1/*	$OpenBSD: s_fmal.c,v 1.3 2013/11/12 19:00:38 martynas 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 */
42long double
43fmal(long double x, long double y, long double z)
44{
45#if LDBL_MANT_DIG == 64
46	static const long double split = 0x1p32L + 1.0;
47#elif LDBL_MANT_DIG == 113
48	static const long double split = 0x1p57L + 1.0;
49#endif
50	long double xs, ys, zs;
51	long double c, cc, hx, hy, p, q, tx, ty;
52	long double r, rr, s;
53	int oround;
54	int ex, ey, ez;
55	int spread;
56
57	/*
58	 * Handle special cases. The order of operations and the particular
59	 * return values here are crucial in handling special cases involving
60	 * infinities, NaNs, overflows, and signed zeroes correctly.
61	 */
62	if (x == 0.0 || y == 0.0)
63		return (x * y + z);
64	if (z == 0.0)
65		return (x * y);
66	if (!isfinite(x) || !isfinite(y))
67		return (x * y + z);
68	if (!isfinite(z))
69		return (z);
70
71	xs = frexpl(x, &ex);
72	ys = frexpl(y, &ey);
73	zs = frexpl(z, &ez);
74	oround = fegetround();
75	spread = ex + ey - ez;
76
77	/*
78	 * If x * y and z are many orders of magnitude apart, the scaling
79	 * will overflow, so we handle these cases specially.  Rounding
80	 * modes other than FE_TONEAREST are painful.
81	 */
82	if (spread > LDBL_MANT_DIG * 2) {
83		fenv_t env;
84		feraiseexcept(FE_INEXACT);
85		switch(oround) {
86		case FE_TONEAREST:
87			return (x * y);
88		case FE_TOWARDZERO:
89			if ((x > 0.0) ^ (y < 0.0) ^ (z < 0.0))
90				return (x * y);
91			feholdexcept(&env);
92			r = x * y;
93			if (!fetestexcept(FE_INEXACT))
94				r = nextafterl(r, 0);
95			feupdateenv(&env);
96			return (r);
97		case FE_DOWNWARD:
98			if (z > 0.0)
99				return (x * y);
100			feholdexcept(&env);
101			r = x * y;
102			if (!fetestexcept(FE_INEXACT))
103				r = nextafterl(r, -INFINITY);
104			feupdateenv(&env);
105			return (r);
106		default:	/* FE_UPWARD */
107			if (z < 0.0)
108				return (x * y);
109			feholdexcept(&env);
110			r = x * y;
111			if (!fetestexcept(FE_INEXACT))
112				r = nextafterl(r, INFINITY);
113			feupdateenv(&env);
114			return (r);
115		}
116	}
117	if (spread < -LDBL_MANT_DIG) {
118		feraiseexcept(FE_INEXACT);
119		if (!isnormal(z))
120			feraiseexcept(FE_UNDERFLOW);
121		switch (oround) {
122		case FE_TONEAREST:
123			return (z);
124		case FE_TOWARDZERO:
125			if ((x > 0.0) ^ (y < 0.0) ^ (z < 0.0))
126				return (z);
127			else
128				return (nextafterl(z, 0));
129		case FE_DOWNWARD:
130			if ((x > 0.0) ^ (y < 0.0))
131				return (z);
132			else
133				return (nextafterl(z, -INFINITY));
134		default:	/* FE_UPWARD */
135			if ((x > 0.0) ^ (y < 0.0))
136				return (nextafterl(z, INFINITY));
137			else
138				return (z);
139		}
140	}
141
142	/*
143	 * Use Dekker's algorithm to perform the multiplication and
144	 * subsequent addition in twice the machine precision.
145	 * Arrange so that x * y = c + cc, and x * y + z = r + rr.
146	 */
147	fesetround(FE_TONEAREST);
148
149	p = xs * split;
150	hx = xs - p;
151	hx += p;
152	tx = xs - hx;
153
154	p = ys * split;
155	hy = ys - p;
156	hy += p;
157	ty = ys - hy;
158
159	p = hx * hy;
160	q = hx * ty + tx * hy;
161	c = p + q;
162	cc = p - c + q + tx * ty;
163
164	zs = ldexpl(zs, -spread);
165	r = c + zs;
166	s = r - c;
167	rr = (c - (r - s)) + (zs - s) + cc;
168
169	spread = ex + ey;
170	if (spread + ilogbl(r) > -16383) {
171		fesetround(oround);
172		r = r + rr;
173	} else {
174		/*
175		 * The result is subnormal, so we round before scaling to
176		 * avoid double rounding.
177		 */
178		p = ldexpl(copysignl(0x1p-16382L, r), -spread);
179		c = r + p;
180		s = c - r;
181		cc = (r - (c - s)) + (p - s) + rr;
182		fesetround(oround);
183		r = (c + cc) - p;
184	}
185	return (ldexpl(r, spread));
186}
187