1 /* $NetBSD: s_fma.c,v 1.6 2013/02/14 09:24:50 matt Exp $ */
2
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 #if 0
31 __FBSDID("$FreeBSD: src/lib/msun/src/s_fma.c,v 1.8 2011/10/21 06:30:43 das Exp $");
32 #else
33 __RCSID("$NetBSD: s_fma.c,v 1.6 2013/02/14 09:24:50 matt Exp $");
34 #endif
35
36 #include <machine/ieee.h>
37 #include <fenv.h>
38 #include <float.h>
39 #include <math.h>
40
41 #include "math_private.h"
42
43 #ifndef __HAVE_LONG_DOUBLE
44 __strong_alias(fmal, fma)
45 #endif
46
47 /*
48 * A struct dd represents a floating-point number with twice the precision
49 * of a double. We maintain the invariant that "hi" stores the 53 high-order
50 * bits of the result.
51 */
52 struct dd {
53 double hi;
54 double lo;
55 };
56
57 /*
58 * Compute a+b exactly, returning the exact result in a struct dd. We assume
59 * that both a and b are finite, but make no assumptions about their relative
60 * magnitudes.
61 */
62 static inline struct dd
dd_add(double a,double b)63 dd_add(double a, double b)
64 {
65 struct dd ret;
66 double s;
67
68 ret.hi = a + b;
69 s = ret.hi - a;
70 ret.lo = (a - (ret.hi - s)) + (b - s);
71 return (ret);
72 }
73
74 /*
75 * Compute a+b, with a small tweak: The least significant bit of the
76 * result is adjusted into a sticky bit summarizing all the bits that
77 * were lost to rounding. This adjustment negates the effects of double
78 * rounding when the result is added to another number with a higher
79 * exponent. For an explanation of round and sticky bits, see any reference
80 * on FPU design, e.g.,
81 *
82 * J. Coonen. An Implementation Guide to a Proposed Standard for
83 * Floating-Point Arithmetic. Computer, vol. 13, no. 1, Jan 1980.
84 */
85 static inline double
add_adjusted(double a,double b)86 add_adjusted(double a, double b)
87 {
88 struct dd sum;
89 uint64_t hibits, lobits;
90
91 sum = dd_add(a, b);
92 if (sum.lo != 0) {
93 EXTRACT_WORD64(hibits, sum.hi);
94 if ((hibits & 1) == 0) {
95 /* hibits += (int)copysign(1.0, sum.hi * sum.lo) */
96 EXTRACT_WORD64(lobits, sum.lo);
97 hibits += 1 - ((hibits ^ lobits) >> 62);
98 INSERT_WORD64(sum.hi, hibits);
99 }
100 }
101 return (sum.hi);
102 }
103
104 /*
105 * Compute ldexp(a+b, scale) with a single rounding error. It is assumed
106 * that the result will be subnormal, and care is taken to ensure that
107 * double rounding does not occur.
108 */
109 static inline double
add_and_denormalize(double a,double b,int scale)110 add_and_denormalize(double a, double b, int scale)
111 {
112 struct dd sum;
113 uint64_t hibits, lobits;
114 int bits_lost;
115
116 sum = dd_add(a, b);
117
118 /*
119 * If we are losing at least two bits of accuracy to denormalization,
120 * then the first lost bit becomes a round bit, and we adjust the
121 * lowest bit of sum.hi to make it a sticky bit summarizing all the
122 * bits in sum.lo. With the sticky bit adjusted, the hardware will
123 * break any ties in the correct direction.
124 *
125 * If we are losing only one bit to denormalization, however, we must
126 * break the ties manually.
127 */
128 if (sum.lo != 0) {
129 EXTRACT_WORD64(hibits, sum.hi);
130 bits_lost = -((int)(hibits >> 52) & 0x7ff) - scale + 1;
131 if ((bits_lost != 1) ^ (int)(hibits & 1)) {
132 /* hibits += (int)copysign(1.0, sum.hi * sum.lo) */
133 EXTRACT_WORD64(lobits, sum.lo);
134 hibits += 1 - (((hibits ^ lobits) >> 62) & 2);
135 INSERT_WORD64(sum.hi, hibits);
136 }
137 }
138 return (ldexp(sum.hi, scale));
139 }
140
141 /*
142 * Compute a*b exactly, returning the exact result in a struct dd. We assume
143 * that both a and b are normalized, so no underflow or overflow will occur.
