env);
233 /* Result is a1 + u.d. */
234 return a1 + u.d;
235 }
236 else if (__glibc_likely (adjust > 0))
237 {
238 if ((u.ieee.mantissa3 & 1) == 0 && u.ieee.exponent != 0x7fff)
239 u.ieee.mantissa3 |= fetestexcept (FE_INEXACT) != 0;
240 feupdateenv (&env);
241 /* Result is a1 + u.d, scaled up. */
242 return (a1 + u.d) * L(0x1p113);
243 }
244 else
245 {
246 if ((u.ieee.mantissa3 & 1) == 0)
247 u.ieee.mantissa3 |= fetestexcept (FE_INEXACT) != 0;
248 v.d = a1 + u.d;
249 /* Ensure the addition is not scheduled after fetestexcept call. */
250 math_force_eval (v.d);
251 int j = fetestexcept (FE_INEXACT) != 0;
252 feupdateenv (&env);
253 /* Ensure the following computations are performed in default rounding
254 mode instead of just reusing the round to zero computation. */
255 asm volatile ("" : "=m" (u) : "m" (u));
256 /* If a1 + u.d is exact, the only rounding happens during
257 scaling down. */
258 if (j == 0)
259 return v.d * L(0x1p-228);
260 /* If result rounded to zero is not subnormal, no double
261 rounding will occur. */
262 if (v.ieee.exponent > 228)
263 return (a1 + u.d) * L(0x1p-228);
264 /* If v.d * 0x1p-228L with round to zero is a subnormal above
265 or equal to LDBL_MIN / 2, then v.d * 0x1p-228L shifts mantissa
266 down just by 1 bit, which means v.ieee.mantissa3 |= j would
267 change the round bit, not sticky or guard bit.
268 v.d * 0x1p-228L never normalizes by shifting up,
269 so round bit plus sticky bit should be already enough
270 for proper rounding. */
271 if (v.ieee.exponent == 228)
272 {
273 /* If the exponent would be in the normal range when
274 rounding to normal precision with unbounded exponent
275 range, the exact result is known and spurious underflows
276 must be avoided on systems detecting tininess after
277 rounding. */
278 if (TININESS_AFTER_ROUNDING)
279 {
280 w.d = a1 + u.d;
281 if (w.ieee.exponent == 229)
282 return w.d * L(0x1p-228);
283 }
284 /* v.ieee.mantissa3 & 2 is LSB bit of the result before rounding,
285 v.ieee.mantissa3 & 1 is the round bit and j is our sticky
286 bit. */
287 w.d = 0;
288 w.ieee.mantissa3 = ((v.ieee.mantissa3 & 3) << 1) | j;
289 w.ieee.negative = v.ieee.negative;
290 v.ieee.mantissa3 &= ~3U;
291 v.d *= L(0x1p-228);
292 w.d *= L(0x1p-2);
293 return v.d + w.d;
294 }
295 v.ieee.mantissa3 |= j;
296 return v.d * L(0x1p-228);
297 }
298}
299libm_alias_ldouble (__fma, fma)
300