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 "quadmath-imp.h"
21
22 /* This implementation uses rounding to odd to avoid problems with
23 double rounding. See a paper by Boldo and Melquiond:
24 http://www.lri.fr/~melquion/doc/08-tc.pdf */
25
26 __float128
27 fmaq (__float128 x, __float128 y, __float128 z)
28 {
29 ieee854_float128 u, v, w;
30 int adjust = 0;
31 u.value = x;
32 v.value = y;
33 w.value = z;
34 if (__builtin_expect (u.ieee.exponent + v.ieee.exponent
35 >= 0x7fff + IEEE854_FLOAT128_BIAS
36 - FLT128_MANT_DIG, 0)
37 || __builtin_expect (u.ieee.exponent >= 0x7fff - FLT128_MANT_DIG, 0)
38 || __builtin_expect (v.ieee.exponent >= 0x7fff - FLT128_MANT_DIG, 0)
39 || __builtin_expect (w.ieee.exponent >= 0x7fff - FLT128_MANT_DIG, 0)
40 || __builtin_expect (u.ieee.exponent + v.ieee.exponent
41 <= IEEE854_FLOAT128_BIAS + FLT128_MANT_DIG, 0))
42 {
43 /* If z is Inf, but x and y are finite, the result should be
44 z rather than NaN. */
45 if (w.ieee.exponent == 0x7fff
46 && u.ieee.exponent != 0x7fff
47 && v.ieee.exponent != 0x7fff)
48 return (z + x) + y;
49 /* If z is zero and x are y are nonzero, compute the result
50 as x * y to avoid the wrong sign of a zero result if x * y
51 underflows to 0. */
52 if (z == 0 && x != 0 && y != 0)
53 return x * y;
54 /* If x or y or z is Inf/NaN, or if x * y is zero, compute as
55 x * y + z. */
56 if (u.ieee.exponent == 0x7fff
57 || v.ieee.exponent == 0x7fff
58 || w.ieee.exponent == 0x7fff
59 || x == 0
60 || y == 0)
61 return x * y + z;
62 /* If fma will certainly overflow, compute as x * y. */
63 if (u.ieee.exponent + v.ieee.exponent
64 > 0x7fff + IEEE854_FLOAT128_BIAS)
65 return x * y;
66 /* If x * y is less than 1/4 of FLT128_TRUE_MIN, neither the
67 result nor whether there is underflow depends on its exact
68 value, only on its sign. */
69 if (u.ieee.exponent + v.ieee.exponent
70 < IEEE854_FLOAT128_BIAS - FLT128_MANT_DIG - 2)
71 {
72 int neg = u.ieee.negative ^ v.ieee.negative;
73 __float128 tiny = neg ? -0x1p-16494Q : 0x1p-16494Q;
74 if (w.ieee.exponent >= 3)
75 return tiny + z;
76 /* Scaling up, adding TINY and scaling down produces the
77 correct result, because in round-to-nearest mode adding
78 TINY has no effect and in other modes double rounding is
79 harmless. But it may not produce required underflow
80 exceptions. */
81 v.value = z * 0x1p114Q + tiny;
82 if (TININESS_AFTER_ROUNDING
83 ? v.ieee.exponent < 115
84 : (w.ieee.exponent == 0
85 || (w.ieee.exponent == 1
86 && w.ieee.negative != neg
87 && w.ieee.mantissa3 == 0
88 && w.ieee.mantissa2 == 0
89 && w.ieee.mantissa1 == 0
90 && w.ieee.mantissa0 == 0)))
91 {
92 __float128 force_underflow = x * y;
93 math_force_eval (force_underflow);
94 }
95 return v.value * 0x1p-114Q;
96 }
97 if (u.ieee.exponent + v.ieee.exponent
98 >= 0x7fff + IEEE854_FLOAT128_BIAS - FLT128_MANT_DIG)
99 {
100 /* Compute 1p-113 times smaller result and multiply
101 at the end. */
102 if (u.ieee.exponent > v.ieee.exponent)
103 u.ieee.exponent -= FLT128_MANT_DIG;
104 else
105 v.ieee.exponent -= FLT128_MANT_DIG;
106 /* If x + y exponent is very large and z exponent is very small,
107 it doesn't matter if we don't adjust it. */
108 if (w.ieee.exponent > FLT128_MANT_DIG)
109 w.ieee.exponent -= FLT128_MANT_DIG;
110 adjust = 1;
111 }
112 else if (w.ieee.exponent >= 0x7fff - FLT128_MANT_DIG)
113 {
114 /* Similarly.
