1 /* Compute x * y + z as ternary operation.
2 Copyright (C) 2010-2023 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 #define dfmal __hide_dfmal
22 #define f32xfmaf64 __hide_f32xfmaf64
23 #include <math.h>
24 #undef dfmal
25 #undef f32xfmaf64
26 #include <fenv.h>
27 #include <ieee754.h>
28 #include <math-barriers.h>
29 #include <fenv_private.h>
30 #include <libm-alias-double.h>
31 #include <math-narrow-alias.h>
32 #include <tininess.h>
33 #include <math-use-builtins.h>
34
35 /* This implementation uses rounding to odd to avoid problems with
36 double rounding. See a paper by Boldo and Melquiond:
37 http://www.lri.fr/~melquion/doc/08-tc.pdf */
38
39 double
40 __fma (double x, double y, double z)
41 {
42 #if USE_FMA_BUILTIN
43 return __builtin_fma (x, y, z);
44 #else
45 /* Use generic implementation. */
46 union ieee754_double u, v, w;
47 int adjust = 0;
48 u.d = x;
49 v.d = y;
50 w.d = z;
51 if (__builtin_expect (u.ieee.exponent + v.ieee.exponent
52 >= 0x7ff + IEEE754_DOUBLE_BIAS - DBL_MANT_DIG, 0)
53 || __builtin_expect (u.ieee.exponent >= 0x7ff - DBL_MANT_DIG, 0)
54 || __builtin_expect (v.ieee.exponent >= 0x7ff - DBL_MANT_DIG, 0)
55 || __builtin_expect (w.ieee.exponent >= 0x7ff - DBL_MANT_DIG, 0)
56 || __builtin_expect (u.ieee.exponent + v.ieee.exponent
57 <= IEEE754_DOUBLE_BIAS + DBL_MANT_DIG, 0))
58 {
59 /* If z is Inf, but x and y are finite, the result should be
60 z rather than NaN. */
61 if (w.ieee.exponent == 0x7ff
62 && u.ieee.exponent != 0x7ff
63 && v.ieee.exponent != 0x7ff)
64 return (z + x) + y;
65 /* If z is zero and x are y are nonzero, compute the result
66 as x * y to avoid the wrong sign of a zero result if x * y
67 underflows to 0. */
68 if (z == 0 && x != 0 && y != 0)
69 return x * y;
70 /* If x or y or z is Inf/NaN, or if x * y is zero, compute as
71 x * y + z. */
72 if (u.ieee.exponent == 0x7ff
73 || v.ieee.exponent == 0x7ff
74 || w.ieee.exponent == 0x7ff
75 || x == 0
76 || y == 0)
77 return x * y + z;
78 /* If fma will certainly overflow, compute as x * y. */
79 if (u.ieee.exponent + v.ieee.exponent > 0x7ff + IEEE754_DOUBLE_BIAS)
80 return x * y;
81 /* If x * y is less than 1/4 of DBL_TRUE_MIN, neither the
82 result nor whether there is underflow depends on its exact
83 value, only on its sign. */
84 if (u.ieee.exponent + v.ieee.exponent
85 < IEEE754_DOUBLE_BIAS - DBL_MANT_DIG - 2)
86 {
87 int neg = u.ieee.negative ^ v.ieee.negative;
88 double tiny = neg ? -0x1p-1074 : 0x1p-1074;
89 if (w.ieee.exponent >= 3)
90 return tiny + z;
91 /* Scaling up, adding TINY and scaling down produces the
92 correct result, because in round-to-nearest mode adding
93 TINY has no effect and in other modes double rounding is
94 harmless. But it may not produce required underflow
95 exceptions. */
96 v.d = z * 0x1p54 + tiny;
97 if (TININESS_AFTER_ROUNDING
98 ? v.ieee.exponent < 55
99 : (w.ieee.exponent == 0
100 || (w.ieee.exponent == 1
101 && w.ieee.negative != neg
102 && w.ieee.mantissa1 == 0
103 && w.ieee.mantissa0 == 0)))
104 {
105 double force_underflow = x * y;
106 math_force_eval (force_underflow);
107 }
108 return v.d * 0x1p-54;
109 }
110 if (u.ieee.exponent + v.ieee.exponent
111 >= 0x7ff + IEEE754_DOUBLE_BIAS - DBL_MANT_DIG)
112 {
113 /* Compute 1p-53 times smaller result and multiply
114 at the end. */
115 if (u.ieee.exponent > v.ieee.exponent)
116 u.ieee.exponent -= DBL_MANT_DIG;
117 else
118 v.ieee.exponent -= DBL_MANT_DIG;
119 /* If x + y exponent is very large and z exponent is very small,
120 it doesn't matter if we don't adjust it. */
121 if (w.ieee.exponent > DBL_MANT_DIG)
122 w.ieee.exponent -= DBL_MANT_DIG;
123 adjust = 1;
124 }
125 else if (w.ieee.exponent >= 0x7ff - DBL_MANT_DIG)
126 {
127 /* Similarly.
