1/* Compute x * y + z as ternary operation.
2 Copyright (C) 2010-2022 Free Software Foundation, Inc.
3 This file is part of the GNU C Library.
4
5 The GNU C Library is free software; you can redistribute it and/or
6 modify it under the terms of the GNU Lesser General Public
7 License as published by the Free Software Foundation; either
8 version 2.1 of the License, or (at your option) any later version.
9
10 The GNU C Library is distributed in the hope that it will be useful,
11 but WITHOUT ANY WARRANTY; without even the implied warranty of
12 MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
13 Lesser General Public License for more details.
14
15 You should have received a copy of the GNU Lesser General Public
16 License along with the GNU C Library; if not, see
17 <https://www.gnu.org/licenses/>. */
18
19#define NO_MATH_REDIRECT
20#include <float.h>
21#include <math.h>
22#include <fenv.h>
23#include <ieee754.h>
24#include <math-barriers.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
32long double
33__fmal (long double x, long double y, long double 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 long double tiny = neg ? -0x1p-16445L : 0x1p-16445L;
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 * 0x1p65L + tiny;
88 if (TININESS_AFTER_ROUNDING
89 ? v.ieee.exponent < 66
90 : (w.ieee.exponent == 0
91 || (w.ieee.exponent == 1
92 && w.ieee.negative != neg
93 && w.ieee.mantissa1 == 0
94 && w.ieee.mantissa0 == 0x80000000)))
95 {
96 long double force_underflow = x * y;
97 math_force_eval (force_underflow);
98 }
99 return v.d * 0x1p-65L;
100 }
101 if (u.ieee.exponent + v.ieee.exponent
102 >= 0x7fff + IEEE854_LONG_DOUBLE_BIAS - LDBL_MANT_DIG)
103 {
104 /* Compute 1p-64 times smaller result and multiply
105 at the end. */
106 if (u.ieee.exponent > v.ieee.exponent)
107 u.ieee.exponent -= LDBL_MANT_DIG;
108 else
109 v.ieee.exponent -= LDBL_MANT_DIG;
110 /* If x + y exponent is very large and z exponent is very small,
111 it doesn't matter if we don't adjust it. */
112 if (w.ieee.exponent > LDBL_MANT_DIG)
113 w.ieee.exponent -= LDBL_MANT_DIG;
114 adjust = 1;
115 }
116 else if (w.ieee.exponent >= 0x7fff - LDBL_MANT_DIG)
117 {
118 /* Similarly.
119 If z exponent is very large and x and y exponents are
120 very small, adjust them up to avoid spurious underflows,
121 rather than down. */
122 if (u.ieee.exponent + v.ieee.exponent
123 <= IEEE854_LONG_DOUBLE_BIAS + 2 * LDBL_MANT_DIG)
124 {
125 if (u.ieee.exponent > v.ieee.exponent)
126 u.ieee.exponent += 2 * LDBL_MANT_DIG + 2;
127 else
128 v.ieee.exponent += 2 * LDBL_MANT_DIG + 2;
129 }
130 else if (u.ieee.exponent > v.ieee.exponent)
131 {
132 if (u.ieee.exponent > LDBL_MANT_DIG)
133 u.ieee.exponent -= LDBL_MANT_DIG;
134 }
135 else if (v.ieee.exponent > LDBL_MANT_DIG)
136 v.ieee.exponent -= LDBL_MANT_DIG;
137 w.ieee.exponent -= LDBL_MANT_DIG;
138 adjust = 1;
139 }
140 else if (u.ieee.exponent >= 0x7fff - LDBL_MANT_DIG)
141 {
142 u.ieee.exponent -= LDBL_MANT_DIG;
143 if (v.ieee.exponent)
144 v.ieee.exponent += LDBL_MANT_DIG;
145 else
146 v.d *= 0x1p64L;
147 }
148 else if (v.ieee.exponent >= 0x7fff - LDBL_MANT_DIG)
149 {
150 v.ieee.exponent -= LDBL_MANT_DIG;
151 if (u.ieee.exponent)
152 u.ieee.exponent += LDBL_MANT_DIG;
153 else
154 u.d *= 0x1p64L;
155 }
156 else /* if (u.ieee.exponent + v.ieee.exponent
157 <= IEEE854_LONG_DOUBLE_BIAS + LDBL_MANT_DIG) */
158 {
159 if (u.ieee.exponent > v.ieee.exponent)
160 u.ieee.exponent += 2 * LDBL_MANT_DIG + 2;
161 else
162 v.ieee.exponent += 2 * LDBL_MANT_DIG + 2;
163 if (w.ieee.exponent <= 4 * LDBL_MANT_DIG + 6)
164 {
165 if (w.ieee.exponent)
166 w.ieee.exponent += 2 * LDBL_MANT_DIG + 2;
167 else
168 w.d *= 0x1p130L;
169 adjust = -1;
170 }
171 /* Otherwise x * y should just affect inexact
172 and nothing else. */
173 }
174 x = u.d;
175 y = v.d;
176 z = w.d;
177 }
178
179 /* Ensure correct sign of exact 0 + 0. */
