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qX_real.hpp
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#pragma once
#include "mX_real.hpp"
namespace mX_real {
namespace qX_real {
template < typename T, Algorithm A=Algorithm::Accurate >
struct qx_real {
static_assert( fp<T>::value,
"The base type must be a pre-defined type such as float or double." );
//
//
//
using base_T = T;
static Algorithm constexpr base_A = A;
//
// {DX,TX,QX}_REAL are shortcut type only within qx_real
//
//
private:
template < Algorithm _A_=A >
using DX_REAL = dX_real::dx_real<T,_A_>;
template < Algorithm _A_=A >
using TX_REAL = tX_real::tx_real<T,_A_>;
template < Algorithm _A_=A >
using QX_REAL = qX_real::qx_real<T,_A_>;
//
//
//
public:
static int constexpr L = 4;
T x[L];
//
//
//
template < Algorithm _A_ >
using type_with_Algorithm = QX_REAL<_A_>;
//
using typeAccurate = type_with_Algorithm<Algorithm::Accurate>;
using typeWeakAccurate = type_with_Algorithm<Algorithm::WeakAccurate>;
using typeSloppy = type_with_Algorithm<Algorithm::Sloppy>;
using typeQuasi = type_with_Algorithm<Algorithm::Quasi>;
//
using accurateType = type_with_Algorithm<accurateAlgorithm<A>::algorithm>;
using inaccurateType = type_with_Algorithm<inaccurateAlgorithm<A>::algorithm>;
using narrowerType = typename std::conditional_t< 4>=4, tX_real::tx_real<T,A>, dX_real::dx_real<T,A> >;
using widerType = typename std::conditional_t< 4==2, tX_real::tx_real<T,A>, qX_real::qx_real<T,A> >;
//
// A special member function utilized in constructors
//
//
INLINE T constexpr quick_Normalized () const NOEXCEPT {
return mX_real::quick_Normalized( *this );
}
INLINE QX_REAL<> constexpr element_rotate () const NOEXCEPT {
T y[L]; for(auto i=0; i<L; i++) { y[i] = x[i]; }
for(auto i=0; i<L-1; i++) {
if ( ! fp<T>::is_zero( y[0] ) ) { return QX_REAL<>( y ); }
T t = y[0]; for(auto j=0; j<L-1; j++) { y[j] = y[j+1]; } y[L-1] = t;
}
return QX_REAL<>( y );
}
//
// Simple constructors
//
//
INLINE constexpr qx_real() NOEXCEPT {
for(auto i=0; i<L; i++) { x[i] = fp<T>::zero(); }
}
INLINE constexpr qx_real( T const& x0, T const& x1, T const& x2, T const& x3 ) NOEXCEPT {
#if MX_REAL_USE_INF_NAN_EXCEPTION
auto t = x0 + x1 + x2 + x3;
if ( fp<T>::isinf( t ) || fp<T>::isnan( t ) ) {
for(auto i=0; i<L; i++) { x[i] = t; }
}
#else
if ( false ) { }
#endif
else {
x[0] = x0; x[1] = x1; x[2] = x2; x[3] = x3;
}
}
INLINE constexpr qx_real( T const &h ) NOEXCEPT {
{ x[0] = h; for(auto i=1; i<L; i++) { x[i] = fp<T>::zero(); } }
}
template < typename Ts, T_scalar(Ts), T_neq_Ts(T,Ts) >
INLINE qx_real( Ts const &h ) NOEXCEPT {
{ x[0] = T(h); for(auto i=1; i<L; i++) { x[i] = fp<T>::zero(); } }
}
INLINE constexpr qx_real( T const *d ) NOEXCEPT {
#if MX_REAL_USE_INF_NAN_EXCEPTION
auto t = d[0]; for(auto i=1; i<L; i++) { t += d[1]; }
if ( fp<T>::isinf( t ) || fp<T>::isnan( t ) ) {
for(auto i=0; i<L; i++) { x[i] = t; }
}
#else
if ( false ) { }
#endif
else {
for(auto i=0; i<L; i++) { x[i] = d[i]; }
}
}
//
// Copy constructor with (or without) any Algorithm switch
//
//
template < template < typename __T__, Algorithm __A__ > class _TX_, Algorithm _A_, T_mX(_TX_<T,_A_>) >
INLINE constexpr qx_real( _TX_<T,_A_> const& h ) NOEXCEPT {
using TX = _TX_<T,_A_>;
#if MX_REAL_USE_INF_NAN_EXCEPTION
auto const t = h.quick_Normalized();
if ( fp<T>::isinf( t ) || fp<T>::isnan( t ) ) {
for(auto i=0; i<L; i++) { x[i] = t; }
}
#else
if ( false ) { }
#endif
else {
// rewritten in the copya and normalization process
auto constexpr LL = TX::L;
auto constexpr need_normalize =
( A != Algorithm::Quasi && _A_ == Algorithm::Quasi );
if ( L < LL ) {
if ( _A_ == Algorithm::Quasi ) {
auto s = h;
mX_real::Normalize<NormalizeOption::Accurate>( s );
mX_real::copy_with_rounding( x, s.x, L, LL );
} else {
mX_real::copy_with_rounding( x, h.x, L, LL );
}
if ( need_normalize ) { mX_real::Normalize<NormalizeOption::Accurate>( *this ); }
} else {
mX_real::copy_with_rounding( x, h.x, L, LL );
if ( need_normalize ) { mX_real::Normalize<NormalizeOption::Regular>( *this ); }
}
}
}
//
// Copy-assignment operator
//
//
