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/*-
* Copyright (c) 2005 David Schultz <das@FreeBSD.ORG>
* All rights reserved.
*
* Redistribution and use in source and binary forms, with or without
* modification, are permitted provided that the following conditions
* are met:
* 1. Redistributions of source code must retain the above copyright
* notice, this list of conditions and the following disclaimer.
* 2. Redistributions in binary form must reproduce the above copyright
* notice, this list of conditions and the following disclaimer in the
* documentation and/or other materials provided with the distribution.
*
* THIS SOFTWARE IS PROVIDED BY THE AUTHOR AND CONTRIBUTORS ``AS IS'' AND
* ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
* IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
* ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR OR CONTRIBUTORS BE LIABLE
* FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
* DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
* OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
* HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
* LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
* OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
* SUCH DAMAGE.
*/
#include <sys/cdefs.h>
__FBSDID("$FreeBSD: src/lib/msun/src/s_fmal.c,v 1.4.2.1.2.1 2009/10/25 01:10:29 kensmith Exp $");
#include <fenv.h>
#include <float.h>
#include <math.h>
/*
* Fused multiply-add: Compute x * y + z with a single rounding error.
*
* We use scaling to avoid overflow/underflow, along with the
* canonical precision-doubling technique adapted from:
*
* Dekker, T. A Floating-Point Technique for Extending the
* Available Precision. Numer. Math. 18, 224-242 (1971).
*/
long double
fmal(long double x, long double y, long double z)
{
#if LDBL_MANT_DIG == 64
static const long double split = 0x1p32L + 1.0;
#elif LDBL_MANT_DIG == 113
static const long double split = 0x1p57L + 1.0;
#endif
long double xs, ys, zs;
long double c, cc, hx, hy, p, q, tx, ty;
long double r, rr, s;
int oround;
int ex, ey, ez;
int spread;
/*
* Handle special cases. The order of operations and the particular
* return values here are crucial in handling special cases involving
* infinities, NaNs, overflows, and signed zeroes correctly.
*/
if (x == 0.0 || y == 0.0)
return (x * y + z);
if (z == 0.0)
return (x * y);
if (!isfinite(x) || !isfinite(y))
return (x * y + z);
if (!isfinite(z))
return (z);
xs = frexpl(x, &ex);
ys = frexpl(y, &ey);
zs = frexpl(z, &ez);
oround = fegetround();
spread = ex + ey - ez;
/*
* If x * y and z are many orders of magnitude apart, the scaling
* will overflow, so we handle these cases specially. Rounding
* modes other than FE_TONEAREST are painful.
*/
if (spread > LDBL_MANT_DIG * 2) {
fenv_t env;
feraiseexcept(FE_INEXACT);
switch(oround) {
case FE_TONEAREST:
return (x * y);
case FE_TOWARDZERO:
if (x > 0.0 ^ y < 0.0 ^ z < 0.0)
return (x * y);
feholdexcept(&env);
r = x * y;
if (!fetestexcept(FE_INEXACT))
r = nextafterl(r, 0);
feupdateenv(&env);
return (r);
case FE_DOWNWARD:
if (z > 0.0)
return (x * y);
feholdexcept(&env);
r = x * y;
if (!fetestexcept(FE_INEXACT))
r = nextafterl(r, -INFINITY);
feupdateenv(&env);
return (r);
default: /* FE_UPWARD */
if (z < 0.0)
return (x * y);
feholdexcept(&env);
r = x * y;
if (!fetestexcept(FE_INEXACT))
r = nextafterl(r, INFINITY);
feupdateenv(&env);
return (r);
}
}
if (spread < -LDBL_MANT_DIG) {
feraiseexcept(FE_INEXACT);
if (!isnormal(z))
feraiseexcept(FE_UNDERFLOW);
switch (oround) {
case FE_TONEAREST:
return (z);
case FE_TOWARDZERO:
if (x > 0.0 ^ y < 0.0 ^ z < 0.0)
return (z);
else
return (nextafterl(z, 0));
case FE_DOWNWARD:
if (x > 0.0 ^ y < 0.0)
return (z);
else
return (nextafterl(z, -INFINITY));
default: /* FE_UPWARD */
if (x > 0.0 ^ y < 0.0)
return (nextafterl(z, INFINITY));
else
return (z);
}
}
/*
* Use Dekker's algorithm to perform the multiplication and
* subsequent addition in twice the machine precision.
* Arrange so that x * y = c + cc, and x * y + z = r + rr.
*/
fesetround(FE_TONEAREST);
p = xs * split;
hx = xs - p;
hx += p;
tx = xs - hx;
p = ys * split;
hy = ys - p;
hy += p;
ty = ys - hy;
p = hx * hy;
q = hx * ty + tx * hy;
c = p + q;
cc = p - c + q + tx * ty;
zs = ldexpl(zs, -spread);
r = c + zs;
s = r - c;
rr = (c - (r - s)) + (zs - s) + cc;
spread = ex + ey;
if (spread + ilogbl(r) > -16383) {
fesetround(oround);
r = r + rr;
} else {
/*
* The result is subnormal, so we round before scaling to
* avoid double rounding.
*/
p = ldexpl(copysignl(0x1p-16382L, r), -spread);
c = r + p;
s = c - r;
cc = (r - (c - s)) + (p - s) + rr;
fesetround(oround);
r = (c + cc) - p;
}
return (ldexpl(r, spread));
}
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