115 If z exponent is very large and x and y exponents are
116 very small, adjust them up to avoid spurious underflows,
117 rather than down. */
118 if (u.ieee.exponent + v.ieee.exponent
119 <= IEEE854_FLOAT128_BIAS + 2 * FLT128_MANT_DIG)
120 {
121 if (u.ieee.exponent > v.ieee.exponent)
122 u.ieee.exponent += 2 * FLT128_MANT_DIG + 2;
123 else
124 v.ieee.exponent += 2 * FLT128_MANT_DIG + 2;
125 }
126 else if (u.ieee.exponent > v.ieee.exponent)
127 {
128 if (u.ieee.exponent > FLT128_MANT_DIG)
129 u.ieee.exponent -= FLT128_MANT_DIG;
130 }
131 else if (v.ieee.exponent > FLT128_MANT_DIG)
132 v.ieee.exponent -= FLT128_MANT_DIG;
133 w.ieee.exponent -= FLT128_MANT_DIG;
134 adjust = 1;
135 }
136 else if (u.ieee.exponent >= 0x7fff - FLT128_MANT_DIG)
137 {
138 u.ieee.exponent -= FLT128_MANT_DIG;
139 if (v.ieee.exponent)
140 v.ieee.exponent += FLT128_MANT_DIG;
141 else
142 v.value *= 0x1p113Q;
143 }
144 else if (v.ieee.exponent >= 0x7fff - FLT128_MANT_DIG)
145 {
146 v.ieee.exponent -= FLT128_MANT_DIG;
147 if (u.ieee.exponent)
148 u.ieee.exponent += FLT128_MANT_DIG;
149 else
150 u.value *= 0x1p113Q;
151 }
152 else /* if (u.ieee.exponent + v.ieee.exponent
153 <= IEEE854_FLOAT128_BIAS + FLT128_MANT_DIG) */
154 {
155 if (u.ieee.exponent > v.ieee.exponent)
156 u.ieee.exponent += 2 * FLT128_MANT_DIG + 2;
157 else
158 v.ieee.exponent += 2 * FLT128_MANT_DIG + 2;
159 if (w.ieee.exponent <= 4 * FLT128_MANT_DIG + 6)
160 {
161 if (w.ieee.exponent)
162 w.ieee.exponent += 2 * FLT128_MANT_DIG + 2;
163 else
164 w.value *= 0x1p228Q;
165 adjust = -1;
166 }
167 /* Otherwise x * y should just affect inexact
168 and nothing else. */
169 }
170 x = u.value;
171 y = v.value;
172 z = w.value;
173 }
174
175 /* Ensure correct sign of exact 0 + 0. */
176 if (__glibc_unlikely ((x == 0 || y == 0) && z == 0))
177 {
178 x = math_opt_barrier (x);
179 return x * y + z;
180 }
181
182 fenv_t env;
183 feholdexcept (&env);
184 fesetround (FE_TONEAREST);
185
186 /* Multiplication m1 + m2 = x * y using Dekker's algorithm. */
187 #define C ((1LL << (FLT128_MANT_DIG + 1) / 2) + 1)
188 __float128 x1 = x * C;
189 __float128 y1 = y * C;
190 __float128 m1 = x * y;
191 x1 = (x - x1) + x1;
192 y1 = (y - y1) + y1;
193 __float128 x2 = x - x1;
194 __float128 y2 = y - y1;
195 __float128 m2 = (((x1 * y1 - m1) + x1 * y2) + x2 * y1) + x2 * y2;
196
197 /* Addition a1 + a2 = z + m1 using Knuth's algorithm. */
198 __float128 a1 = z + m1;
199 __float128 t1 = a1 - z;
200 __float128 t2 = a1 - t1;
201 t1 = m1 - t1;
202 t2 = z - t2;
203 __float128 a2 = t1 + t2;
204 /* Ensure the arithmetic is not scheduled after feclearexcept call. */
205 math_force_eval (m2);
206 math_force_eval (a2);
207 feclearexcept (FE_INEXACT);
208
209 /* If the result is an exact zero, ensure it has the correct sign. */
210 if (a1 == 0 && m2 == 0)
211 {
212 feupdateenv (&env);