INLINE QX_REAL<> constexpr &operator=( T const& h )& NOEXCEPT {
x[0] = h; for(auto i=1; i<L; i++) { x[i] = fp<T>::zero(); }
return *this;
}
template < typename Ts, T_scalar(Ts), T_neq_Ts(T,Ts) >
INLINE QX_REAL<> constexpr &operator=( Ts const& h )& NOEXCEPT {
x[0] = T(h); for(auto i=1; i<L; i++) { x[i] = fp<T>::zero(); }
return *this;
}
template < template < typename __T__, Algorithm __A__ > class _TX_, Algorithm _A_, T_mX(_TX_<T,_A_>) >
INLINE QX_REAL<> constexpr &operator=( _TX_<T,_A_> const& h )& NOEXCEPT {
if ( std::is_same< QX_REAL<A>, _TX_<T,_A_> >::value &&
this == (QX_REAL<>*)(&h) ) { return *this; }
#if MX_REAL_USE_INF_NAN_EXCEPTION
auto const t = h.quick_Normalized();
if ( fp<T>::isinf( t ) || fp<T>::isnan( t ) ) {
for(auto i=0; i<L; i++) { x[i] = t; }
}
#else
if ( false ) { }
#endif
else {
// rewritten in the copya and normalization process
auto constexpr LL = _TX_<T,_A_>::L;
auto constexpr need_normalize =
( A != Algorithm::Quasi && _A_ == Algorithm::Quasi );
if ( L < LL ) {
if ( _A_ == Algorithm::Quasi ) {
auto s = h;
mX_real::Normalize<NormalizeOption::Accurate>( s );
mX_real::copy_with_rounding( x, s.x, L, LL );
} else {
mX_real::copy_with_rounding( x, h.x, L, LL );
}
if ( need_normalize ) { mX_real::Normalize<NormalizeOption::Regular>( *this ); }
} else {
mX_real::copy_with_rounding( x, h.x, L, LL );
if ( need_normalize ) { mX_real::Normalize<NormalizeOption::Accurate>( *this ); }
}
}
return *this;
}
//
// Casting to other datatypes
//
//
explicit
INLINE constexpr operator T() const NOEXCEPT {
if ( A == Algorithm::Quasi ) {
return this->quick_Normalized();
} else {
return x[0];
}
}
template < typename Ts, T_fp(Ts), T_neq_Ts(T,Ts) >
explicit
INLINE constexpr operator Ts() const NOEXCEPT {
if ( A == Algorithm::Quasi ) {
return Ts(this->quick_Normalized());
} else {
return Ts(x[0]);
}
}
template < template < typename __T__, Algorithm __A__ > class _TX_, Algorithm _A_, T_mX(_TX_<T,_A_>) >
explicit
INLINE constexpr operator _TX_<T,_A_>() const NOEXCEPT {
using TX = _TX_<T,_A_>;
TX c;
auto constexpr LL = TX::L;
#if MX_REAL_USE_INF_NAN_EXCEPTION
auto const t = this->quick_Normalized();
if ( fp<T>::isinf( t ) || fp<T>::isnan( t ) ) {
for(auto i=0; i<LL; i++) { c.x[i] = t; }
}
#else
if ( false ) { }
#endif
else {
// rewritten in the copya and normalization process
auto constexpr need_normalize =
( _A_ != Algorithm::Quasi && A == Algorithm::Quasi );
if ( LL < L ) {
if ( A == Algorithm::Quasi ) {
auto s = *this;
mX_real::Normalize<NormalizeOption::Accurate>( s );
mX_real::copy_with_rounding( c.x, s.x, LL, L );
} else {
mX_real::copy_with_rounding( c.x, x, LL, L );
}
if ( need_normalize ) { mX_real::Normalize<NormalizeOption::Regular>( c ); }
} else {
mX_real::copy_with_rounding( c.x, x, LL, L );
if ( need_normalize ) { mX_real::Normalize<NormalizeOption::Accurate>( c ); }
}
}
return c;
}
//
// unneccessary operators are invalidaded
//
//
auto operator! () = delete;
auto operator~ () = delete;
auto operator++ () = delete;
auto operator-- () = delete;
template < typename _T_ > auto operator&& ( _T_ a ) = delete;
template < typename _T_ > auto operator|| ( _T_ a ) = delete;
template < typename _T_ > auto operator&= ( _T_ a ) = delete;
template < typename _T_ > auto operator^= ( _T_ a ) = delete;
template < typename _T_ > auto operator<<= ( _T_ a ) = delete;
template < typename _T_ > auto operator>>= ( _T_ a ) = delete;
template < typename _T_ > auto operator, ( _T_ a ) = delete;
template < typename _T_ > auto operator[] ( _T_ a ) = delete;
auto operator() (...) = delete;
//
// static constexpr member functions or constatns
// contents are always in Accurate format
// but they follow in the Algorithm argument 'A'
//
// Some constants such as NAN, INF are compatible to the format
// defined in the QD library by Bailey and Hida
//
// For example,
// NAN = { NAN, NAN, ... } could be aligned with the same sign
// case of mixtured sign, it presents NAN...
// INF = { INF, INF, ... } must align with the same sign
// case of mixtured sign, it turns to -NAN
// after the Normalization.
