1 /*-
2 * SPDX-License-Identifier: BSD-2-Clause-FreeBSD
3 *
4 * Copyright (c) 2005-2011 David Schultz <das@FreeBSD.ORG>
5 * All rights reserved.
6 *
7 * Redistribution and use in source and binary forms, with or without
8 * modification, are permitted provided that the following conditions
9 * are met:
10 * 1. Redistributions of source code must retain the above copyright
11 * notice, this list of conditions and the following disclaimer.
12 * 2. Redistributions in binary form must reproduce the above copyright
13 * notice, this list of conditions and the following disclaimer in the
14 * documentation and/or other materials provided with the distribution.
15 *
16 * THIS SOFTWARE IS PROVIDED BY THE AUTHOR AND CONTRIBUTORS ``AS IS'' AND
17 * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
18 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
19 * ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR OR CONTRIBUTORS BE LIABLE
20 * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
21 * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
22 * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
23 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
24 * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
25 * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
26 * SUCH DAMAGE.
27 */
28
29 #include <sys/cdefs.h>
30 __FBSDID("$FreeBSD$");
31
32 #include <fenv.h>
33 #include <float.h>
34 #include <math.h>
35
36 #include "math_private.h"
37
38 #ifdef USE_BUILTIN_FMA
39 double
fma(double x,double y,double z)40 fma(double x, double y, double z)
41 {
42 return (__builtin_fma(x, y, z));
43 }
44 #else
45 /*
46 * A struct dd represents a floating-point number with twice the precision
47 * of a double. We maintain the invariant that "hi" stores the 53 high-order
48 * bits of the result.
49 */
50 struct dd {
51 double hi;
52 double lo;
53 };
54
55 /*
56 * Compute a+b exactly, returning the exact result in a struct dd. We assume
57 * that both a and b are finite, but make no assumptions about their relative
58 * magnitudes.
59 */
60 static inline struct dd
dd_add(double a,double b)61 dd_add(double a, double b)
62 {
63 struct dd ret;
64 double s;
65
66 ret.hi = a + b;
67 s = ret.hi - a;
68 ret.lo = (a - (ret.hi - s)) + (b - s);
69 return (ret);
70 }
71
72 /*
73 * Compute a+b, with a small tweak: The least significant bit of the
74 * result is adjusted into a sticky bit summarizing all the bits that
75 * were lost to rounding. This adjustment negates the effects of double
76 * rounding when the result is added to another number with a higher
77 * exponent. For an explanation of round and sticky bits, see any reference
78 * on FPU design, e.g.,
79 *
80 * J. Coonen. An Implementation Guide to a Proposed Standard for
81 * Floating-Point Arithmetic. Computer, vol. 13, no. 1, Jan 1980.
82 */
83 static inline double
add_adjusted(double a,double b)84 add_adjusted(double a, double b)
85 {
86 struct dd sum;
87 uint64_t hibits, lobits;
88
89 sum = dd_add(a, b);
90 if (sum.lo != 0) {
91 EXTRACT_WORD64(hibits, sum.hi);
92 if ((hibits & 1) == 0) {
93 /* hibits += (int)copysign(1.0, sum.hi * sum.lo) */
94 EXTRACT_WORD64(lobits, sum.lo);
95 hibits += 1 - ((hibits ^ lobits) >> 62);
96 INSERT_WORD64(sum.hi, hibits);
97 }
98 }
99 return (sum.hi);
100 }
101
102 /*
103 * Compute ldexp(a+b, scale) with a single rounding error. It is assumed
104 * that the result will be subnormal, and care is taken to ensure that
105 * double rounding does not occur.
106 */
107 static inline double
add_and_denormalize(double a,double b,int scale)108 add_and_denormalize(double a, double b, int scale)
109 {
110 struct dd sum;
111 uint64_t hibits, lobits;
112 int bits_lost;
113
114 sum = dd_add(a, b);
115
116 /*
117 * If we are losing at least two bits of accuracy to denormalization,
118 * then the first lost bit becomes a round bit, and we adjust the
119 * lowest bit of sum.hi to make it a sticky bit summarizing all the
120 * bits in sum.lo. With the sticky bit adjusted, the hardware will
121 * break any ties in the correct direction.
122 *
123 * If we are losing only one bit to denormalization, however, we must
124 * break the ties manually.
125 */
126 if (sum.lo != 0) {
127 EXTRACT_WORD64(hibits, sum.hi);
128 bits_lost = -((int)(hibits >> 52) & 0x7ff) - scale + 1;
129 if ((bits_lost != 1) ^ (int)(hibits & 1)) {
130 /* hibits += (int)copysign(1.0, sum.hi * sum.lo) */
131 EXTRACT_WORD64(lobits, sum.lo);
132 hibits += 1 - (((hibits ^ lobits) >> 62) & 2);
133 INSERT_WORD64(sum.hi, hibits);
134 }
135 }
136 return (ldexp(sum.hi, scale));
137 }
138
139 /*
140 * Compute a*b exactly, returning the exact result in a struct dd. We assume
141 * that both a and b are normalized, so no underflow or overflow will occur.
142 * The current rounding mode must be round-to-nearest.
