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/*
** Copyright (C) 2018 Martin Brain
**
** See the file LICENSE for licensing information.
*/
/*
** compare.h
**
** Martin Brain
** martin.brain@cs.ox.ac.uk
** 25/08/14
**
** Comparison between floating-point numbers
**
*/
#include "symfpu/core/unpackedFloat.h"
#include "symfpu/core/ite.h"
#ifndef SYMFPU_COMPARE
#define SYMFPU_COMPARE
namespace symfpu {
// SMT-LIB equality
template <class t>
typename t::prop smtlibEqual (const typename t::fpt &format,
const unpackedFloat<t> &left,
const unpackedFloat<t> &right) {
typedef typename t::prop prop;
PRECONDITION(left.valid(format));
PRECONDITION(right.valid(format));
// Relies on a number of properties of the unpacked format
// particularly the use of default exponents, significands and signs
prop flagsEqual((left.getNaN() == right.getNaN()) &&
(left.getInf() == right.getInf()) &&
(left.getZero() == right.getZero()) &&
(left.getSign() == right.getSign()));
prop flagsAndExponent(flagsEqual && left.getExponent() == right.getExponent());
// Avoid comparing (and thus instantiating) the significand unless necessary
probabilityAnnotation<t,prop>(flagsAndExponent, UNLIKELY);
prop res(ITE(flagsAndExponent,
left.getSignificand() == right.getSignificand(),
prop(false)));
return res;
}
// IEEE-754 Equality (not actually an equivalence relation but ...)
template <class t>
typename t::prop ieee754Equal (const typename t::fpt &format,
const unpackedFloat<t> &left,
const unpackedFloat<t> &right) {
typedef typename t::prop prop;
PRECONDITION(left.valid(format));
PRECONDITION(right.valid(format));
prop neitherNan(!left.getNaN() && !right.getNaN()); // All comparison with NaN are false
prop bothZero(left.getZero() && right.getZero()); // Both zeros are equal
prop neitherZero(!left.getZero() && !right.getZero());
prop flagsAndExponent(neitherNan &&
(bothZero || (neitherZero &&
(left.getInf() == right.getInf() &&
left.getSign() == right.getSign() &&
left.getExponent() == right.getExponent()))));
// Avoid comparing (and thus instantiating) the significand unless necessary
probabilityAnnotation<t,prop>(flagsAndExponent, UNLIKELY);
prop res(ITE(flagsAndExponent, left.getSignificand() == right.getSignificand(), prop(false)));
return res;
}
// Share the common comparison code between functions
// equality == true if the equal case returns true
// IEEE-754 semantics for ordering with NaN
// (i.e. unordered with everything, not even equal to itself)
template <class t>
typename t::prop ordering (const typename t::fpt &format,
const unpackedFloat<t> &left,
const unpackedFloat<t> &right,
const typename t::prop equality) {
typedef typename t::prop prop;
PRECONDITION(left.valid(format));
PRECONDITION(right.valid(format));
// All comparison with NaN are false
prop neitherNaN(!left.getNaN() && !right.getNaN());
// Either is an infinity (wrong in the case of NaN but will be corrected)
prop infCase( (left.isNegativeInf() && ITE(equality, prop(true), !right.isNegativeInf()) ) ||
(right.isPositiveInf() && ITE(equality, prop(true), !left.isPositiveInf()) ) ||
(ITE(equality, left.getInf() && right.getInf() && left.getSign() == right.getSign(), prop(false))) );
// Either is a zero (wrong in the case of NaN but will be corrected)
prop zeroCase( ( left.getZero() && !right.getZero() && !right.getSign()) ||
(right.getZero() && !left.getZero() && left.getSign()) ||
(ITE(equality, left.getZero() && right.getZero(), prop(false))) );
// Normal and subnormal case
prop normalOrSubnormal(!left.getNaN() && !right.getNaN() &&
!left.getInf() && !right.getInf() &&
!left.getZero() && !right.getZero());
prop negativeLessThanPositive(normalOrSubnormal && left.getSign() && !right.getSign());
prop exponentNeeded(normalOrSubnormal && left.getSign() == right.getSign());
probabilityAnnotation<t>(exponentNeeded, UNLIKELY);
prop positiveCase(!left.getSign() && !right.getSign() &&
left.getExponent() < right.getExponent());
prop negativeCase( left.getSign() && right.getSign() &&
left.getExponent() > right.getExponent());
prop exponentEqual(left.getExponent() == right.getExponent());
prop significandNeeded(exponentNeeded && exponentEqual);
probabilityAnnotation<t>(significandNeeded, VERYUNLIKELY);
prop positiveExEqCase(!left.getSign() && !right.getSign() &&
left.getSignificand() < right.getSignificand());
prop negativeExEqCase( left.getSign() && right.getSign() &&
left.getSignificand() > right.getSignificand());
prop positiveExEqCaseEq(!left.getSign() && !right.getSign() &&
left.getSignificand() <= right.getSignificand());
prop negativeExEqCaseEq( left.getSign() && right.getSign() &&