//
static INLINE QX_REAL<> constexpr zero () NOEXCEPT {
return QX_REAL<>{ fp<T>::zero() };
}
static INLINE QX_REAL<> constexpr one () NOEXCEPT {
return QX_REAL<>{ fp<T>::one() };
}
static INLINE QX_REAL<> constexpr two () NOEXCEPT {
return QX_REAL<>{ fp<T>::two() };
}
static INLINE QX_REAL<> constexpr nhalf () NOEXCEPT {
return QX_REAL<>{ fp<T>::nhalf() };
}
static INLINE QX_REAL<> constexpr nan () NOEXCEPT {
QX_REAL<> c;
T p = fp<T>::nan();
for(auto i=0; i<L; i++) { c.x[i] = p; }
return c;
}
static INLINE QX_REAL<> constexpr inf () NOEXCEPT {
QX_REAL<> c;
T p = fp<T>::inf();
for(auto i=0; i<L; i++) { c.x[i] = p; }
return c;
}
//
// epsilon() : machine epsilon, specifically,
// it is defined by the ulp of a contiguous full-bit
// representation
//
// 1.11B+0 + 1.11B-{E+1} + 1.11B-{2(E+1)}
// EPS = 0.01B-{2(E+1)} = 1.00B-{2(E+1)+E}
//
static INLINE QX_REAL<> constexpr epsilon () NOEXCEPT {
QX_REAL<> c;
T p = fp<T>::connect_fp(); p = ( p * p* p* p) * 2;
T q = fp<T>::zero();
c.x[0] = p; for(auto i=1; i<L; i++) { c.x[i] = q; }
return c;
}
//
// denormal_min() : minimum of the denormal floating point
// representation with sign aligment
// it is always the same as denormal_min of the
// single word
//
static INLINE QX_REAL<> constexpr denorm_min () NOEXCEPT {
QX_REAL<> c;
T p = fp<T>::denorm_min();
T q = fp<T>::zero();
c.x[0] = p; for(auto i=1; i<L; i++) { c.x[i] = q; }
return c;
}
//
// min() : minimum of the normalized floating point representation
// if its last component is regarded as min<T>() with a full-bit
// data concatination and normalization
//
// for example,
// 1.11B+{K} + 1.11B-{K+(E+1)} ... + std::min<T>()
// 1.00B-{K} * (EPS/2)^{L-1} = min<T>()
// min() = 1.00B-{K} = min<T>()*(EPS/2)^{L-1}
// = min<T>() / (EPS/2)^{L-1}
// case of dX
// X = min() / EPS * 2 = min() * epslioni() * 2
//
static INLINE QX_REAL<> constexpr min () NOEXCEPT {
QX_REAL<> c;
T p = fp<T>::min();
T q = fp<T>::zero();
for(auto i=1; i<L; i++) { p *= ( fp<T>::disconnect_fp() ); }
c.x[0] = p; for(auto i=1; i<L; i++) { c.x[i] = q; }
return c;
}
//
// maxmax() : maximum of the normalized floating point representation
// it could be defined by the full bit pattern with the same
// exponent component of the max<T>(), however,
// carry-up by the nearest-even rounding violates
// max() + zero() == max()
// thus, max() and mamax() are seperately defined.
//
static INLINE QX_REAL<> constexpr maxmax () NOEXCEPT {
QX_REAL<> c;
T p = fp<T>::max();
T q = fp<T>::connect_fp();
c.x[0] = p; for(auto i=1; i<L; i++) { c.x[i] = c.x[i-1] * q; }
return c;
}
//
// max() : maximum of the normalized floating point representation
// it does not violate the essential equality such that
//
// max() + zero() == max().
//
static INLINE QX_REAL<> constexpr max () NOEXCEPT {
QX_REAL<> c;
T p = fp<T>::max();
T q = fp<T>::connect_fp() * fp<T>::nhalf();
c.x[0] = p; for(auto i=1; i<L; i++) { c.x[i] = c.x[i-1] * q; }
return c;
}
//
// safe_min() : the smallest positive normalized number such that
// 1/safe_min does not overflow.
// it is always the same as min of T of the single word
//
static INLINE QX_REAL<> constexpr safe_min () NOEXCEPT {
QX_REAL<> c;
T p = fp<T>::min();
T q = fp<T>::zero();
c.x[0] = p; for(auto i=1; i<L; i++) { c.x[i] = q; }
return c;
}
//
// safe_max() : the largest positive number such that
// 1/safe_max is normalized.
//
static INLINE QX_REAL<> constexpr safe_max () NOEXCEPT {
QX_REAL<> c;
auto exp_size = QxW::fp_const<T>::MASK - QxW::fp_const<T>::RINF - 1;
exp_size >>= QxW::fp_const<T>::SOFF;
T p = (1 << exp_size) * fp<T>::one() / fp<T>::min();
T q = fp<T>::zero();
c.x[0] = p; for(auto i=1; i<L; i++) { c.x[i] = q; }
return c;
}
//
// static member funtions
// their definitions are below outside of the struct definition block
//
static INLINE QX_REAL<> constexpr sqrt ( DX_REAL<> const& a ) NOEXCEPT;
static INLINE QX_REAL<> constexpr sqrt ( TX_REAL<> const& a ) NOEXCEPT;
static INLINE QX_REAL<> constexpr sqrt ( QX_REAL<> const& a ) NOEXCEPT;
static INLINE QX_REAL<> constexpr abs ( DX_REAL<> const& a ) NOEXCEPT;
static INLINE QX_REAL<> constexpr abs ( TX_REAL<> const& a ) NOEXCEPT;
