1 /* $NetBSD: s_fmal.c,v 1.3 2013/02/12 21:40:19 martin 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 1.3 2013/02/12 21:40:19 martin 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 #ifdef __HAVE_LONG_DOUBLE
44 /*
45 * A struct dd represents a floating-point number with twice the precision
46 * of a long double. We maintain the invariant that "hi" stores the high-order
47 * bits of the result.
48 */
49 struct dd {
50 long double hi;
51 long double lo;
52 };
53
54 /*
55 * Compute a+b exactly, returning the exact result in a struct dd. We assume
56 * that both a and b are finite, but make no assumptions about their relative
57 * magnitudes.
58 */
59 static inline struct dd
dd_add(long double a,long double b)60 dd_add(long double a, long double b)
61 {
62 struct dd ret;
63 long double s;
64
65 ret.hi = a + b;
66 s = ret.hi - a;
67 ret.lo = (a - (ret.hi - s)) + (b - s);
68 return (ret);
69 }
70
71 /*
72 * Compute a+b, with a small tweak: The least significant bit of the
73 * result is adjusted into a sticky bit summarizing all the bits that
74 * were lost to rounding. This adjustment negates the effects of double
75 * rounding when the result is added to another number with a higher
76 * exponent. For an explanation of round and sticky bits, see any reference
77 * on FPU design, e.g.,
78 *
79 * J. Coonen. An Implementation Guide to a Proposed Standard for
80 * Floating-Point Arithmetic. Computer, vol. 13, no. 1, Jan 1980.
81 */
82 static inline long double
add_adjusted(long double a,long double b)83 add_adjusted(long double a, long double b)
84 {
85 struct dd sum;
86 union ieee_ext_u u;
87
88 sum = dd_add(a, b);
89 if (sum.lo != 0) {
90 u.extu_ld = sum.hi;
91 if ((u.extu_ext.ext_fracl & 1) == 0)
92 sum.hi = nextafterl(sum.hi, INFINITY * sum.lo);
93 }
94 return (sum.hi);
95 }
96
97 /*
98 * Compute ldexp(a+b, scale) with a single rounding error. It is assumed
99 * that the result will be subnormal, and care is taken to ensure that
100 * double rounding does not occur.
101 */
102 static inline long double
add_and_denormalize(long double a,long double b,int scale)103 add_and_denormalize(long double a, long double b, int scale)
104 {
105 struct dd sum;
106 int bits_lost;
107 union ieee_ext_u u;
108
109 sum = dd_add(a, b);
110
111 /*
112 * If we are losing at least two bits of accuracy to denormalization,
113 * then the first lost bit becomes a round bit, and we adjust the
114 * lowest bit of sum.hi to make it a sticky bit summarizing all the
115 * bits in sum.lo. With the sticky bit adjusted, the hardware will
116 * break any ties in the correct direction.
117 *
118 * If we are losing only one bit to denormalization, however, we must
119 * break the ties manually.
120 */
121 if (sum.lo != 0) {
122 u.extu_ld = sum.hi;
123 bits_lost = -u.extu_ext.ext_exp - scale + 1;
124 if ((bits_lost != 1) ^ (int)(u.extu_ext.ext_fracl & 1))
125 sum.hi = nextafterl(sum.hi, INFINITY * sum.lo);
126 }
127 return (ldexp((double)sum.hi, scale));
128 }
129
130 /*
131 * Compute a*b exactly, returning the exact result in a struct dd. We assume
132 * that both a and b are normalized, so no underflow or overflow will occur.
133 * The current rounding mode must be round-to-nearest.
134 */
135 static inline struct dd
dd_mul(long double a,long double b)136 dd_mul(long double a, long double b)
137 {
138 #if LDBL_MANT_DIG == 64
139 static const long double split = 0x1p32L + 1.0;
140 #elif LDBL_MANT_DIG == 113
141 static const long double split = 0x1p57L + 1.0;
142 #endif
143 struct dd ret;
144 long double ha, hb, la, lb, p, q;
145
146 p = a * split;
147 ha = a - p;
148 ha += p;
149 la = a - ha;
150
151 p = b * split;
152 hb = b - p;
153 hb += p;
154 lb = b - hb;
155
156 p = ha * hb;
157 q = ha * lb + la * hb;
158
159 ret.hi = p + q;
160 ret.lo = p - ret.hi + q + la * lb;
161 return (ret);
162 }
163
164 /*
165 * Fused multiply-add: Compute x * y + z with a single rounding error.