213 /* Ensure that round-to-nearest value of z + m1 is not reused. */
214 z = math_opt_barrier (z);
215 return z + m1;
216 }
217
218 fesetround (FE_TOWARDZERO);
219 /* Perform m2 + a2 addition with round to odd. */
220 u.value = a2 + m2;
221
222 if (__glibc_likely (adjust == 0))
223 {
224 if ((u.ieee.mantissa3 & 1) == 0 && u.ieee.exponent != 0x7fff)
225 u.ieee.mantissa3 |= fetestexcept (FE_INEXACT) != 0;
226 feupdateenv (&env);
227 /* Result is a1 + u.value. */
228 return a1 + u.value;
229 }
230 else if (__glibc_likely (adjust > 0))
231 {
232 if ((u.ieee.mantissa3 & 1) == 0 && u.ieee.exponent != 0x7fff)
233 u.ieee.mantissa3 |= fetestexcept (FE_INEXACT) != 0;
234 feupdateenv (&env);
235 /* Result is a1 + u.value, scaled up. */
236 return (a1 + u.value) * 0x1p113Q;
237 }
238 else
239 {
240 if ((u.ieee.mantissa3 & 1) == 0)
241 u.ieee.mantissa3 |= fetestexcept (FE_INEXACT) != 0;
242 v.value = a1 + u.value;
243 /* Ensure the addition is not scheduled after fetestexcept call. */
244 math_force_eval (v.value);
245 int j = fetestexcept (FE_INEXACT) != 0;
246 feupdateenv (&env);
247 /* Ensure the following computations are performed in default rounding
248 mode instead of just reusing the round to zero computation. */
249 asm volatile ("" : "=m" (u) : "m" (u));
250 /* If a1 + u.value is exact, the only rounding happens during
251 scaling down. */
252 if (j == 0)
253 return v.value * 0x1p-228Q;
254 /* If result rounded to zero is not subnormal, no double
255 rounding will occur. */
256 if (v.ieee.exponent > 228)
257 return (a1 + u.value) * 0x1p-228Q;
258 /* If v.value * 0x1p-228L with round to zero is a subnormal above
259 or equal to FLT128_MIN / 2, then v.value * 0x1p-228L shifts mantissa
260 down just by 1 bit, which means v.ieee.mantissa3 |= j would
261 change the round bit, not sticky or guard bit.
262 v.value * 0x1p-228L never normalizes by shifting up,
263 so round bit plus sticky bit should be already enough
264 for proper rounding. */
265 if (v.ieee.exponent == 228)
266 {
267 /* If the exponent would be in the normal range when
268 rounding to normal precision with unbounded exponent
269 range, the exact result is known and spurious underflows
270 must be avoided on systems detecting tininess after
271 rounding. */
272 if (TININESS_AFTER_ROUNDING)
273 {
274 w.value = a1 + u.value;
275 if (w.ieee.exponent == 229)
276 return w.value * 0x1p-228Q;
277 }
278 /* v.ieee.mantissa3 & 2 is LSB bit of the result before rounding,
279 v.ieee.mantissa3 & 1 is the round bit and j is our sticky
280 bit. */
281 w.value = 0;
282 w.ieee.mantissa3 = ((v.ieee.mantissa3 & 3) << 1) | j;
283 w.ieee.negative = v.ieee.negative;
284 v.ieee.mantissa3 &= ~3U;
285 v.value *= 0x1p-228Q;
286 w.value *= 0x1p-2Q;
287 return v.value + w.value;
288 }
289 v.ieee.mantissa3 |= j;
290 return v.value * 0x1p-228Q;
291 }
292 }