2 * Copyright (c) 2005 David Schultz <das@FreeBSD.ORG>
5 * Redistribution and use in source and binary forms, with or without
6 * modification, are permitted provided that the following conditions
8 * 1. Redistributions of source code must retain the above copyright
9 * notice, this list of conditions and the following disclaimer.
10 * 2. Redistributions in binary form must reproduce the above copyright
11 * notice, this list of conditions and the following disclaimer in the
12 * documentation and/or other materials provided with the distribution.
14 * THIS SOFTWARE IS PROVIDED BY THE AUTHOR AND CONTRIBUTORS ``AS IS'' AND
15 * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
16 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
17 * ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR OR CONTRIBUTORS BE LIABLE
18 * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
19 * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
20 * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
21 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
22 * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
23 * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
27 #include <sys/cdefs.h>
28 __FBSDID("$FreeBSD$");
35 * Fused multiply-add: Compute x * y + z with a single rounding error.
37 * We use scaling to avoid overflow/underflow, along with the
38 * canonical precision-doubling technique adapted from:
40 * Dekker, T. A Floating-Point Technique for Extending the
41 * Available Precision. Numer. Math. 18, 224-242 (1971).
44 fmal(long double x, long double y, long double z)
46 #if LDBL_MANT_DIG == 64
47 static const long double split = 0x1p32L + 1.0;
48 #elif LDBL_MANT_DIG == 113
49 static const long double split = 0x1p57L + 1.0;
51 long double xs, ys, zs;
52 long double c, cc, hx, hy, p, q, tx, ty;
60 if (x == 0.0 || y == 0.0)
63 /* Results of frexp() are undefined for these cases. */
64 if (!isfinite(x) || !isfinite(y) || !isfinite(z))
70 oround = fegetround();
71 spread = ex + ey - ez;
74 * If x * y and z are many orders of magnitude apart, the scaling
75 * will overflow, so we handle these cases specially. Rounding
76 * modes other than FE_TONEAREST are painful.
78 if (spread > LDBL_MANT_DIG * 2) {
80 feraiseexcept(FE_INEXACT);
85 if (x > 0.0 ^ y < 0.0 ^ z < 0.0)
89 if (!fetestexcept(FE_INEXACT))
98 if (!fetestexcept(FE_INEXACT))
99 r = nextafterl(r, -INFINITY);
102 default: /* FE_UPWARD */
107 if (!fetestexcept(FE_INEXACT))
108 r = nextafterl(r, INFINITY);
113 if (spread < -LDBL_MANT_DIG) {
114 feraiseexcept(FE_INEXACT);
116 feraiseexcept(FE_UNDERFLOW);
121 if (x > 0.0 ^ y < 0.0 ^ z < 0.0)
124 return (nextafterl(z, 0));
126 if (x > 0.0 ^ y < 0.0)
129 return (nextafterl(z, -INFINITY));
130 default: /* FE_UPWARD */
131 if (x > 0.0 ^ y < 0.0)
132 return (nextafterl(z, INFINITY));
139 * Use Dekker's algorithm to perform the multiplication and
140 * subsequent addition in twice the machine precision.
141 * Arrange so that x * y = c + cc, and x * y + z = r + rr.
143 fesetround(FE_TONEAREST);
156 q = hx * ty + tx * hy;
158 cc = p - c + q + tx * ty;
160 zs = ldexpl(zs, -spread);
163 rr = (c - (r - s)) + (zs - s) + cc;
166 if (spread + ilogbl(r) > -16383) {
171 * The result is subnormal, so we round before scaling to
172 * avoid double rounding.
174 p = ldexpl(copysignl(0x1p-16382L, r), -spread);
177 cc = (r - (c - s)) + (p - s) + rr;
181 return (ldexpl(r, spread));