static INLINE QX_REAL<> constexpr abs ( QX_REAL<> const& a ) NOEXCEPT;
static INLINE QX_REAL<> constexpr fabs ( DX_REAL<> const& a ) NOEXCEPT {
return QX_REAL<>::abs(a);
}
static INLINE QX_REAL<> constexpr fabs ( TX_REAL<> const& a ) NOEXCEPT {
return QX_REAL<>::abs(a);
}
static INLINE QX_REAL<> constexpr fabs ( QX_REAL<> const& a ) NOEXCEPT {
return QX_REAL<>::abs(a);
}
static INLINE QX_REAL<> rand () NOEXCEPT;
static INLINE bool constexpr isnan ( QX_REAL<> const& a ) NOEXCEPT;
static INLINE bool constexpr is_positive ( QX_REAL<> const& a ) NOEXCEPT;
static INLINE bool constexpr is_negative ( QX_REAL<> const& a ) NOEXCEPT;
static INLINE bool constexpr signbit ( QX_REAL<> const& a ) NOEXCEPT;
static INLINE bool constexpr is_zero ( QX_REAL<> const& a ) NOEXCEPT;
static INLINE bool constexpr isinf ( QX_REAL<> const& a ) NOEXCEPT;
//
//
// member functions (aka methods)
//
//
INLINE void constexpr Normalize () NOEXCEPT { mX_real::Normalize( *this ); }
//
INLINE bool constexpr isnan () const NOEXCEPT { return QX_REAL<>::isnan( *this ); }
INLINE bool constexpr is_positive () const NOEXCEPT { return QX_REAL<>::is_positive( *this ); }
INLINE bool constexpr is_negative () const NOEXCEPT { return QX_REAL<>::is_negative( *this ); }
INLINE bool constexpr signbit () const NOEXCEPT { return QX_REAL<>::signbit( *this ); }
INLINE bool constexpr is_zero () const NOEXCEPT { return QX_REAL<>::is_zero( *this ); }
INLINE bool constexpr isinf () const NOEXCEPT { return QX_REAL<>::isinf( *this ); }
//
// Friend
template < typename _T_, Algorithm _A_ >
friend std::ostream& operator<< ( std::ostream& stream, qX_real::qx_real<_T_,_A_> const& a );
};
//
// get a new type qX_real<T,A> from TX(={m}X_real<T,A>)
//
//
template < typename TX, T_mX(TX) >
struct mX_real_impl {
static Algorithm constexpr A = TX::base_A;
using T = typename TX::base_T;
using type = typename qX_real::qx_real<T,A>;
};
template < typename TX >
using mX_real = typename mX_real_impl<TX>::type;
//
//
//
template < typename TX, Algorithm A, T_mX(TX) >
struct type_with_Algorithm_impl {
using type = typename qX_real::qx_real<typename TX::base_T,A>;
};
template < typename TX, Algorithm A >
using type_with_Algorithm = typename type_with_Algorithm_impl<TX,A>::type;
//
template < typename TX >
using type_Accurate = type_with_Algorithm_impl<TX, Algorithm::Accurate>;
template < typename TX >
using type_WeakAccurate = type_with_Algorithm_impl<TX, Algorithm::WeakAccurate>;
template < typename TX >
using type_Sloppy = type_with_Algorithm_impl<TX, Algorithm::Sloppy>;
template < typename TX >
using type_Quasi = type_with_Algorithm_impl<TX, Algorithm::Quasi>;
//
// Alias names
//
//
template < typename T >
using qX_real_accurate = qX_real::qx_real<T,Algorithm::Accurate>;
template < typename T >
using qX_real_weakaccurate = qX_real::qx_real<T,Algorithm::WeakAccurate>;
template < typename T >
using qX_real_sloppy = qX_real::qx_real<T,Algorithm::Sloppy>;
template < typename T >
using qX_real_quasi = qX_real::qx_real<T,Algorithm::Quasi>;
//
// Bitwise or FP-bit-specific ops
//
//
template < typename T, Algorithm Aa >
INLINE auto constexpr isinf ( qX_real::qx_real<T,Aa> const& a ) NOEXCEPT {
return fp<T>::isinf( a.quick_Normalized() );
}
template < typename T, Algorithm Aa >
INLINE auto constexpr signbit ( qX_real::qx_real<T,Aa> const& a ) NOEXCEPT {
return fp<T>::signbit( a.quick_Normalized() );
}
template < typename T, Algorithm Aa >
INLINE auto constexpr isnan ( qX_real::qx_real<T,Aa> const& a ) NOEXCEPT {
return fp<T>::isnan( a.quick_Normalized() );
}
template < typename T, Algorithm Aa >
INLINE bool constexpr is_positive ( qX_real::qx_real<T,Aa> const& a ) NOEXCEPT {
return a.quick_Normalized() > fp<T>::zero();
}
template < typename T, Algorithm Aa >
INLINE bool constexpr is_negative ( qX_real::qx_real<T,Aa> const& a ) NOEXCEPT {
return a.quick_Normalized() < fp<T>::zero();
}
template < typename T, Algorithm Aa >
INLINE bool constexpr is_zero ( qX_real::qx_real<T,Aa> const& a ) NOEXCEPT {
return a.quick_Normalized() == fp<T>::zero();
}
//
template < typename T, Algorithm A >
INLINE bool constexpr qX_real::qx_real<T,A>::isinf ( qX_real::qx_real<T,A> const& a ) NOEXCEPT {
