1/*-
2 * Copyright (c) 2005 David Schultz <das@FreeBSD.ORG>
3 * All rights reserved.
4 *
5 * Redistribution and use in source and binary forms, with or without
6 * modification, are permitted provided that the following conditions
7 * are met:
8 * 1. Redistributions of source code must retain the above copyright
9 *    notice, this list of conditions and the following disclaimer.
10 * 2. Redistributions in binary form must reproduce the above copyright
11 *    notice, this list of conditions and the following disclaimer in the
12 *    documentation and/or other materials provided with the distribution.
13 *
14 * THIS SOFTWARE IS PROVIDED BY THE AUTHOR AND CONTRIBUTORS ``AS IS'' AND
15 * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
16 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
17 * ARE DISCLAIMED.  IN NO EVENT SHALL THE AUTHOR OR CONTRIBUTORS BE LIABLE
18 * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
19 * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
20 * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
21 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
22 * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
23 * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
24 * SUCH DAMAGE.
25 */
26
27#include <sys/cdefs.h>
28/* __FBSDID("$FreeBSD: src/lib/msun/src/s_fma.c,v 1.4 2005/03/18 02:27:59 das Exp $"); */
29
30#include <fenv.h>
31#include <float.h>
32#include <math.h>
33
34/*
35 * Fused multiply-add: Compute x * y + z with a single rounding error.
36 *
37 * We use scaling to avoid overflow/underflow, along with the
38 * canonical precision-doubling technique adapted from:
39 *
40 *	Dekker, T.  A Floating-Point Technique for Extending the
41 *	Available Precision.  Numer. Math. 18, 224-242 (1971).
42 *
43 * This algorithm is sensitive to the rounding precision.  FPUs such
44 * as the i387 must be set in double-precision mode if variables are
45 * to be stored in FP registers in order to avoid incorrect results.
46 * This is the default on FreeBSD, but not on many other systems.
47 *
48 * Hardware instructions should be used on architectures that support it,
49 * since this implementation will likely be several times slower.
50 */
51#if LDBL_MANT_DIG != 113
52double
53fma(double x, double y, double z)
54{
55	static const double split = 0x1p27 + 1.0;
56	double xs, ys, zs;
57	double c, cc, hx, hy, p, q, tx, ty;
58	double r, rr, s;
59	int oround;
60	int ex, ey, ez;
61	int spread;
62
63	if (z == 0.0)
64		return (x * y);
65	if (x == 0.0 || y == 0.0)
66		return (x * y + z);
67
68	/* Results of frexp() are undefined for these cases. */
69	if (!isfinite(x) || !isfinite(y) || !isfinite(z))
70		return (x * y + z);
71
72	xs = frexp(x, &ex);
73	ys = frexp(y, &ey);
74	zs = frexp(z, &ez);
75	oround = fegetround();
76	spread = ex + ey - ez;
77
78	/*
79	 * If x * y and z are many orders of magnitude apart, the scaling
80	 * will overflow, so we handle these cases specially.  Rounding
81	 * modes other than FE_TONEAREST are painful.
82	 */
83	if (spread > DBL_MANT_DIG * 2) {
84		fenv_t env;
85		feraiseexcept(FE_INEXACT);
86		switch(oround) {
87		case FE_TONEAREST:
88			return (x * y);
89		case FE_TOWARDZERO:
90			if (x > 0.0 ^ y < 0.0 ^ z < 0.0)
91				return (x * y);
92			feholdexcept(&env);
93			r = x * y;
94			if (!fetestexcept(FE_INEXACT))
95				r = nextafter(r, 0);
96			feupdateenv(&env);
97			return (r);
98		case FE_DOWNWARD:
99			if (z > 0.0)
100				return (x * y);
101			feholdexcept(&env);
102			r = x * y;
103			if (!fetestexcept(FE_INEXACT))
104				r = nextafter(r, -INFINITY);
105			feupdateenv(&env);
106			return (r);
107		default:	/* FE_UPWARD */
108			if (z < 0.0)
109				return (x * y);
110			feholdexcept(&env);
111			r = x * y;
112			if (!fetestexcept(FE_INEXACT))
113				r = nextafter(r, INFINITY);
114			feupdateenv(&env);
115			return (r);
116		}
117	}
118	if (spread < -DBL_MANT_DIG) {
119		feraiseexcept(FE_INEXACT);
120		if (!isnormal(z))
121			feraiseexcept(FE_UNDERFLOW);
122		switch (oround) {
123		case FE_TONEAREST:
124			return (z);
125		case FE_TOWARDZERO:
126			if (x > 0.0 ^ y < 0.0 ^ z < 0.0)
127				return (z);
128			else
129				return (nextafter(z, 0));
130		case FE_DOWNWARD:
131			if (x > 0.0 ^ y < 0.0)
132				return (z);
133			else
134				return (nextafter(z, -INFINITY));
135		default:	/* FE_UPWARD */
136			if (x > 0.0 ^ y < 0.0)
137				return (nextafter(z, INFINITY));
138			else
139				return (z);
140		}
141	}
142
143	/*
144	 * Use Dekker's algorithm to perform the multiplication and
145	 * subsequent addition in twice the machine precision.
146	 * Arrange so that x * y = c + cc, and x * y + z = r + rr.
147	 */
148	fesetround(FE_TONEAREST);
149
150	p = xs * split;
151	hx = xs - p;
152	hx += p;
153	tx = xs - hx;
154
155	p = ys * split;
156	hy = ys - p;
157	hy += p;
158	ty = ys - hy;
159
160	p = hx * hy;
161	q = hx * ty + tx * hy;
162	c = p + q;
163	cc = p - c + q + tx * ty;
164
165	zs = ldexp(zs, -spread);
166	r = c + zs;
167	s = r - c;
168	rr = (c - (r - s)) + (zs - s) + cc;
169
170	spread = ex + ey;
171	if (spread + ilogb(r) > -1023) {
172		fesetround(oround);
173		r = r + rr;
174	} else {
175		/*
176		 * The result is subnormal, so we round before scaling to
177		 * avoid double rounding.
178		 */
179		p = ldexp(copysign(0x1p-1022, r), -spread);
180		c = r + p;
181		s = c - r;
182		cc = (r - (c - s)) + (p - s) + rr;
183		fesetround(oround);
184		r = (c + cc) - p;
185	}
186	return (ldexp(r, spread));
187}
188#else	/* LDBL_MANT_DIG == 113 */
189/*
190 * 113 bits of precision is more than twice the precision of a double,
191 * so it is enough to represent the intermediate product exactly.
192 */
193double
194fma(double x, double y, double z)
195{
196	return ((long double)x * y + z);
197}
198#endif	/* LDBL_MANT_DIG != 113 */
199
200#if (LDBL_MANT_DIG == 53)
201__weak_reference(fma, fmal);
202#endif
203