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 "quadmath-imp.h"
21#include <math.h>
22#include <float.h>
23#ifdef HAVE_FENV_H
24# include <fenv.h>
25# if defined HAVE_FEHOLDEXCEPT && defined HAVE_FESETROUND \
26 && defined HAVE_FEUPDATEENV && defined HAVE_FETESTEXCEPT \
27 && defined FE_TOWARDZERO && defined FE_INEXACT
28# define USE_FENV_H
29# endif
30#endif
31
32/* This implementation uses rounding to odd to avoid problems with
33 double rounding. See a paper by Boldo and Melquiond:
34 http://www.lri.fr/~melquion/doc/08-tc.pdf */
35
36__float128
37fmaq (__float128 x, __float128 y, __float128 z)
38{
39 ieee854_float128 u, v, w;
40 int adjust = 0;
41 u.value = x;
42 v.value = y;
43 w.value = z;
44 if (__builtin_expect (u.ieee.exponent + v.ieee.exponent
45 >= 0x7fff + IEEE854_FLOAT128_BIAS
46 - FLT128_MANT_DIG, 0)
47 || __builtin_expect (u.ieee.exponent >= 0x7fff - FLT128_MANT_DIG, 0)
48 || __builtin_expect (v.ieee.exponent >= 0x7fff - FLT128_MANT_DIG, 0)
49 || __builtin_expect (w.ieee.exponent >= 0x7fff - FLT128_MANT_DIG, 0)
50 || __builtin_expect (u.ieee.exponent + v.ieee.exponent
51 <= IEEE854_FLOAT128_BIAS + FLT128_MANT_DIG, 0))
52 {
53 /* If z is Inf, but x and y are finite, the result should be
54 z rather than NaN. */
55 if (w.ieee.exponent == 0x7fff
56 && u.ieee.exponent != 0x7fff
57 && v.ieee.exponent != 0x7fff)
58 return (z + x) + y;
59 /* If z is zero and x are y are nonzero, compute the result
60 as x * y to avoid the wrong sign of a zero result if x * y
61 underflows to 0. */
62 if (z == 0 && x != 0 && y != 0)
63 return x * y;
64 /* If x or y or z is Inf/NaN, or if x * y is zero, compute as
65 x * y + z. */
66 if (u.ieee.exponent == 0x7fff
67 || v.ieee.exponent == 0x7fff
68 || w.ieee.exponent == 0x7fff
69 || x == 0
70 || y == 0)
71 return x * y + z;
72 /* If fma will certainly overflow, compute as x * y. */
73 if (u.ieee.exponent + v.ieee.exponent
74 > 0x7fff + IEEE854_FLOAT128_BIAS)
75 return x * y;
76 /* If x * y is less than 1/4 of FLT128_DENORM_MIN, neither the
77 result nor whether there is underflow depends on its exact
78 value, only on its sign. */
79 if (u.ieee.exponent + v.ieee.exponent
80 < IEEE854_FLOAT128_BIAS - FLT128_MANT_DIG - 2)
81 {
82 int neg = u.ieee.negative ^ v.ieee.negative;
83 __float128 tiny = neg ? -0x1p-16494Q : 0x1p-16494Q;
84 if (w.ieee.exponent >= 3)
85 return tiny + z;
86 /* Scaling up, adding TINY and scaling down produces the
87 correct result, because in round-to-nearest mode adding
88 TINY has no effect and in other modes double rounding is
89 harmless. But it may not produce required underflow
90 exceptions. */
91 v.value = z * 0x1p114Q + tiny;
92 if (TININESS_AFTER_ROUNDING
93 ? v.ieee.exponent < 115
94 : (w.ieee.exponent == 0
95 || (w.ieee.exponent == 1
96 && w.ieee.negative != neg
97 && w.ieee.mant_low == 0
98 && w.ieee.mant_high == 0)))
99 {
100 __float128 force_underflow = x * y;
101 math_force_eval (force_underflow);
102 }
103 return v.value * 0x1p-114Q;
104 }
105 if (u.ieee.exponent + v.ieee.exponent
106 >= 0x7fff + IEEE854_FLOAT128_BIAS - FLT128_MANT_DIG)
107 {
108 /* Compute 1p-113 times smaller result and multiply
109 at the end. */
110 if (u.ieee.exponent > v.ieee.exponent)
111 u.ieee.exponent -= FLT128_MANT_DIG;
112 else
113 v.ieee.exponent -= FLT128_MANT_DIG;
114 /* If x + y exponent is very large and z exponent is very small,
115 it doesn't matter if we don't adjust it. */
116 if (w.ieee.exponent > FLT128_MANT_DIG)
117 w.ieee.exponent -= FLT128_MANT_DIG;
118 adjust = 1;
119 }
120 else if (w.ieee.exponent >= 0x7fff - FLT128_MANT_DIG)
121 {
122 /* Similarly.
