1/* Compute x * y + z as ternary operation.
2 Copyright (C) 2010-2017 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 <tininess.h>
26
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 */
30
31_Float128
32__fmal (_Float128 x, _Float128 y, _Float128 z)
33{
34 union ieee854_long_double u, v, w;
35 int adjust = 0;
36 u.d = x;
37 v.d = y;
38 w.d = z;
39 if (__builtin_expect (u.ieee.exponent + v.ieee.exponent
40 >= 0x7fff + IEEE854_LONG_DOUBLE_BIAS
41 - LDBL_MANT_DIG, 0)
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))
47 {
48 /* If z is Inf, but x and y are finite, the result should be
49 z rather than NaN. */
50 if (w.ieee.exponent == 0x7fff
51 && u.ieee.exponent != 0x7fff
52 && v.ieee.exponent != 0x7fff)
53 return (z + x) + y;
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
56 underflows to 0. */
57 if (z == 0 && x != 0 && y != 0)
58 return x * y;
59 /* If x or y or z is Inf/NaN, or if x * y is zero, compute as
60 x * y + z. */
61 if (u.ieee.exponent == 0x7fff
62 || v.ieee.exponent == 0x7fff
63 || w.ieee.exponent == 0x7fff
64 || x == 0
65 || y == 0)
66 return x * y + z;
67 /* If fma will certainly overflow, compute as x * y. */
68 if (u.ieee.exponent + v.ieee.exponent
69 > 0x7fff + IEEE854_LONG_DOUBLE_BIAS)
70 return x * y;
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)
76 {
77 int neg = u.ieee.negative ^ v.ieee.negative;
78 _Float128 tiny = neg ? L(-0x1p-16494) : L(0x1p-16494);
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 * L(0x1p114) + 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)))
96 {
97 _Float128 force_underflow = x * y;
98 math_force_eval (force_underflow);
99 }
100 return v.d * L(0x1p-114);
101 }
102 if (u.ieee.exponent + v.ieee.exponent
103 >= 0x7fff + IEEE854_LONG_DOUBLE_BIAS - LDBL_MANT_DIG)
104 {
105 /* Compute 1p-113 times smaller result and multiply
106 at the end. */
107 if (u.ieee.exponent > v.ieee.exponent)
108 u.ieee.exponent -= LDBL_MANT_DIG;
109 else
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;
115 adjust = 1;
116 }
117 else if (w.ieee.exponent >= 0x7fff - LDBL_MANT_DIG)
118 {
119 /* Similarly.
120 If z exponent is very large and x and y exponents are
121 very small, adjust them up to avoid spurious underflows,
122 rather than down. */
123 if (u.ieee.exponent + v.ieee.exponent
124 <= IEEE854_LONG_DOUBLE_BIAS + 2 * LDBL_MANT_DIG)
125 {
126 if (u.ieee.exponent > v.ieee.exponent)
127 u.ieee.exponent += 2 * LDBL_MANT_DIG + 2;
128 else
129 v.ieee.exponent += 2 * LDBL_MANT_DIG + 2;
130 }
131 else if (u.ieee.exponent > v.ieee.exponent)
132 {
133 if (u.ieee.exponent > LDBL_MANT_DIG)
134 u.ieee.exponent -= LDBL_MANT_DIG;
135 }
136 else if (v.ieee.exponent > LDBL_MANT_DIG)
137 v.ieee.exponent -= LDBL_MANT_DIG;
138 w.ieee.exponent -= LDBL_MANT_DIG;
139 adjust = 1;
140 }
141 else if (u.ieee.exponent >= 0x7fff - LDBL_MANT_DIG)
142 {
143 u.ieee.exponent -= LDBL_MANT_DIG;
144 if (v.ieee.exponent)
145 v.ieee.exponent += LDBL_MANT_DIG;
146 else
147 v.d *= L(0x1p113);
148 }
149 else if (v.ieee.exponent >= 0x7fff - LDBL_MANT_DIG)
150 {
151 v.ieee.exponent -= LDBL_MANT_DIG;
152 if (u.ieee.exponent)
153 u.ieee.exponent += LDBL_MANT_DIG;
154 else
155 u.d *= L(0x1p113);
156 }
157 else /* if (u.ieee.exponent + v.ieee.exponent
158 <= IEEE854_LONG_DOUBLE_BIAS + LDBL_MANT_DIG) */
159 {
160 if (u.ieee.exponent > v.ieee.exponent)
161 u.ieee.exponent += 2 * LDBL_MANT_DIG + 2;
162 else
163 v.ieee.exponent += 2 * LDBL_MANT_DIG + 2;
164 if (w.ieee.exponent <= 4 * LDBL_MANT_DIG + 6)
165 {
166 if (w.ieee.exponent)
167 w.ieee.exponent += 2 * LDBL_MANT_DIG + 2;
168 else
169 w.d *= L(0x1p228);
170 adjust = -1;
171 }
172 /* Otherwise x * y should just affect inexact
173 and nothing else. */
174 }
175 x = u.d;
176 y = v.d;
177 z = w.d;
178 }
179
180 /* Ensure correct sign of exact 0 + 0. */
181 if (__glibc_unlikely ((x == 0 || y == 0) && z == 0))