166 *
167 * We use scaling to avoid overflow/underflow, along with the
168 * canonical precision-doubling technique adapted from:
169 *
170 * Dekker, T. A Floating-Point Technique for Extending the
171 * Available Precision. Numer. Math. 18, 224-242 (1971).
172 */
173 long double
fmal(long double x,long double y,long double z)174 fmal(long double x, long double y, long double z)
175 {
176 long double xs, ys, zs, adj;
177 struct dd xy, r;
178 int oround;
179 int ex, ey, ez;
180 int spread;
181
182 /*
183 * Handle special cases. The order of operations and the particular
184 * return values here are crucial in handling special cases involving
185 * infinities, NaNs, overflows, and signed zeroes correctly.
186 */
187 if (x == 0.0 || y == 0.0)
188 return (x * y + z);
189 if (z == 0.0)
190 return (x * y);
191 if (!isfinite(x) || !isfinite(y))
192 return (x * y + z);
193 if (!isfinite(z))
194 return (z);
195
196 xs = frexpl(x, &ex);
197 ys = frexpl(y, &ey);
198 zs = frexpl(z, &ez);
199 oround = fegetround();
200 spread = ex + ey - ez;
201
202 /*
203 * If x * y and z are many orders of magnitude apart, the scaling
204 * will overflow, so we handle these cases specially. Rounding
205 * modes other than FE_TONEAREST are painful.
206 */
207 if (spread < -LDBL_MANT_DIG) {
208 feraiseexcept(FE_INEXACT);
209 if (!isnormal(z))
210 feraiseexcept(FE_UNDERFLOW);
211 switch (oround) {
212 case FE_TONEAREST:
213 return (z);
214 case FE_TOWARDZERO:
215 if ((x > 0.0) ^ (y < 0.0) ^ (z < 0.0))
216 return (z);
217 else
218 return (nextafterl(z, 0));
219 case FE_DOWNWARD:
220 if ((x > 0.0) ^ (y < 0.0))
221 return (z);
222 else
223 return (nextafterl(z, (long double)-INFINITY));
224 default: /* FE_UPWARD */
225 if ((x > 0.0) ^ (y < 0.0))
226 return (nextafterl(z, (long double)INFINITY));
227 else
228 return (z);
229 }
230 }
231 if (spread <= LDBL_MANT_DIG * 2)
232 zs = ldexpl(zs, -spread);
233 else
234 zs = copysignl(LDBL_MIN, zs);
235
236 fesetround(FE_TONEAREST);
237
238 /*
239 * Basic approach for round-to-nearest:
240 *
241 * (xy.hi, xy.lo) = x * y (exact)
242 * (r.hi, r.lo) = xy.hi + z (exact)
243 * adj = xy.lo + r.lo (inexact; low bit is sticky)
244 * result = r.hi + adj (correctly rounded)
245 */
246 xy = dd_mul(xs, ys);
247 r = dd_add(xy.hi, zs);
248
249 spread = ex + ey;
250
251 if (r.hi == 0.0) {
252 /*
253 * When the addends cancel to 0, ensure that the result has
254 * the correct sign.
255 */
256 fesetround(oround);
257 {
258 volatile long double vzs = zs; /* XXX gcc CSE bug workaround */
259 return (xy.hi + vzs + ldexpl(xy.lo, spread));
260 }
261 }
262
263 if (oround != FE_TONEAREST) {
264 /*
265 * There is no need to worry about double rounding in directed
266 * rounding modes.
267 */
268 fesetround(oround);
269 adj = r.lo + xy.lo;
270 return (ldexpl(r.hi + adj, spread));
271 }
272
273 adj = add_adjusted(r.lo, xy.lo);
274 if (spread + ilogbl(r.hi) > -16383)
275 return (ldexpl(r.hi + adj, spread));
276 else
277 return (add_and_denormalize(r.hi, adj, spread));
278 }
279 #endif
280