return qX_real::isinf ( a );
}
template < typename T, Algorithm A >
INLINE bool constexpr qX_real::qx_real<T,A>::isnan ( qX_real::qx_real<T,A> const& a ) NOEXCEPT {
return qX_real::isnan ( a );
}
template < typename T, Algorithm A >
INLINE bool constexpr qX_real::qx_real<T,A>::signbit ( qX_real::qx_real<T,A> const& a ) NOEXCEPT {
return qX_real::signbit ( a );
}
template < typename T, Algorithm A >
INLINE bool constexpr qX_real::qx_real<T,A>::is_zero ( qX_real::qx_real<T,A> const& a ) NOEXCEPT {
return qX_real::is_zero ( a );
}
template < typename T, Algorithm A >
INLINE bool constexpr qX_real::qx_real<T,A>::is_positive ( qX_real::qx_real<T,A> const& a ) NOEXCEPT {
return qX_real::is_positive ( a );
}
template < typename T, Algorithm A >
INLINE bool constexpr qX_real::qx_real<T,A>::is_negative ( qX_real::qx_real<T,A> const& a ) NOEXCEPT {
return qX_real::is_negative ( a );
}
//
//
// Comparison, equivalence (==)
//
//
template < typename T, Algorithm A >
INLINE auto constexpr operator_eq ( qX_real::qx_real<T,A> const& a, qX_real::qx_real<T,A> const& b ) NOEXCEPT {
using TX = qX_real::qx_real<T,A>;
auto i=0; for(i=0; i<TX::L-1; i++) {
if ( a.x[i] != b.x[i] ) { return a.x[i] == b.x[i]; }
} return a.x[i] == b.x[i];
}
template < typename T, Algorithm Aa, Algorithm Ab >
INLINE auto constexpr operator== ( qX_real::qx_real<T,Aa> const& a, qX_real::qx_real<T,Ab> const& b ) NOEXCEPT {
Algorithm constexpr A=commonAlgorithm<Aa,Ab>::algorithm;
if ( A == Algorithm::Quasi ) {
using TT = qX_real::qx_real<T,Algorithm::Accurate>;
return qX_real::operator_eq( TT{ a }, TT{ b } );
} else {
return qX_real::operator_eq( a, b );
}
}
template < typename T, Algorithm Aa, Algorithm Ab >
INLINE auto constexpr operator!= ( qX_real::qx_real<T,Aa> const& a, qX_real::qx_real<T,Ab> const& b ) NOEXCEPT {
return !(a == b);
}
//
// Comparison, greater than (>)
//
//
template < typename T, Algorithm A >
INLINE auto constexpr operator_gt ( qX_real::qx_real<T,A> const& a, qX_real::qx_real<T,A> const& b ) NOEXCEPT {
using TX = qX_real::qx_real<T,A>;
auto i=0; for(i=0; i<TX::L-1; i++) {
if ( a.x[i] != b.x[i] ) { return a.x[i] > b.x[i]; }
} return a.x[i] > b.x[i];
}
template < typename T, Algorithm Aa, Algorithm Ab >
INLINE auto constexpr operator> ( qX_real::qx_real<T,Aa> const& a, qX_real::qx_real<T,Ab> const& b ) NOEXCEPT {
Algorithm constexpr A=commonAlgorithm<Aa,Ab>::algorithm;
if ( A == Algorithm::Quasi ) {
using TT = qX_real::qx_real<T,Algorithm::Accurate>;
return qX_real::operator_gt( TT{ a }, TT{ b } );
} else {
return qX_real::operator_gt( a, b );
}
}
template < typename T, Algorithm Aa, Algorithm Ab >
INLINE auto constexpr operator<= ( qX_real::qx_real<T,Aa> const& a, qX_real::qx_real<T,Ab> const& b ) NOEXCEPT {
return !(a > b);
}
//
// Comparison, less than (<)
//
//
template < typename T, Algorithm A >
INLINE auto constexpr operator_lt ( qX_real::qx_real<T,A> const& a, qX_real::qx_real<T,A> const& b ) NOEXCEPT {
using TX = qX_real::qx_real<T,A>;
auto i=0; for(i=0; i<TX::L-1; i++) {
if ( a.x[i] != b.x[i] ) { return a.x[i] < b.x[i]; }
} return a.x[i] < b.x[i];
}
template < typename T, Algorithm Aa, Algorithm Ab >
INLINE auto constexpr operator< ( qX_real::qx_real<T,Aa> const& a, qX_real::qx_real<T,Ab> const& b ) NOEXCEPT {
Algorithm constexpr A=commonAlgorithm<Aa,Ab>::algorithm;
if ( A == Algorithm::Quasi ) {
using TT = qX_real::qx_real<T,Algorithm::Accurate>;
return qX_real::operator_lt( TT{ a }, TT{ b } );
} else {
return qX_real::operator_lt( a, b );
}
}
template < typename T, Algorithm Aa, Algorithm Ab >
INLINE auto constexpr operator>= ( qX_real::qx_real<T,Aa> const& a, qX_real::qx_real<T,Ab> const& b ) NOEXCEPT {
return !(a < b);
}
//
// Unitary operators
//
//
template < typename T, Algorithm Aa >
INLINE auto constexpr reversed_sign ( qX_real::qx_real<T,Aa> const& a ) NOEXCEPT {
return qX_real::qx_real<T,Aa>( -a.x[0],-a.x[1],-a.x[2],-a.x[3] );
}
//
template < typename T, Algorithm Aa >
INLINE auto constexpr operator+ ( qX_real::qx_real<T,Aa> const& a ) NOEXCEPT {
return a;
}
template < typename T, Algorithm Aa >
INLINE auto constexpr operator- ( qX_real::qx_real<T,Aa> const& a ) NOEXCEPT {