144 * The current rounding mode must be round-to-nearest.
145 */
146 static inline struct dd
dd_mul(double a,double b)147 dd_mul(double a, double b)
148 {
149 static const double split = 0x1p27 + 1.0;
150 struct dd ret;
151 double ha, hb, la, lb, p, q;
152
153 p = a * split;
154 ha = a - p;
155 ha += p;
156 la = a - ha;
157
158 p = b * split;
159 hb = b - p;
160 hb += p;
161 lb = b - hb;
162
163 p = ha * hb;
164 q = ha * lb + la * hb;
165
166 ret.hi = p + q;
167 ret.lo = p - ret.hi + q + la * lb;
168 return (ret);
169 }
170
171 /*
172 * Fused multiply-add: Compute x * y + z with a single rounding error.
173 *
174 * We use scaling to avoid overflow/underflow, along with the
175 * canonical precision-doubling technique adapted from:
176 *
177 * Dekker, T. A Floating-Point Technique for Extending the
178 * Available Precision. Numer. Math. 18, 224-242 (1971).
179 *
180 * This algorithm is sensitive to the rounding precision. FPUs such
181 * as the i387 must be set in double-precision mode if variables are
182 * to be stored in FP registers in order to avoid incorrect results.
183 * This is the default on FreeBSD, but not on many other systems.
184 *
185 * Hardware instructions should be used on architectures that support it,
186 * since this implementation will likely be several times slower.
187 */
188 double
fma(double x,double y,double z)189 fma(double x, double y, double z)
190 {
191 double xs, ys, zs, adj;
192 struct dd xy, r;
193 int oround;
194 int ex, ey, ez;
195 int spread;
196
197 /*
198 * Handle special cases. The order of operations and the particular
199 * return values here are crucial in handling special cases involving
200 * infinities, NaNs, overflows, and signed zeroes correctly.
201 */
202 if (x == 0.0 || y == 0.0)
203 return (x * y + z);
204 if (z == 0.0)
205 return (x * y);
206 if (!isfinite(x) || !isfinite(y))
207 return (x * y + z);
208 if (!isfinite(z))
209 return (z);
210
211 xs = frexp(x, &ex);
212 ys = frexp(y, &ey);
213 zs = frexp(z, &ez);
214 oround = fegetround();
215 spread = ex + ey - ez;
216
217 /*
218 * If x * y and z are many orders of magnitude apart, the scaling
219 * will overflow, so we handle these cases specially. Rounding
220 * modes other than FE_TONEAREST are painful.
221 */
222 if (spread < -DBL_MANT_DIG) {
223 feraiseexcept(FE_INEXACT);
224 if (!isnormal(z))
225 feraiseexcept(FE_UNDERFLOW);
226 switch (oround) {
227 case FE_TONEAREST:
228 return (z);
229 case FE_TOWARDZERO:
230 if ((x > 0.0) ^ (y < 0.0) ^ (z < 0.0))
231 return (z);
232 else
233 return (nextafter(z, 0));
234 case FE_DOWNWARD:
235 if ((x > 0.0) ^ (y < 0.0))
236 return (z);
237 else
238 return (nextafter(z, -INFINITY));
239 default: /* FE_UPWARD */
240 if ((x > 0.0) ^ (y < 0.0))
241 return (nextafter(z, INFINITY));
242 else
243 return (z);
244 }
245 }
246 if (spread <= DBL_MANT_DIG * 2)
247 zs = ldexp(zs, -spread);
248 else
249 zs = copysign(DBL_MIN, zs);
250
251 fesetround(FE_TONEAREST);
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(xs, 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 {
273 volatile double vzs = zs; /* XXX gcc CSE bug workaround */
274 return (xy.hi + vzs + ldexp(xy.lo, spread));
275 }
276 }
277
278 if (oround != FE_TONEAREST) {
279 /*
280 * There is no need to worry about double rounding in directed
281 * rounding modes.
282 */
283 fesetround(oround);
284 adj = r.lo + 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