180 if (__glibc_unlikely ((x == 0 || y == 0) && z == 0))
181 {
182 x = math_opt_barrier (x);
183 return x * y + z;
184 }
185
186 fenv_t env;
187 feholdexcept (&env);
188 fesetround (FE_TONEAREST);
189
190 /* Multiplication m1 + m2 = x * y using Dekker's algorithm. */
191#define C ((1LL << (LDBL_MANT_DIG + 1) / 2) + 1)
192 long double x1 = x * C;
193 long double y1 = y * C;
194 long double m1 = x * y;
195 x1 = (x - x1) + x1;
196 y1 = (y - y1) + y1;
197 long double x2 = x - x1;
198 long double y2 = y - y1;
199 long double m2 = (((x1 * y1 - m1) + x1 * y2) + x2 * y1) + x2 * y2;
200
201 /* Addition a1 + a2 = z + m1 using Knuth's algorithm. */
202 long double a1 = z + m1;
203 long double t1 = a1 - z;
204 long double t2 = a1 - t1;
205 t1 = m1 - t1;
206 t2 = z - t2;
207 long double a2 = t1 + t2;
208 /* Ensure the arithmetic is not scheduled after feclearexcept call. */
209 math_force_eval (m2);
210 math_force_eval (a2);
211 feclearexcept (FE_INEXACT);
212
213 /* If the result is an exact zero, ensure it has the correct sign. */
214 if (a1 == 0 && m2 == 0)
215 {
216 feupdateenv (&env);
217 /* Ensure that round-to-nearest value of z + m1 is not reused. */
218 z = math_opt_barrier (z);
219 return z + m1;
220 }
221
222 fesetround (FE_TOWARDZERO);
223 /* Perform m2 + a2 addition with round to odd. */
224 u.d = a2 + m2;
225
226 if (__glibc_likely (adjust == 0))
227 {
228 if ((u.ieee.mantissa1 & 1) == 0 && u.ieee.exponent != 0x7fff)
229 u.ieee.mantissa1 |= fetestexcept (FE_INEXACT) != 0;
230 feupdateenv (&env);
231 /* Result is a1 + u.d. */
232 return a1 + u.d;
233 }
234 else if (__glibc_likely (adjust > 0))
235 {
236 if ((u.ieee.mantissa1 & 1) == 0 && u.ieee.exponent != 0x7fff)
237 u.ieee.mantissa1 |= fetestexcept (FE_INEXACT) != 0;
238 feupdateenv (&env);
239 /* Result is a1 + u.d, scaled up. */
240 return (a1 + u.d) * 0x1p64L;
241 }
242 else
243 {
244 if ((u.ieee.mantissa1 & 1) == 0)
245 u.ieee.mantissa1 |= fetestexcept (FE_INEXACT) != 0;
246 v.d = a1 + u.d;
247 /* Ensure the addition is not scheduled after fetestexcept call. */
248 math_force_eval (v.d);
249 int j = fetestexcept (FE_INEXACT) != 0;
250 feupdateenv (&env);
251 /* Ensure the following computations are performed in default rounding
252 mode instead of just reusing the round to zero computation. */
253 asm volatile ("" : "=m" (u) : "m" (u));
254 /* If a1 + u.d is exact, the only rounding happens during
255 scaling down. */
256 if (j == 0)
257 return v.d * 0x1p-130L;
258 /* If result rounded to zero is not subnormal, no double
259 rounding will occur. */
260 if (v.ieee.exponent > 130)
261 return (a1 + u.d) * 0x1p-130L;
262 /* If v.d * 0x1p-130L with round to zero is a subnormal above
263 or equal to LDBL_MIN / 2, then v.d * 0x1p-130L shifts mantissa
264 down just by 1 bit, which means v.ieee.mantissa1 |= j would
265 change the round bit, not sticky or guard bit.
266 v.d * 0x1p-130L never normalizes by shifting up,
267 so round bit plus sticky bit should be already enough
268 for proper rounding. */
269 if (v.ieee.exponent == 130)
270 {
271 /* If the exponent would be in the normal range when
272 rounding to normal precision with unbounded exponent
273 range, the exact result is known and spurious underflows
274 must be avoided on systems detecting tininess after
275 rounding. */
276 if (TININESS_AFTER_ROUNDING)
277 {
278 w.d = a1 + u.d;
279 if (w.ieee.exponent == 131)
280 return w.d * 0x1p-130L;
281 }
282 /* v.ieee.mantissa1 & 2 is LSB bit of the result before rounding,
283 v.ieee.mantissa1 & 1 is the round bit and j is our sticky
284 bit. */
285 w.d = 0.0L;
286 w.ieee.mantissa1 = ((v.ieee.mantissa1 & 3) << 1) | j;
287 w.ieee.negative = v.ieee.negative;
288 v.ieee.mantissa1 &= ~3U;
289 v.d *= 0x1p-130L;
290 w.d *= 0x1p-2L;
291 return v.d + w.d;
292 }
293 v.ieee.mantissa1 |= j;
294 return v.d * 0x1p-130L;
295 }
296}
297libm_alias_ldouble (__fma, fma)
298