1 /* Compute x * y + z as ternary operation.
2 Copyright (C) 2010-2012 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
fmaq(__float128 x,__float128 y,__float128 z)37 fmaq (__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 volatile __float128 force_underflow = x * y;
101 (void) 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;
165 else
166 v.ieee.exponent += 2 * FLT128_MANT_DIG;
167 if (w.ieee.exponent <= 4 * FLT128_MANT_DIG + 4)
168 {
169 if (w.ieee.exponent)
170 w.ieee.exponent += 2 * FLT128_MANT_DIG;
171 else
172 w.value *= 0x1p226Q;
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 return x * y + z;
186
187 #ifdef USE_FENV_H
188 fenv_t env;
189 feholdexcept (&env);
190 fesetround (FE_TONEAREST);
191 #endif
192
193 /* Multiplication m1 + m2 = x * y using Dekker's algorithm. */
194 #define C ((1LL << (FLT128_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 #ifdef USE_FENV_H
212 feclearexcept (FE_INEXACT);
213 #endif
214
215 /* If the result is an exact zero, ensure it has the correct
216 sign. */
217 if (a1 == 0 && m2 == 0)
218 {
219 #ifdef USE_FENV_H
220 feupdateenv (&env);
221 #endif
222 /* Ensure that round-to-nearest value of z + m1 is not
223 reused. */
224 asm volatile ("" : "=m" (z) : "m" (z));
225 return z + m1;
226 }
227
228
229 #ifdef USE_FENV_H
230 fesetround (FE_TOWARDZERO);
231 #endif
232 /* Perform m2 + a2 addition with round to odd. */
233 u.value = a2 + m2;
234
235 if (__builtin_expect (adjust == 0, 1))
236 {
237 #ifdef USE_FENV_H
238 if ((u.ieee.mant_low & 1) == 0 && u.ieee.exponent != 0x7fff)
239 u.ieee.mant_low |= fetestexcept (FE_INEXACT) != 0;
240 feupdateenv (&env);
241 #endif
242 /* Result is a1 + u.value. */
243 return a1 + u.value;
244 }
245 else if (__builtin_expect (adjust > 0, 1))
246 {
247 #ifdef USE_FENV_H
248 if ((u.ieee.mant_low & 1) == 0 && u.ieee.exponent != 0x7fff)
249 u.ieee.mant_low |= fetestexcept (FE_INEXACT) != 0;
250 feupdateenv (&env);
251 #endif
252 /* Result is a1 + u.value, scaled up. */
253 return (a1 + u.value) * 0x1p113Q;
254 }
255 else
256 {
257 #ifdef USE_FENV_H
258 if ((u.ieee.mant_low & 1) == 0)
259 u.ieee.mant_low |= fetestexcept (FE_INEXACT) != 0;
260 #endif
261 v.value = a1 + u.value;
262 /* Ensure the addition is not scheduled after fetestexcept call. */
263 asm volatile ("" : : "m" (v.value));
264 #ifdef USE_FENV_H
265 int j = fetestexcept (FE_INEXACT) != 0;
266 feupdateenv (&env);
267 #else
268 int j = 0;
269 #endif
270 /* Ensure the following computations are performed in default rounding
271 mode instead of just reusing the round to zero computation. */
272 asm volatile ("" : "=m" (u) : "m" (u));
273 /* If a1 + u.value is exact, the only rounding happens during
274 scaling down. */
275 if (j == 0)
276 return v.value * 0x1p-226Q;
277 /* If result rounded to zero is not subnormal, no double
278 rounding will occur. */
279 if (v.ieee.exponent > 226)
280 return (a1 + u.value) * 0x1p-226Q;
281 /* If v.value * 0x1p-226Q with round to zero is a subnormal above
282 or equal to FLT128_MIN / 2, then v.value * 0x1p-226Q shifts mantissa
283 down just by 1 bit, which means v.ieee.mant_low |= j would
284 change the round bit, not sticky or guard bit.
285 v.value * 0x1p-226Q never normalizes by shifting up,
286 so round bit plus sticky bit should be already enough
287 for proper rounding. */
288 if (v.ieee.exponent == 226)
289 {
290 /* If the exponent would be in the normal range when
291 rounding to normal precision with unbounded exponent
292 range, the exact result is known and spurious underflows
293 must be avoided on systems detecting tininess after
294 rounding. */
295 if (TININESS_AFTER_ROUNDING)
296 {
297 w.value = a1 + u.value;
298 if (w.ieee.exponent == 227)
299 return w.value * 0x1p-226Q;
300 }
301 /* v.ieee.mant_low & 2 is LSB bit of the result before rounding,
302 v.ieee.mant_low & 1 is the round bit and j is our sticky
303 bit. */
304 w.value = 0.0Q;
305 w.ieee.mant_low = ((v.ieee.mant_low & 3) << 1) | j;
306 w.ieee.negative = v.ieee.negative;
307 v.ieee.mant_low &= ~3U;
308 v.value *= 0x1p-226Q;
309 w.value *= 0x1p-2Q;
310 return v.value + w.value;
311 }
312 v.ieee.mant_low |= j;
313 return v.value * 0x1p-226Q;
314 }
315 }
316