left.getSignificand() >= right.getSignificand());
return ITE(!normalOrSubnormal,
neitherNaN && (infCase || zeroCase),
ITE(!exponentNeeded,
negativeLessThanPositive,
ITE(!significandNeeded,
positiveCase || negativeCase,
ITE(equality,
positiveExEqCaseEq || negativeExEqCaseEq,
positiveExEqCase || negativeExEqCase))));
}
template <class t>
typename t::prop lessThan (const typename t::fpt &format,
const unpackedFloat<t> &left,
const unpackedFloat<t> &right) {
PRECONDITION(left.valid(format));
PRECONDITION(right.valid(format));
typedef typename t::prop prop;
return ordering(format, left, right, prop(false));
}
template <class t>
typename t::prop lessThanOrEqual (const typename t::fpt &format,
const unpackedFloat<t> &left,
const unpackedFloat<t> &right) {
PRECONDITION(left.valid(format));
PRECONDITION(right.valid(format));
typedef typename t::prop prop;
return ordering(format, left, right, prop(true));
}
// Note that IEEE-754 says that max(+0,-0) = +/-0 and max(-0,+0) = +/- 0
template <class t>
unpackedFloat<t> max (const typename t::fpt &format,
const unpackedFloat<t> &left,
const unpackedFloat<t> &right,
const typename t::prop &zeroCase) {
return ITE(left.getNaN() || ordering(format, left, right, zeroCase),
right,
left);
}
// Note that IEEE-754 says that min(+0,-0) = +/-0 and min(-0,+0) = +/- 0
// this will always return the left one.
template <class t>
unpackedFloat<t> min (const typename t::fpt &format,
const unpackedFloat<t> &left,
const unpackedFloat<t> &right,
const typename t::prop &zeroCase) {
return ITE(right.getNaN() || ordering(format, left, right, zeroCase),
left,
right);
}
template <class t>
typename t::prop originalLessThan (const typename t::fpt &format,
const unpackedFloat<t> &left,
const unpackedFloat<t> &right) {
typedef typename t::prop prop;
PRECONDITION(left.valid(format));
PRECONDITION(right.valid(format));
// Optimisation : merge < and ==
// All comparison with NaN are false
prop neitherNan(!left.getNaN() && !right.getNaN());
// Infinities are bigger than everything but themself
prop eitherInf(left.getInf() || right.getInf());
prop infCase(( left.isNegativeInf() && !right.isNegativeInf()) ||
(!left.isPositiveInf() && right.isPositiveInf()));
// Both zero are equal
prop eitherZero(left.getZero() || right.getZero());
prop zeroCase(( left.getZero() && !right.getZero() && !right.getSign()) ||
(!left.getZero() && left.getSign() && right.getZero()));
// Normal and subnormal
prop negativeLessThanPositive(left.getSign() && !right.getSign()); // - < +
prop positiveCase(!left.getSign() && !right.getSign() &&
((left.getExponent() < right.getExponent()) ||
(left.getExponent() == right.getExponent() &&
left.getSignificand() < right.getSignificand())));
prop negativeCase(left.getSign() && right.getSign() &&
((left.getExponent() > right.getExponent()) ||
(left.getExponent() == right.getExponent() &&
left.getSignificand() > right.getSignificand())));
return neitherNan &&
ITE(eitherInf,
infCase,
ITE(eitherZero,
zeroCase,
negativeLessThanPositive || positiveCase || negativeCase));
}
// Optimised combination of the two
template <class t>
typename t::prop originalLessThanOrEqual (const typename t::fpt &format,
const unpackedFloat<t> &left,
const unpackedFloat<t> &right) {
typedef typename t::prop prop;
PRECONDITION(left.valid(format));
PRECONDITION(right.valid(format));
// Optimisation : merge < and ==
// All comparison with NaN are false
prop neitherNan(!left.getNaN() && !right.getNaN());
// Infinities are bigger than everything but themself
prop eitherInf(left.getInf() || right.getInf());
prop infCase( (left.getInf() && right.getInf() && left.getSign() == right.getSign()) ||
left.isNegativeInf() ||
right.isPositiveInf());
// Both zero are equal
prop eitherZero(left.getZero() || right.getZero());
prop zeroCase((left.getZero() && right.getZero()) ||
( left.getZero() && !right.getSign()) ||
( left.getSign() && right.getZero()));
// Normal and subnormal
prop negativeLessThanPositive(left.getSign() && !right.getSign()); // - < +
prop positiveCase(!left.getSign() && !right.getSign() &&
((left.getExponent() < right.getExponent()) ||
(left.getExponent() == right.getExponent() &&
left.getSignificand() <= right.getSignificand())));
prop negativeCase(left.getSign() && right.getSign() &&
((left.getExponent() > right.getExponent()) ||
(left.getExponent() == right.getExponent() &&
left.getSignificand() >= right.getSignificand())));
return neitherNan &&
ITE(eitherInf,
infCase,
ITE(eitherZero,
zeroCase,
negativeLessThanPositive || positiveCase || negativeCase));
}
}
#endif
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