1/*	$NetBSD: s_fmal.c,v 1.4 2017/05/06 18:02:52 christos 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_fmal.c,v 1.7 2011/10/21 06:30:43 das Exp $");
32#else
33__RCSID("$NetBSD: s_fmal.c,v 1.4 2017/05/06 18:02:52 christos Exp $");
34#endif
35
36#include "namespace.h"
37
38#include <machine/ieee.h>
39#include <fenv.h>
40#include <float.h>
41#include <math.h>
42
43#include "math_private.h"
44
45#ifdef __HAVE_LONG_DOUBLE
46/*
47 * A struct dd represents a floating-point number with twice the precision
48 * of a long double.  We maintain the invariant that "hi" stores the high-order
49 * bits of the result.
50 */
51struct dd {
52	long double hi;
53	long double lo;
54};
55
56/*
57 * Compute a+b exactly, returning the exact result in a struct dd.  We assume
58 * that both a and b are finite, but make no assumptions about their relative
59 * magnitudes.
60 */
61static inline struct dd
62dd_add(long double a, long double b)
63{
64	struct dd ret;
65	long double s;
66
67	ret.hi = a + b;
68	s = ret.hi - a;
69	ret.lo = (a - (ret.hi - s)) + (b - s);
70	return (ret);
71}
72
73/*
74 * Compute a+b, with a small tweak:  The least significant bit of the
75 * result is adjusted into a sticky bit summarizing all the bits that
76 * were lost to rounding.  This adjustment negates the effects of double
77 * rounding when the result is added to another number with a higher
78 * exponent.  For an explanation of round and sticky bits, see any reference
79 * on FPU design, e.g.,
80 *
81 *     J. Coonen.  An Implementation Guide to a Proposed Standard for
82 *     Floating-Point Arithmetic.  Computer, vol. 13, no. 1, Jan 1980.
83 */
84static inline long double
85add_adjusted(long double a, long double b)
86{
87	struct dd sum;
88	union ieee_ext_u u;
89
90	sum = dd_add(a, b);
91	if (sum.lo != 0) {
92		u.extu_ld = sum.hi;
93		if ((u.extu_ext.ext_fracl & 1) == 0)
94			sum.hi = nextafterl(sum.hi, INFINITY * sum.lo);
95	}
96	return (sum.hi);
97}
98
99/*
100 * Compute ldexp(a+b, scale) with a single rounding error. It is assumed
101 * that the result will be subnormal, and care is taken to ensure that
102 * double rounding does not occur.
103 */
104static inline long double
105add_and_denormalize(long double a, long double b, int scale)
106{
107	struct dd sum;
108	int bits_lost;
109	union ieee_ext_u u;
110
111	sum = dd_add(a, b);
112
113	/*
114	 * If we are losing at least two bits of accuracy to denormalization,
115	 * then the first lost bit becomes a round bit, and we adjust the
116	 * lowest bit of sum.hi to make it a sticky bit summarizing all the
117	 * bits in sum.lo. With the sticky bit adjusted, the hardware will
118	 * break any ties in the correct direction.
119	 *
120	 * If we are losing only one bit to denormalization, however, we must
121	 * break the ties manually.
122	 */
123	if (sum.lo != 0) {
124		u.extu_ld = sum.hi;
125		bits_lost = -u.extu_ext.ext_exp - scale + 1;
126		if ((bits_lost != 1) ^ (int)(u.extu_ext.ext_fracl & 1))
127			sum.hi = nextafterl(sum.hi, INFINITY * sum.lo);
128	}
129	return (ldexp((double)sum.hi, scale));
130}
131
132/*
133 * Compute a*b exactly, returning the exact result in a struct dd.  We assume
134 * that both a and b are normalized, so no underflow or overflow will occur.
135 * The current rounding mode must be round-to-nearest.
136 */
137static inline struct dd
138dd_mul(long double a, long double b)
139{
140#if LDBL_MANT_DIG == 64
141	static const long double split = 0x1p32L + 1.0;
142#elif LDBL_MANT_DIG == 113
143	static const long double split = 0x1p57L + 1.0;
144#endif
145	struct dd ret;
146	long double ha, hb, la, lb, p, q;
147
148	p = a * split;
149	ha = a - p;
150	ha += p;
151	la = a - ha;
152
153	p = b * split;
154	hb = b - p;
155	hb += p;
156	lb = b - hb;
157
158	p = ha * hb;
159	q = ha * lb + la * hb;
160
161	ret.hi = p + q;
162	ret.lo = p - ret.hi + q + la * lb;
163	return (ret);
164}
165
166/*
167 * Fused multiply-add: Compute x * y + z with a single rounding error.
