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 */
51 struct 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 */
61 static inline struct dd
dd_add(long double a,long double b)62 dd_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 */
84 static inline long double
add_adjusted(long double a,long double b)85 add_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 */
104 static inline long double
add_and_denormalize(long double a,long double b,int scale)105 add_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 */
137 static inline struct dd
dd_mul(long double a,long double b)138 dd_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 */
175 long double
fmal(long double x,long double y,long double z)176 fmal(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