return qX_real::reversed_sign( a );
}
//
//
// Addition
//
//
template < typename T, Algorithm Aa, Algorithm Ab >
INLINE auto constexpr operator_add_body ( qX_real::qx_real<T,Aa> const& a, qX_real::qx_real<T,Ab> const& b ) NOEXCEPT {
Algorithm constexpr A = commonAlgorithm<Aa,Ab>::algorithm;
using TX = qX_real::qx_real<T,A>;
TX c;
if ( A <= Algorithm::WeakAccurate ) {
QxW::add_QW_QW_QW ( _QX_(a), _QX_(b), _QX_(c) );
} else {
QxW::add_QQW_QQW_QQW ( _QX_(a), _QX_(b), _QX_(c) );
if ( A != Algorithm::Quasi ) { Normalize( c ); }
}
return c;
}
template < typename T, Algorithm Aa, Algorithm Ab >
INLINE auto constexpr operator_add_body ( qX_real::qx_real<T,Aa> const& a, tX_real::tx_real<T,Ab> const& b ) NOEXCEPT {
Algorithm constexpr A = commonAlgorithm<Aa,Ab>::algorithm;
using TX = qX_real::qx_real<T,A>;
TX c;
if ( A <= Algorithm::WeakAccurate ) {
QxW::add_QW_TW_QW ( _QX_(a), _TX_(b), _QX_(c) );
} else {
QxW::add_QQW_QTW_QQW ( _QX_(a), _TX_(b), _QX_(c) );
if ( A != Algorithm::Quasi ) { Normalize( c ); }
}
return c;
}
template < typename T, Algorithm Aa, Algorithm Ab >
INLINE auto constexpr operator_add_body ( qX_real::qx_real<T,Aa> const& a, dX_real::dx_real<T,Ab> const& b ) NOEXCEPT {
Algorithm constexpr A = commonAlgorithm<Aa,Ab>::algorithm;
using TX = qX_real::qx_real<T,A>;
TX c;
if ( A <= Algorithm::WeakAccurate ) {
QxW::add_QW_DW_QW ( _QX_(a), _DX_(b), _QX_(c) );
} else {
QxW::add_QQW_PA_QQW ( _QX_(a), _DX_(b), _QX_(c) );
if ( A != Algorithm::Quasi ) { Normalize( c ); }
}
return c;
}
template < typename T, Algorithm Aa >
INLINE auto constexpr operator_add_body ( qX_real::qx_real<T,Aa> const& a, T const& b ) NOEXCEPT {
using TX = qX_real::qx_real<T,Aa>;
TX c;
if ( Aa <= Algorithm::WeakAccurate ) {
QxW::add_QW_SW_QW ( _QX_(a), _SX_(b), _QX_(c) );
} else {
QxW::add_QQW_SW_QQW ( _QX_(a), _SX_(b), _QX_(c) );
if ( Aa != Algorithm::Quasi ) { Normalize( c ); }
}
return c;
}
template < typename T, Algorithm Aa, Algorithm Ab >
INLINE auto constexpr operator_add_body ( tX_real::tx_real<T,Aa> const& a, qX_real::qx_real<T,Ab> const& b ) NOEXCEPT {
return operator_add_body ( b, a );
}
template < typename T, Algorithm Aa, Algorithm Ab >
INLINE auto constexpr operator_add_body ( tX_real::tx_real<T,Aa> const& a, tX_real::tx_real<T,Ab> const& b ) NOEXCEPT {
Algorithm constexpr A = commonAlgorithm<Aa,Ab>::algorithm;
using TX = qX_real::qx_real<T,A>;
TX c;
if ( A <= Algorithm::WeakAccurate ) {
QxW::add_TW_TW_QW ( _TX_(a), _TX_(b), _QX_(c) );
} else {
QxW::add_QTW_QTW_QQW ( _TX_(a), _TX_(b), _QX_(c) );
if ( A != Algorithm::Quasi ) { Normalize( c ); }
}
return c;
}
template < typename T, Algorithm Aa, Algorithm Ab >
INLINE auto constexpr operator_add_body ( tX_real::tx_real<T,Aa> const& a, dX_real::dx_real<T,Ab> const& b ) NOEXCEPT {
Algorithm constexpr A = commonAlgorithm<Aa,Ab>::algorithm;
using TX = qX_real::qx_real<T,A>;
TX c;
if ( A <= Algorithm::WeakAccurate ) {
QxW::add_TW_DW_QW ( _TX_(a), _DX_(b), _QX_(c) );
} else {
QxW::add_QTW_PA_QQW ( _TX_(a), _DX_(b), _QX_(c) );
if ( A != Algorithm::Quasi ) { Normalize( c ); }
}
return c;
}
template < typename T, Algorithm Aa >
INLINE auto constexpr operator_add_body ( tX_real::tx_real<T,Aa> const& a, T const& b ) NOEXCEPT {
using TX = qX_real::qx_real<T,Aa>;
TX c;
if ( Aa <= Algorithm::WeakAccurate ) {
QxW::add_TW_SW_QW ( _TX_(a), _SX_(b), _QX_(c) );
} else {
QxW::add_QTW_SW_QQW ( _TX_(a), _SX_(b), _QX_(c) );
if ( Aa != Algorithm::Quasi ) { Normalize( c ); }
}
return c;
}
template < typename T, Algorithm Aa, Algorithm Ab >
INLINE auto constexpr operator_add_body ( dX_real::dx_real<T,Aa> const& a, qX_real::qx_real<T,Ab> const& b ) NOEXCEPT {
return operator_add_body ( b, a );
}
template < typename T, Algorithm Aa, Algorithm Ab >
INLINE auto constexpr operator_add_body ( dX_real::dx_real<T,Aa> const& a, tX_real::tx_real<T,Ab> const& b ) NOEXCEPT {
return operator_add_body ( b, a );
}
template < typename T, Algorithm Aa, Algorithm Ab >
INLINE auto constexpr operator_add_body ( dX_real::dx_real<T,Aa> const& a, dX_real::dx_real<T,Ab> const& b ) NOEXCEPT {
Algorithm constexpr A = commonAlgorithm<Aa,Ab>::algorithm;
