1/*	$OpenBSD: s_fma.c,v 1.7 2016/09/12 19:47:02 guenther Exp $	*/
2
3/*-
4 * Copyright (c) 2005 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 <fenv.h>
30#include <float.h>
31#include <math.h>
32
33/*
34 * Fused multiply-add: Compute x * y + z with a single rounding error.
35 *
36 * We use scaling to avoid overflow/underflow, along with the
37 * canonical precision-doubling technique adapted from:
38 *
39 *	Dekker, T.  A Floating-Point Technique for Extending the
40 *	Available Precision.  Numer. Math. 18, 224-242 (1971).
41 *
42 * This algorithm is sensitive to the rounding precision.  FPUs such
43 * as the i387 must be set in double-precision mode if variables are
44 * to be stored in FP registers in order to avoid incorrect results.
45 * This is the default on FreeBSD, but not on many other systems.
46 *
47 * Hardware instructions should be used on architectures that support it,
48 * since this implementation will likely be several times slower.
49 */
50#if LDBL_MANT_DIG != 113
51double
52fma(double x, double y, double z)
53{
54	static const double split = 0x1p27 + 1.0;
55	double xs, ys, zs;
56	double c, cc, hx, hy, p, q, tx, ty;
57	double r, rr, s;
58	int oround;
59	int ex, ey, ez;
60	int spread;
61
62	/*
63	 * Handle special cases. The order of operations and the particular
64	 * return values here are crucial in handling special cases involving
65	 * infinities, NaNs, overflows, and signed zeroes correctly.
66	 */
67	if (x == 0.0 || y == 0.0)
68		return (x * y + z);
69	if (z == 0.0)
70		return (x * y);
71	if (!isfinite(x) || !isfinite(y))
72		return (x * y + z);
73	if (!isfinite(z))
74		return (z);
75
76	xs = frexp(x, &ex);
77	ys = frexp(y, &ey);
78	zs = frexp(z, &ez);
79	oround = fegetround();
80	spread = ex + ey - ez;
81
82	/*
83	 * If x * y and z are many orders of magnitude apart, the scaling
84	 * will overflow, so we handle these cases specially.  Rounding
85	 * modes other than FE_TONEAREST are painful.
86	 */
87	if (spread > DBL_MANT_DIG * 2) {
88		fenv_t env;
89		feraiseexcept(FE_INEXACT);
90		switch(oround) {
91		case FE_TONEAREST:
92			return (x * y);
93		case FE_TOWARDZERO:
94			if ((x > 0.0) ^ (y < 0.0) ^ (z < 0.0))
95				return (x * y);
96			feholdexcept(&env);
97			r = x * y;
98			if (!fetestexcept(FE_INEXACT))
99				r = nextafter(r, 0);
100			feupdateenv(&env);
101			return (r);
102		case FE_DOWNWARD:
103			if (z > 0.0)
104				return (x * y);
105			feholdexcept(&env);
106			r = x * y;
107			if (!fetestexcept(FE_INEXACT))
108				r = nextafter(r, -INFINITY);
109			feupdateenv(&env);
110			return (r);
111		default:	/* FE_UPWARD */
112			if (z < 0.0)
113				return (x * y);
114			feholdexcept(&env);
115			r = x * y;
116			if (!fetestexcept(FE_INEXACT))
117				r = nextafter(r, INFINITY);
118			feupdateenv(&env);
119			return (r);
120		}
121	}
122	if (spread < -DBL_MANT_DIG) {
123		feraiseexcept(FE_INEXACT);
124		if (!isnormal(z))
125			feraiseexcept(FE_UNDERFLOW);
126		switch (oround) {
127		case FE_TONEAREST:
128			return (z);
129		case FE_TOWARDZERO:
130			if ((x > 0.0) ^ (y < 0.0) ^ (z < 0.0))
131				return (z);
132			else
133				return (nextafter(z, 0));
134		case FE_DOWNWARD:
135			if ((x > 0.0) ^ (y < 0.0))
136				return (z);
137			else
138				return (nextafter(z, -INFINITY));
139		default:	/* FE_UPWARD */
140			if ((x > 0.0) ^ (y < 0.0))
141				return (nextafter(z, INFINITY));
142			else
143				return (z);
144		}
145	}
146
147	/*
148	 * Use Dekker's algorithm to perform the multiplication and
149	 * subsequent addition in twice the machine precision.
150	 * Arrange so that x * y = c + cc, and x * y + z = r + rr.
151	 */
152	fesetround(FE_TONEAREST);
153
154	p = xs * split;
155	hx = xs - p;
156	hx += p;
157	tx = xs - hx;
158
159	p = ys * split;
160	hy = ys - p;
161	hy += p;
162	ty = ys - hy;
163
164	p = hx * hy;
165	q = hx * ty + tx * hy;
166	c = p + q;
167	cc = p - c + q + tx * ty;
168
169	zs = ldexp(zs, -spread);
170	r = c + zs;
171	s = r - c;
172	rr = (c - (r - s)) + (zs - s) + cc;
173
174	spread = ex + ey;
175	if (spread + ilogb(r) > -1023) {
176		fesetround(oround);
177		r = r + rr;
178	} else {
179		/*
180		 * The result is subnormal, so we round before scaling to
181		 * avoid double rounding.
182		 */
183		p = ldexp(copysign(0x1p-1022, r), -spread);
184		c = r + p;
185		s = c - r;
186		cc = (r - (c - s)) + (p - s) + rr;
187		fesetround(oround);
188		r = (c + cc) - p;
189	}
190	return (ldexp(r, spread));
191}
192#else	/* LDBL_MANT_DIG == 113 */
193/*
194 * 113 bits of precision is more than twice the precision of a double,
195 * so it is enough to represent the intermediate product exactly.
196 */
197double
198fma(double x, double y, double z)
199{
200	return ((long double)x * y + z);
201}
202#endif	/* LDBL_MANT_DIG != 113 */
203DEF_STD(fma);
204LDBL_MAYBE_UNUSED_CLONE(fma);
205