168 *
169 * We use scaling to avoid overflow/underflow, along with the
170 * canonical precision-doubling technique adapted from:
171 *
172 *	Dekker, T.  A Floating-Point Technique for Extending the
173 *	Available Precision.  Numer. Math. 18, 224-242 (1971).
174 */
175long double
176fmal(long double x, long double y, long double z)
177{
178	long double xs, ys, zs, adj;
179	struct dd xy, r;
180	int oround;
181	int ex, ey, ez;
182	int spread;
183
184	/*
185	 * Handle special cases. The order of operations and the particular
186	 * return values here are crucial in handling special cases involving
187	 * infinities, NaNs, overflows, and signed zeroes correctly.
188	 */
189	if (x == 0.0 || y == 0.0)
190		return (x * y + z);
191	if (z == 0.0)
192		return (x * y);
193	if (!isfinite(x) || !isfinite(y))
194		return (x * y + z);
195	if (!isfinite(z))
196		return (z);
197
198	xs = frexpl(x, &ex);
199	ys = frexpl(y, &ey);
200	zs = frexpl(z, &ez);
201	oround = fegetround();
202	spread = ex + ey - ez;
203
204	/*
205	 * If x * y and z are many orders of magnitude apart, the scaling
206	 * will overflow, so we handle these cases specially.  Rounding
207	 * modes other than FE_TONEAREST are painful.
208	 */
209	if (spread < -LDBL_MANT_DIG) {
210		feraiseexcept(FE_INEXACT);
211		if (!isnormal(z))
212			feraiseexcept(FE_UNDERFLOW);
213		switch (oround) {
214		case FE_TONEAREST:
215			return (z);
216		case FE_TOWARDZERO:
217			if ((x > 0.0) ^ (y < 0.0) ^ (z < 0.0))
218				return (z);
219			else
220				return (nextafterl(z, 0));
221		case FE_DOWNWARD:
222			if ((x > 0.0) ^ (y < 0.0))
223				return (z);
224			else
225				return (nextafterl(z, (long double)-INFINITY));
226		default:	/* FE_UPWARD */
227			if ((x > 0.0) ^ (y < 0.0))
228				return (nextafterl(z, (long double)INFINITY));
229			else
230				return (z);
231		}
232	}
233	if (spread <= LDBL_MANT_DIG * 2)
234		zs = ldexpl(zs, -spread);
235	else
236		zs = copysignl(LDBL_MIN, zs);
237
238	fesetround(FE_TONEAREST);
239
240	/*
241	 * Basic approach for round-to-nearest:
242	 *
243	 *     (xy.hi, xy.lo) = x * y		(exact)
244	 *     (r.hi, r.lo)   = xy.hi + z	(exact)
245	 *     adj = xy.lo + r.lo		(inexact; low bit is sticky)
246	 *     result = r.hi + adj		(correctly rounded)
247	 */
248	xy = dd_mul(xs, ys);
249	r = dd_add(xy.hi, zs);
250
251	spread = ex + ey;
252
253	if (r.hi == 0.0) {
254		/*
255		 * When the addends cancel to 0, ensure that the result has
256		 * the correct sign.
257		 */
258		fesetround(oround);
259		{
260		volatile long double vzs = zs; /* XXX gcc CSE bug workaround */
261		return (xy.hi + vzs + ldexpl(xy.lo, spread));
262		}
263	}
264
265	if (oround != FE_TONEAREST) {
266		/*
267		 * There is no need to worry about double rounding in directed
268		 * rounding modes.
269		 */
270		fesetround(oround);
271		adj = r.lo + xy.lo;
272		return (ldexpl(r.hi + adj, spread));
273	}
274
275	adj = add_adjusted(r.lo, xy.lo);
276	if (spread + ilogbl(r.hi) > -16383)
277		return (ldexpl(r.hi + adj, spread));
278	else
279		return (add_and_denormalize(r.hi, adj, spread));
280}
281#endif
282