using TX = qX_real::qx_real<T,A>;
TX c;
if ( A <= Algorithm::WeakAccurate ) {
QxW::add_DW_DW_QW ( _DX_(a), _DX_(b), _QX_(c) );
} else {
QxW::add_PA_PA_QQW ( _DX_(a), _DX_(b), _QX_(c) );
if ( A != Algorithm::Quasi ) { Normalize( c ); }
}
return c;
}
template < typename T, Algorithm Aa >
INLINE auto constexpr operator_add_body ( dX_real::dx_real<T,Aa> const& a, T const& b ) NOEXCEPT {
using TX = qX_real::qx_real<T,Aa>;
TX c;
if ( Aa <= Algorithm::WeakAccurate ) {
QxW::add_DW_SW_QW ( _DX_(a), _SX_(b), _QX_(c) );
} else {
QxW::add_PA_SW_QQW ( _DX_(a), _SX_(b), _QX_(c) );
if ( Aa != Algorithm::Quasi ) { Normalize( c ); }
}
return c;
}
template < typename T, Algorithm Ab >
INLINE auto constexpr operator_add_body ( T const& a, qX_real::qx_real<T,Ab> const& b ) NOEXCEPT {
return operator_add_body ( b, a );
}
template < typename T, Algorithm Ab >
INLINE auto constexpr operator_add_body ( T const& a, tX_real::tx_real<T,Ab> const& b ) NOEXCEPT {
return operator_add_body ( b, a );
}
template < typename T, Algorithm Ab >
INLINE auto constexpr operator_add_body ( T const& a, dX_real::dx_real<T,Ab> const& b ) NOEXCEPT {
return operator_add_body ( b, a );
}
template < typename T, Algorithm A, T_fp(T) >
INLINE auto constexpr operator_add_body ( T const& a, T const& b ) NOEXCEPT {
using TX = qX_real::qx_real<T,A>;
TX c;
if ( A <= Algorithm::WeakAccurate ) {
QxW::add_SW_SW_QW ( _SX_(a), _SX_(b), _QX_(c) );
} else {
QxW::add_SW_SW_QQW ( _SX_(a), _SX_(b), _QX_(c) );
if ( A != Algorithm::Quasi ) { Normalize( c ); }
}
return c;
}
//
#if MX_REAL_USE_INF_NAN_EXCEPTION
template < Algorithm A, typename T, T_fp(T) >
INLINE auto constexpr operator_add_exception ( T const& a, T const& b, bool & flag ) NOEXCEPT {
using TX = qX_real::qx_real<T,A>;
{
if ( fp<T>::isnan( a ) || fp<T>::isnan( b ) ) {
auto c = fp<T>::copysign( fp<T>::copysign( fp<T>::nan(), a ), b );
flag = true; return TX{ c,c,c,c };
}
if ( fp<T>::isinf( a ) && fp<T>::isinf( b ) ) {
auto sign = std::signbit( a ) ^ std::signbit( b );
auto c = sign ? fp<T>::nan() : fp<T>::copysign( fp<T>::inf(), a );
flag = true; return TX{ c,c,c,c };
}
if ( fp<T>::isinf( a ) ) {
auto c = fp<T>::copysign( fp<T>::inf(), a );
flag = true; return TX{ c,c,c,c };
}
if ( fp<T>::isinf( b ) ) {
auto c = fp<T>::copysign( fp<T>::inf(), b );
flag = true; return TX{ c,c,c,c };
}
}
//
flag = false; return TX::zero();
}
template < typename TXa, typename TXb, T_mX2(TXa,TXb) >
INLINE auto constexpr operator_add_exception ( TXa const& a, TXb const& b, bool & flag ) NOEXCEPT {
Algorithm constexpr A = commonAlgorithm< TXa::base_A, TXb::base_A >::algorithm;
auto sa = a.quick_Normalized();
auto sb = b.quick_Normalized();
return qX_real::operator_add_exception<A> ( sa, sb, flag );
}
template < typename TXa, typename T, T_mX_fp(TXa,T) >
INLINE auto constexpr operator_add_exception ( TXa const& a, T const& b, bool & flag ) NOEXCEPT {
Algorithm constexpr A = TXa::base_A;
auto sa = a.quick_Normalized();
return qX_real::operator_add_exception<A> ( sa, b, flag );
}
template < typename T, typename TXb, T_mX_fp(TXb,T) >
INLINE auto constexpr operator_add_exception ( T const& a, TXb const& b, bool & flag ) NOEXCEPT {
return qX_real::operator_add_exception ( b, a, flag );
}
#endif
//
template < typename TXa, typename TXb, T_mX2(TXa,TXb), T_assert( TXa::L >= TXb::L ) >
INLINE auto constexpr operator_add ( TXa const& a, TXb const& b ) NOEXCEPT {
#if MX_REAL_USE_INF_NAN_EXCEPTION
auto flag = false;
auto e = qX_real::operator_add_exception ( a, b, flag );
if ( flag ) { return e; }
#endif
return qX_real::operator_add_body ( a, b );
}
template < typename TXa, typename TXb, T_mX2(TXa,TXb), T_assert( TXa::L < TXb::L ) >
INLINE auto constexpr operator_add ( TXa const& a, TXb const& b ) NOEXCEPT {
return qX_real::operator_add ( b, a );
}
template < typename TXa, typename T, T_mX_fp(TXa,T) >
INLINE auto constexpr operator_add ( TXa const& a, T const& b ) NOEXCEPT {
#if MX_REAL_USE_INF_NAN_EXCEPTION
auto flag = false;
auto e = qX_real::operator_add_exception ( a, b, flag );
if ( flag ) { return e; }
#endif
return qX_real::operator_add_body ( a, b );
}
template < typename T, typename TXb, T_mX_fp(TXb,T) >