123 If z exponent is very large and x and y exponents are
124 very small, adjust them up to avoid spurious underflows,
125 rather than down. */
126 if (u.ieee.exponent + v.ieee.exponent
127 <= IEEE854_FLOAT128_BIAS + FLT128_MANT_DIG)
128 {
129 if (u.ieee.exponent > v.ieee.exponent)
130 u.ieee.exponent += 2 * FLT128_MANT_DIG + 2;
131 else
132 v.ieee.exponent += 2 * FLT128_MANT_DIG + 2;
133 }
134 else if (u.ieee.exponent > v.ieee.exponent)
135 {
136 if (u.ieee.exponent > FLT128_MANT_DIG)
137 u.ieee.exponent -= FLT128_MANT_DIG;
138 }
139 else if (v.ieee.exponent > FLT128_MANT_DIG)
140 v.ieee.exponent -= FLT128_MANT_DIG;
141 w.ieee.exponent -= FLT128_MANT_DIG;
142 adjust = 1;
143 }
144 else if (u.ieee.exponent >= 0x7fff - FLT128_MANT_DIG)
145 {
146 u.ieee.exponent -= FLT128_MANT_DIG;
147 if (v.ieee.exponent)
148 v.ieee.exponent += FLT128_MANT_DIG;
149 else
150 v.value *= 0x1p113Q;
151 }
152 else if (v.ieee.exponent >= 0x7fff - FLT128_MANT_DIG)
153 {
154 v.ieee.exponent -= FLT128_MANT_DIG;
155 if (u.ieee.exponent)
156 u.ieee.exponent += FLT128_MANT_DIG;
157 else
158 u.value *= 0x1p113Q;
159 }
160 else /* if (u.ieee.exponent + v.ieee.exponent
161 <= IEEE854_FLOAT128_BIAS + FLT128_MANT_DIG) */
162 {
163 if (u.ieee.exponent > v.ieee.exponent)
164 u.ieee.exponent += 2 * FLT128_MANT_DIG + 2;
165 else
166 v.ieee.exponent += 2 * FLT128_MANT_DIG + 2;
167 if (w.ieee.exponent <= 4 * FLT128_MANT_DIG + 6)
168 {
169 if (w.ieee.exponent)
170 w.ieee.exponent += 2 * FLT128_MANT_DIG + 2;
171 else
172 w.value *= 0x1p228Q;
173 adjust = -1;
174 }
175 /* Otherwise x * y should just affect inexact
176 and nothing else. */
177 }
178 x = u.value;
179 y = v.value;
180 z = w.value;
181 }
182
183 /* Ensure correct sign of exact 0 + 0. */
184 if (__builtin_expect ((x == 0 || y == 0) && z == 0, 0))
185 {
186 x = math_opt_barrier (x);
187 return x * y + z;
188 }
189
190#ifdef USE_FENV_H
191 fenv_t env;
192 feholdexcept (&env);
193 fesetround (FE_TONEAREST);
194#endif
195
196 /* Multiplication m1 + m2 = x * y using Dekker's algorithm. */
197#define C ((1LL << (FLT128_MANT_DIG + 1) / 2) + 1)
198 __float128 x1 = x * C;
199 __float128 y1 = y * C;
200 __float128 m1 = x * y;
201 x1 = (x - x1) + x1;
202 y1 = (y - y1) + y1;
203 __float128 x2 = x - x1;
204 __float128 y2 = y - y1;
205 __float128 m2 = (((x1 * y1 - m1) + x1 * y2) + x2 * y1) + x2 * y2;
206
207 /* Addition a1 + a2 = z + m1 using Knuth's algorithm. */
208 __float128 a1 = z + m1;
209 __float128 t1 = a1 - z;
210 __float128 t2 = a1 - t1;
211 t1 = m1 - t1;
212 t2 = z - t2;
213 __float128 a2 = t1 + t2;
214 /* Ensure the arithmetic is not scheduled after feclearexcept call. */
215 math_force_eval (m2);
216 math_force_eval (a2);
217#ifdef USE_FENV_H
218 feclearexcept (FE_INEXACT);
219#endif
220
221 /* If the result is an exact zero, ensure it has the correct sign. */
222 if (a1 == 0 && m2 == 0)
223 {
224#ifdef USE_FENV_H
225 feupdateenv (&env);
226#endif
227 /* Ensure that round-to-nearest value of z + m1 is not reused. */
