/*-
* Copyright (c) 2005-2011 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("$FreeBSD$");
#include <fenv.h>
#include <float.h>
#include <math.h>
#include "math_private.h"
/*
* A struct dd represents a floating-point number with twice the precision
* of a double. We maintain the invariant that "hi" stores the 53 high-order
* bits of the result.
*/
struct dd {
double hi;
double lo;
};
/*
* Compute a+b exactly, returning the exact result in a struct dd. We assume
* that both a and b are finite, but make no assumptions about their relative
* magnitudes.
*/
static inline struct dd
dd_add(double a, double b)
{
struct dd ret;
double s;
ret.hi = a + b;
s = ret.hi - a;
ret.lo = (a - (ret.hi - s)) + (b - s);
return (ret);
}
/*
* Compute a+b, with a small tweak: The least significant bit of the
* result is adjusted into a sticky bit summarizing all the bits that
* were lost to rounding. This adjustment negates the effects of double
* rounding when the result is added to another number with a higher
* exponent. For an explanation of round and sticky bits, see any reference
* on FPU design, e.g.,
*
* J. Coonen. An Implementation Guide to a Proposed Standard for
* Floating-Point Arithmetic. Computer, vol. 13, no. 1, Jan 1980.
*/
static inline double
add_adjusted(double a, double b)
{
struct dd sum;
uint64_t hibits, lobits;
sum = dd_add(a, b);
if (sum.lo != 0) {
EXTRACT_WORD64(hibits, sum.hi);
if ((hibits & 1) == 0) {
/* hibits += (int)copysign(1.0, sum.hi * sum.lo) */
EXTRACT_WORD64(lobits, sum.lo);
hibits += 1 - ((hibits ^ lobits) >> 62);
INSERT_WORD64(sum.hi, hibits);
}
}
return (sum.hi);
}
/*
* Compute ldexp(a+b, scale) with a single rounding error. It is assumed
* that the result will be subnormal, and care is taken to ensure that
* double rounding does not occur.
*/
static inline double
add_and_denormalize(double a, double b, int scale)
{
struct dd sum;
uint64_t hibits, lobits;
int bits_lost;
sum = dd_add(a, b);
/*
* If we are losing at least two bits of accuracy to denormalization,
* then the first lost bit becomes a round bit, and we adjust the
* lowest bit of sum.hi to make it a sticky bit summarizing all the
* bits in sum.lo. With the sticky bit adjusted, the hardware will
* break any ties in the correct direction.
*
* If we are losing only one bit to denormalization, however, we must
* break the ties manually.
*/
if (sum.lo != 0) {
EXTRACT_WORD64(hibits, sum.hi);
bits_lost = -((int)(hibits >> 52) & 0x7ff) - scale + 1;
if ((bits_lost != 1) ^ (int)(hibits & 1)) {
/* hibits += (int)copysign(1.0, sum.hi * sum.lo) */
EXTRACT_WORD64(lobits, sum.lo);
hibits += 1 - (((hibits ^ lobits) >> 62) & 2);
INSERT_WORD64(sum.hi, hibits);
}
}
return (ldexp(sum.hi, scale));
}
/*
* Compute a*b exactly, returning the exact result in a struct dd. We assume
* that both a and b are normalized, so no underflow or overflow will occur.
* The current rounding mode must be round-to-nearest.
*/
static inline struct dd
dd_mul(double a, double b)
{
static const double split = 0x1p27 + 1.0;
struct dd ret;
double ha, hb, la, lb, p, q;
p = a * split;
ha = a - p;
ha += p;
la = a - ha;
p = b * split;
hb = b - p;
hb += p;
lb = b - hb;
p = ha * hb;
q = ha * lb + la * hb;
ret.hi = p + q;
ret.lo = p - ret.hi + q + la * lb;
return (ret);
}
/*
* 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).
*
* This algorithm is sensitive to the rounding precision. FPUs such
* as the i387 must be set in double-precision mode if variables are
* to be stored in FP registers in order to avoid incorrect results.
* This is the default on FreeBSD, but not on many other systems.
*
* Hardware instructions should be used on architectures that support it,
* since this implementation will likely be several times slower.
*/
double
fma(double x, double y, double z)
{
double xs, ys, zs, adj;
struct dd xy, r;
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 = frexp(x, &ex);
ys = frexp(y, &ey);
zs = frexp(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 < -DBL_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 (nextafter(z, 0));
case FE_DOWNWARD:
if (x > 0.0 ^ y < 0.0)
return (z);
else
return (nextafter(z, -INFINITY));
default: /* FE_UPWARD */
if (x > 0.0 ^ y < 0.0)
return (nextafter(z, INFINITY));
else
return (z);
}
}
if (spread <= DBL_MANT_DIG * 2)
zs = ldexp(zs, -spread);
else
zs = copysign(DBL_MIN, zs);
fesetround(FE_TONEAREST);
/* work around clang bug 8100 */
volatile double vxs = xs;
/*
* Basic approach for round-to-nearest:
*
* (xy.hi, xy.lo) = x * y (exact)
* (r.hi, r.lo) = xy.hi + z (exact)
* adj = xy.lo + r.lo (inexact; low bit is sticky)
* result = r.hi + adj (correctly rounded)
*/
xy = dd_mul(vxs, ys);
r = dd_add(xy.hi, zs);
spread = ex + ey;
if (r.hi == 0.0) {
/*
* When the addends cancel to 0, ensure that the result has
* the correct sign.
*/
fesetround(oround);
volatile double vzs = zs; /* XXX gcc CSE bug workaround */
return (xy.hi + vzs + ldexp(xy.lo, spread));
}
if (oround != FE_TONEAREST) {
/*
* There is no need to worry about double rounding in directed
* rounding modes.
*/
fesetround(oround);
/* work around clang bug 8100 */
volatile double vrlo = r.lo;
adj = vrlo + xy.lo;
return (ldexp(r.hi + adj, spread));
}
adj = add_adjusted(r.lo, xy.lo);
if (spread + ilogb(r.hi) > -1023)
return (ldexp(r.hi + adj, spread));
else
return (add_and_denormalize(r.hi, adj, spread));
}
#if (LDBL_MANT_DIG == 53)
__weak_reference(fma, fmal);
#endif