INLINE auto constexpr operator_add ( T const& a, TXb const& b ) NOEXCEPT {
return qX_real::operator_add ( b, a );
}
template < Algorithm A=Algorithm::Accurate, typename T, T_fp(T) >
INLINE auto constexpr operator_add ( T const& a, T const& b ) NOEXCEPT {
#if MX_REAL_USE_INF_NAN_EXCEPTION
auto flag = false;
auto e = qX_real::operator_add_exception<A> ( a, b, flag );
if ( flag ) { return e; }
#endif
return qX_real::operator_add_body<T,A> ( a, b );
}
//
template < typename TXa, typename TXb, T_mX2(TXa,TXb), T_assert( std::max( TXa::L, TXb::L ) == 4 ) >
INLINE auto constexpr operator+ ( TXa const& a, TXb const& b ) NOEXCEPT {
return qX_real::operator_add ( a, b );
}
template < typename T, Algorithm A >
INLINE auto constexpr operator+ ( qX_real::qx_real<T,A> const& a, T const& b ) NOEXCEPT {
return qX_real::operator_add ( a, b );
}
template < typename T, Algorithm A, typename Ts, T_scalar(Ts), T_neq_Ts(T,Ts) >
INLINE auto constexpr operator+ ( qX_real::qx_real<T,A> const& a, Ts const& b ) NOEXCEPT {
return qX_real::operator_add ( a, T(b) );
}
template < typename T, Algorithm A >
INLINE auto constexpr operator+ ( T const& a, qX_real::qx_real<T,A> const& b ) NOEXCEPT {
return b + a;
}
template < typename T, Algorithm A, typename Ts, T_scalar(Ts), T_neq_Ts(T,Ts) >
INLINE auto constexpr operator+ ( Ts const& a, qX_real::qx_real<T,A> const& b ) NOEXCEPT {
return b + a;
}
//
//
template < typename T, Algorithm Aa, Algorithm Ab >
INLINE auto constexpr operator_add_ow_body ( qX_real::qx_real<T,Aa> & a, qX_real::qx_real<T,Ab> const& b ) NOEXCEPT {
Algorithm constexpr A=commonAlgorithm<Aa,Ab>::algorithm;
if ( A <= Algorithm::WeakAccurate ) {
QxW::add_QW_QW_QW ( _QX_(a), _QX_(b), _QX_(a) );
} else {
QxW::add_QQW_QQW_QQW ( _QX_(a), _QX_(b), _QX_(a) );
if ( A != Algorithm::Quasi ) { Normalize( a ); }
}
return a;
}
template < typename T, Algorithm Aa, Algorithm Ab >
INLINE auto constexpr operator_add_ow_body ( qX_real::qx_real<T,Aa> & a, tX_real::tx_real<T,Ab> const& b ) NOEXCEPT {
Algorithm constexpr A=commonAlgorithm<Aa,Ab>::algorithm;
if ( A <= Algorithm::WeakAccurate ) {
QxW::add_QW_TW_QW ( _QX_(a), _TX_(b), _QX_(a) );
} else {
QxW::add_QQW_QTW_QQW ( _QX_(a), _TX_(b), _QX_(a) );
if ( A != Algorithm::Quasi ) { Normalize( a ); }
}
return a;
}
template < typename T, Algorithm Aa, Algorithm Ab >
INLINE auto constexpr operator_add_ow_body ( qX_real::qx_real<T,Aa> & a, dX_real::dx_real<T,Ab> const& b ) NOEXCEPT {
Algorithm constexpr A=commonAlgorithm<Aa,Ab>::algorithm;
if ( A <= Algorithm::WeakAccurate ) {
QxW::add_QW_DW_QW ( _QX_(a), _DX_(b), _QX_(a) );
} else {
QxW::add_QQW_PA_QQW ( _QX_(a), _DX_(b), _QX_(a) );
if ( A != Algorithm::Quasi ) { Normalize( a ); }
}
return a;
}
template < typename T, Algorithm Aa >
INLINE auto constexpr operator_add_ow_body ( qX_real::qx_real<T,Aa> & a, T const& b ) NOEXCEPT {
if ( Aa <= Algorithm::WeakAccurate ) {
QxW::add_QW_SW_QW ( _QX_(a), _SX_(b), _QX_(a) );
} else {
QxW::add_QQW_SW_QQW ( _QX_(a), _SX_(b), _QX_(a) );
if ( Aa != Algorithm::Quasi ) { Normalize( a ); }
}
return a;
}
//
template < typename T, Algorithm Aa, template < typename _Tb_, Algorithm _Ab_ > class TXb, Algorithm Ab, T_mX(TXb<T,Ab>), A_owAble(Aa,Ab) >
INLINE auto constexpr operator_add_ow ( qX_real::qx_real<T,Aa> & a, TXb<T,Ab> const& b ) NOEXCEPT {
#if MX_REAL_USE_INF_NAN_EXCEPTION
auto flag = false;
auto e = qX_real::operator_add_exception ( a, b, flag );
if ( flag ) { return e; }
#endif
return qX_real::operator_add_ow_body ( a, b );
}
template < typename T, Algorithm A >
INLINE auto constexpr operator_add_ow ( qX_real::qx_real<T,A> & a, T const& b ) NOEXCEPT {
#if MX_REAL_USE_INF_NAN_EXCEPTION
auto flag = false;
auto e = qX_real::operator_add_exception ( a, b, flag );
if ( flag ) { return e; }
#endif
return qX_real::operator_add_ow_body ( a, b );
}
//
template < typename T, Algorithm Aa, template < typename _Tb_, Algorithm _Ab_ > class TXb, Algorithm Ab, T_mX(TXb<T,Ab>), A_owAble(Aa,Ab) >
INLINE auto constexpr operator+= ( qX_real::qx_real<T,Aa> & a, TXb<T,Ab> const& b ) NOEXCEPT {
return qX_real::operator_add_ow ( a, b );
}
template < typename T, Algorithm A >