228 z = math_opt_barrier (z);
229 return z + m1;
230 }
231
232#ifdef USE_FENV_H
233 fesetround (FE_TOWARDZERO);
234#endif
235 /* Perform m2 + a2 addition with round to odd. */
236 u.value = a2 + m2;
237
238 if (__builtin_expect (adjust == 0, 1))
239 {
240#ifdef USE_FENV_H
241 if ((u.ieee.mant_low & 1) == 0 && u.ieee.exponent != 0x7fff)
242 u.ieee.mant_low |= fetestexcept (FE_INEXACT) != 0;
243 feupdateenv (&env);
244#endif
245 /* Result is a1 + u.value. */
246 return a1 + u.value;
247 }
248 else if (__builtin_expect (adjust > 0, 1))
249 {
250#ifdef USE_FENV_H
251 if ((u.ieee.mant_low & 1) == 0 && u.ieee.exponent != 0x7fff)
252 u.ieee.mant_low |= fetestexcept (FE_INEXACT) != 0;
253 feupdateenv (&env);
254#endif
255 /* Result is a1 + u.value, scaled up. */
256 return (a1 + u.value) * 0x1p113Q;
257 }
258 else
259 {
260#ifdef USE_FENV_H
261 if ((u.ieee.mant_low & 1) == 0)
262 u.ieee.mant_low |= fetestexcept (FE_INEXACT) != 0;
263#endif
264 v.value = a1 + u.value;
265 /* Ensure the addition is not scheduled after fetestexcept call. */
266 asm volatile ("" : : "m" (v.value));
267#ifdef USE_FENV_H
268 int j = fetestexcept (FE_INEXACT) != 0;
269 feupdateenv (&env);
270#else
271 int j = 0;
272#endif
273 /* Ensure the following computations are performed in default rounding
274 mode instead of just reusing the round to zero computation. */
275 asm volatile ("" : "=m" (u) : "m" (u));
276 /* If a1 + u.value is exact, the only rounding happens during
277 scaling down. */
278 if (j == 0)
279 return v.value * 0x1p-228Q;
280 /* If result rounded to zero is not subnormal, no double
281 rounding will occur. */
282 if (v.ieee.exponent > 228)
283 return (a1 + u.value) * 0x1p-228Q;
284 /* If v.value * 0x1p-228Q with round to zero is a subnormal above
285 or equal to FLT128_MIN / 2, then v.value * 0x1p-228Q shifts mantissa
286 down just by 1 bit, which means v.ieee.mant_low |= j would
287 change the round bit, not sticky or guard bit.
288 v.value * 0x1p-228Q never normalizes by shifting up,
289 so round bit plus sticky bit should be already enough
290 for proper rounding. */
291 if (v.ieee.exponent == 228)
292 {
293 /* If the exponent would be in the normal range when
294 rounding to normal precision with unbounded exponent
295 range, the exact result is known and spurious underflows
296 must be avoided on systems detecting tininess after
297 rounding. */
298 if (TININESS_AFTER_ROUNDING)
299 {
300 w.value = a1 + u.value;
301 if (w.ieee.exponent == 229)
302 return w.value * 0x1p-228Q;
303 }
304 /* v.ieee.mant_low & 2 is LSB bit of the result before rounding,
305 v.ieee.mant_low & 1 is the round bit and j is our sticky
306 bit. */
307 w.value = 0.0Q;
308 w.ieee.mant_low = ((v.ieee.mant_low & 3) << 1) | j;
309 w.ieee.negative = v.ieee.negative;
310 v.ieee.mant_low &= ~3U;
311 v.value *= 0x1p-228Q;
312 w.value *= 0x1p-2Q;
313 return v.value + w.value;
314 }
315 v.ieee.mant_low |= j;
316 return v.value * 0x1p-228Q;
317 }
318}
319