128 If z exponent is very large and x and y exponents are
129 very small, adjust them up to avoid spurious underflows,
130 rather than down. */
131 if (u.ieee.exponent + v.ieee.exponent
132 <= IEEE754_DOUBLE_BIAS + 2 * DBL_MANT_DIG)
133 {
134 if (u.ieee.exponent > v.ieee.exponent)
135 u.ieee.exponent += 2 * DBL_MANT_DIG + 2;
136 else
137 v.ieee.exponent += 2 * DBL_MANT_DIG + 2;
138 }
139 else if (u.ieee.exponent > v.ieee.exponent)
140 {
141 if (u.ieee.exponent > DBL_MANT_DIG)
142 u.ieee.exponent -= DBL_MANT_DIG;
143 }
144 else if (v.ieee.exponent > DBL_MANT_DIG)
145 v.ieee.exponent -= DBL_MANT_DIG;
146 w.ieee.exponent -= DBL_MANT_DIG;
147 adjust = 1;
148 }
149 else if (u.ieee.exponent >= 0x7ff - DBL_MANT_DIG)
150 {
151 u.ieee.exponent -= DBL_MANT_DIG;
152 if (v.ieee.exponent)
153 v.ieee.exponent += DBL_MANT_DIG;
154 else
155 v.d *= 0x1p53;
156 }
157 else if (v.ieee.exponent >= 0x7ff - DBL_MANT_DIG)
158 {
159 v.ieee.exponent -= DBL_MANT_DIG;
160 if (u.ieee.exponent)
161 u.ieee.exponent += DBL_MANT_DIG;
162 else
163 u.d *= 0x1p53;
164 }
165 else /* if (u.ieee.exponent + v.ieee.exponent
166 <= IEEE754_DOUBLE_BIAS + DBL_MANT_DIG) */
167 {
168 if (u.ieee.exponent > v.ieee.exponent)
169 u.ieee.exponent += 2 * DBL_MANT_DIG + 2;
170 else
171 v.ieee.exponent += 2 * DBL_MANT_DIG + 2;
172 if (w.ieee.exponent <= 4 * DBL_MANT_DIG + 6)
173 {
174 if (w.ieee.exponent)
175 w.ieee.exponent += 2 * DBL_MANT_DIG + 2;
176 else
177 w.d *= 0x1p108;
178 adjust = -1;
179 }
180 /* Otherwise x * y should just affect inexact
181 and nothing else. */
182 }
183 x = u.d;
184 y = v.d;
185 z = w.d;
186 }
187
188 /* Ensure correct sign of exact 0 + 0. */
189 if (__glibc_unlikely ((x == 0 || y == 0) && z == 0))
190 {
191 x = math_opt_barrier (x);
192 return x * y + z;
193 }
194
195 fenv_t env;
196 libc_feholdexcept_setround (&env, FE_TONEAREST);
197
198 /* Multiplication m1 + m2 = x * y using Dekker's algorithm. */
199 #define C ((1 << (DBL_MANT_DIG + 1) / 2) + 1)
200 double x1 = x * C;
201 double y1 = y * C;
202 double m1 = x * y;
203 x1 = (x - x1) + x1;
204 y1 = (y - y1) + y1;
205 double x2 = x - x1;
206 double y2 = y - y1;
207 double m2 = (((x1 * y1 - m1) + x1 * y2) + x2 * y1) + x2 * y2;
208
209 /* Addition a1 + a2 = z + m1 using Knuth's algorithm. */
210 double a1 = z + m1;
211 double t1 = a1 - z;
212 double t2 = a1 - t1;
213 t1 = m1 - t1;
214 t2 = z - t2;
215 double a2 = t1 + t2;
216 /* Ensure the arithmetic is not scheduled after feclearexcept call. */
217 math_force_eval (m2);
218 math_force_eval (a2);
219 feclearexcept (FE_INEXACT);
220
221 /* If the result is an exact zero, ensure it has the correct sign. */
222 if (a1 == 0 && m2 == 0)
223 {
224 libc_feupdateenv (&env);
