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 */
42 long double
fmal(long double x,long double y,long double z)43 fmal(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