143 */
144 static inline struct dd
dd_mul(double a,double b)145 dd_mul(double a, double b)
146 {
147 static const double split = 0x1p27 + 1.0;
148 struct dd ret;
149 double ha, hb, la, lb, p, q;
150
151 p = a * split;
152 ha = a - p;
153 ha += p;
154 la = a - ha;
155
156 p = b * split;
157 hb = b - p;
158 hb += p;
159 lb = b - hb;
160
161 p = ha * hb;
162 q = ha * lb + la * hb;
163
164 ret.hi = p + q;
165 ret.lo = p - ret.hi + q + la * lb;
166 return (ret);
167 }
168
169 /*
170 * Fused multiply-add: Compute x * y + z with a single rounding error.
171 *
172 * We use scaling to avoid overflow/underflow, along with the
173 * canonical precision-doubling technique adapted from:
174 *
175 * Dekker, T. A Floating-Point Technique for Extending the
176 * Available Precision. Numer. Math. 18, 224-242 (1971).
177 *
178 * This algorithm is sensitive to the rounding precision. FPUs such
179 * as the i387 must be set in double-precision mode if variables are
180 * to be stored in FP registers in order to avoid incorrect results.
181 * This is the default on FreeBSD, but not on many other systems.
182 *
183 * Hardware instructions should be used on architectures that support it,
184 * since this implementation will likely be several times slower.
185 */
186 double
fma(double x,double y,double z)187 fma(double x, double y, double z)
188 {
189 double xs, ys, zs, adj;
190 struct dd xy, r;
191 int oround;
192 int ex, ey, ez;
193 int spread;
194
195 /*
196 * Handle special cases. The order of operations and the particular
197 * return values here are crucial in handling special cases involving
198 * infinities, NaNs, overflows, and signed zeroes correctly.
199 */
200 if (x == 0.0 || y == 0.0)
201 return (x * y + z);
202 if (z == 0.0)
203 return (x * y);
204 if (!isfinite(x) || !isfinite(y))
205 return (x * y + z);
206 if (!isfinite(z))
207 return (z);
208
209 xs = frexp(x, &ex);
210 ys = frexp(y, &ey);
211 zs = frexp(z, &ez);
212 oround = fegetround();
213 spread = ex + ey - ez;
214
215 /*
216 * If x * y and z are many orders of magnitude apart, the scaling
217 * will overflow, so we handle these cases specially. Rounding
218 * modes other than FE_TONEAREST are painful.
219 */
220 if (spread < -DBL_MANT_DIG) {
221 feraiseexcept(FE_INEXACT);
222 if (!isnormal(z))
223 feraiseexcept(FE_UNDERFLOW);
224 switch (oround) {
225 case FE_TONEAREST:
226 return (z);
227 case FE_TOWARDZERO:
228 if (x > 0.0 ^ y < 0.0 ^ z < 0.0)
229 return (z);
230 else
231 return (nextafter(z, 0));
232 case FE_DOWNWARD:
233 if (x > 0.0 ^ y < 0.0)
234 return (z);
235 else
236 return (nextafter(z, -INFINITY));
237 default: /* FE_UPWARD */
238 if (x > 0.0 ^ y < 0.0)
239 return (nextafter(z, INFINITY));
240 else
241 return (z);
242 }
243 }
244 if (spread <= DBL_MANT_DIG * 2)
245 zs = ldexp(zs, -spread);
246 else
247 zs = copysign(DBL_MIN, zs);
248
249 fesetround(FE_TONEAREST);
250 /* work around clang bug 8100 */
251 volatile double vxs = xs;
252
253 /*
254 * Basic approach for round-to-nearest:
255 *
256 * (xy.hi, xy.lo) = x * y (exact)
257 * (r.hi, r.lo) = xy.hi + z (exact)
258 * adj = xy.lo + r.lo (inexact; low bit is sticky)
259 * result = r.hi + adj (correctly rounded)
260 */
261 xy = dd_mul(vxs, ys);
262 r = dd_add(xy.hi, zs);
263
264 spread = ex + ey;
265
266 if (r.hi == 0.0) {
267 /*
268 * When the addends cancel to 0, ensure that the result has
269 * the correct sign.
270 */
271 fesetround(oround);
272 volatile double vzs = zs; /* XXX gcc CSE bug workaround */
273 return (xy.hi + vzs + ldexp(xy.lo, spread));
274 }
275
276 if (oround != FE_TONEAREST) {
277 /*
278 * There is no need to worry about double rounding in directed
279 * rounding modes.
280 */
281 fesetround(oround);
282 /* work around clang bug 8100 */
283 volatile double vrlo = r.lo;
284 adj = vrlo + xy.lo;
285 return (ldexp(r.hi + adj, spread));
286 }
287
288 adj = add_adjusted(r.lo, xy.lo);
289 if (spread + ilogb(r.hi) > -1023)
290 return (ldexp(r.hi + adj, spread));
291 else
292 return (add_and_denormalize(r.hi, adj, spread));
293 }
294 #endif /* !USE_BUILTIN_FMA */
295
296 #if (LDBL_MANT_DIG == 53)
297 __weak_reference(fma, fmal);
298 #endif
299