225 /* Ensure that round-to-nearest value of z + m1 is not reused. */
226 z = math_opt_barrier (z);
227 return z + m1;
228 }
229
230 libc_fesetround (FE_TOWARDZERO);
231
232 /* Perform m2 + a2 addition with round to odd. */
233 u.d = a2 + m2;
234
235 if (__glibc_unlikely (adjust < 0))
236 {
237 if ((u.ieee.mantissa1 & 1) == 0)
238 u.ieee.mantissa1 |= libc_fetestexcept (FE_INEXACT) != 0;
239 v.d = a1 + u.d;
240 /* Ensure the addition is not scheduled after fetestexcept call. */
241 math_force_eval (v.d);
242 }
243
244 /* Reset rounding mode and test for inexact simultaneously. */
245 int j = libc_feupdateenv_test (&env, FE_INEXACT) != 0;
246
247 if (__glibc_likely (adjust == 0))
248 {
249 if ((u.ieee.mantissa1 & 1) == 0 && u.ieee.exponent != 0x7ff)
250 u.ieee.mantissa1 |= j;
251 /* Result is a1 + u.d. */
252 return a1 + u.d;
253 }
254 else if (__glibc_likely (adjust > 0))
255 {
256 if ((u.ieee.mantissa1 & 1) == 0 && u.ieee.exponent != 0x7ff)
257 u.ieee.mantissa1 |= j;
258 /* Result is a1 + u.d, scaled up. */
259 return (a1 + u.d) * 0x1p53;
260 }
261 else
262 {
263 /* If a1 + u.d is exact, the only rounding happens during
264 scaling down. */
265 if (j == 0)
266 return v.d * 0x1p-108;
267 /* If result rounded to zero is not subnormal, no double
268 rounding will occur. */
269 if (v.ieee.exponent > 108)
270 return (a1 + u.d) * 0x1p-108;
271 /* If v.d * 0x1p-108 with round to zero is a subnormal above
272 or equal to DBL_MIN / 2, then v.d * 0x1p-108 shifts mantissa
273 down just by 1 bit, which means v.ieee.mantissa1 |= j would
274 change the round bit, not sticky or guard bit.
275 v.d * 0x1p-108 never normalizes by shifting up,
276 so round bit plus sticky bit should be already enough
277 for proper rounding. */
278 if (v.ieee.exponent == 108)
279 {
280 /* If the exponent would be in the normal range when
281 rounding to normal precision with unbounded exponent
282 range, the exact result is known and spurious underflows
283 must be avoided on systems detecting tininess after
284 rounding. */
285 if (TININESS_AFTER_ROUNDING)
286 {
287 w.d = a1 + u.d;
288 if (w.ieee.exponent == 109)
289 return w.d * 0x1p-108;
290 }
291 /* v.ieee.mantissa1 & 2 is LSB bit of the result before rounding,
292 v.ieee.mantissa1 & 1 is the round bit and j is our sticky
293 bit. */
294 w.d = 0.0;
295 w.ieee.mantissa1 = ((v.ieee.mantissa1 & 3) << 1) | j;
296 w.ieee.negative = v.ieee.negative;
297 v.ieee.mantissa1 &= ~3U;
298 v.d *= 0x1p-108;
299 w.d *= 0x1p-2;
300 return v.d + w.d;
301 }
302 v.ieee.mantissa1 |= j;
303 return v.d * 0x1p-108;
304 }
305 #endif /* ! USE_FMA_BUILTIN */
306 }
307 #ifndef __fma
308 libm_alias_double (__fma, fma)
309 libm_alias_double_narrow (__fma, fma)
310 #endif