182 {
183 x = math_opt_barrier (x);
184 return x * y + z;
185 }
186
187 fenv_t env;
188 feholdexcept (&env);
189 fesetround (FE_TONEAREST);
190
191 /* Multiplication m1 + m2 = x * y using Dekker's algorithm. */
192#define C ((1LL << (LDBL_MANT_DIG + 1) / 2) + 1)
193 _Float128 x1 = x * C;
194 _Float128 y1 = y * C;
195 _Float128 m1 = x * y;
196 x1 = (x - x1) + x1;
197 y1 = (y - y1) + y1;
198 _Float128 x2 = x - x1;
199 _Float128 y2 = y - y1;
200 _Float128 m2 = (((x1 * y1 - m1) + x1 * y2) + x2 * y1) + x2 * y2;
201
202 /* Addition a1 + a2 = z + m1 using Knuth's algorithm. */
203 _Float128 a1 = z + m1;
204 _Float128 t1 = a1 - z;
205 _Float128 t2 = a1 - t1;
206 t1 = m1 - t1;
207 t2 = z - t2;
208 _Float128 a2 = t1 + t2;
209 /* Ensure the arithmetic is not scheduled after feclearexcept call. */
210 math_force_eval (m2);
211 math_force_eval (a2);
212 feclearexcept (FE_INEXACT);
213
214 /* If the result is an exact zero, ensure it has the correct sign. */
215 if (a1 == 0 && m2 == 0)
216 {
217 feupdateenv (&env);
218 /* Ensure that round-to-nearest value of z + m1 is not reused. */
219 z = math_opt_barrier (z);
220 return z + m1;
221 }
222
223 fesetround (FE_TOWARDZERO);
224 /* Perform m2 + a2 addition with round to odd. */
225 u.d = a2 + m2;
226
227 if (__glibc_likely (adjust == 0))
228 {
229 if ((u.ieee.mantissa3 & 1) == 0 && u.ieee.exponent != 0x7fff)
230 u.ieee.mantissa3 |= fetestexcept (FE_INEXACT) != 0;
231 feupdateenv (&env);
232 /* Result is a1 + u.d. */
233 return a1 + u.d;
234 }
235 else if (__glibc_likely (adjust > 0))
236 {
237 if ((u.ieee.mantissa3 & 1) == 0 && u.ieee.exponent != 0x7fff)
238 u.ieee.mantissa3 |= fetestexcept (FE_INEXACT) != 0;
239 feupdateenv (&env);
240 /* Result is a1 + u.d, scaled up. */
241 return (a1 + u.d) * L(0x1p113);
242 }
243 else
244 {
245 if ((u.ieee.mantissa3 & 1) == 0)
246 u.ieee.mantissa3 |= fetestexcept (FE_INEXACT) != 0;
247 v.d = a1 + u.d;
248 /* Ensure the addition is not scheduled after fetestexcept call. */
249 math_force_eval (v.d);
250 int j = fetestexcept (FE_INEXACT) != 0;
251 feupdateenv (&env);
252 /* Ensure the following computations are performed in default rounding
253 mode instead of just reusing the round to zero computation. */
254 asm volatile ("" : "=m" (u) : "m" (u));
255 /* If a1 + u.d is exact, the only rounding happens during
256 scaling down. */
257 if (j == 0)
258 return v.d * L(0x1p-228);
259 /* If result rounded to zero is not subnormal, no double
260 rounding will occur. */
261 if (v.ieee.exponent > 228)
262 return (a1 + u.d) * L(0x1p-228);
263 /* If v.d * 0x1p-228L with round to zero is a subnormal above
264 or equal to LDBL_MIN / 2, then v.d * 0x1p-228L shifts mantissa
265 down just by 1 bit, which means v.ieee.mantissa3 |= j would
266 change the round bit, not sticky or guard bit.
267 v.d * 0x1p-228L never normalizes by shifting up,
268 so round bit plus sticky bit should be already enough
269 for proper rounding. */
270 if (v.ieee.exponent == 228)
271 {
272 /* If the exponent would be in the normal range when
273 rounding to normal precision with unbounded exponent
274 range, the exact result is known and spurious underflows
275 must be avoided on systems detecting tininess after
276 rounding. */
277 if (TININESS_AFTER_ROUNDING)
278 {
279 w.d = a1 + u.d;
280 if (w.ieee.exponent == 229)
281 return w.d * L(0x1p-228);
282 }
283 /* v.ieee.mantissa3 & 2 is LSB bit of the result before rounding,
284 v.ieee.mantissa3 & 1 is the round bit and j is our sticky
285 bit. */
286 w.d = 0;
287 w.ieee.mantissa3 = ((v.ieee.mantissa3 & 3) << 1) | j;
288 w.ieee.negative = v.ieee.negative;
289 v.ieee.mantissa3 &= ~3U;
290 v.d *= L(0x1p-228);
291 w.d *= L(0x1p-2);
292 return v.d + w.d;
293 }
294 v.ieee.mantissa3 |= j;
295 return v.d * L(0x1p-228);
296 }
297}
298weak_alias (__fmal, fmal)
299