cnl Namespace Reference

compositional numeric library More...

Namespaces
	literals
	user-defined literals
	numbers
	compositional numeric library, numbers header/namespace

Classes
struct	constant
	type whose value is also its non-type template parameter More...
struct	convert
	converts a value from one type to another More...
struct	custom_operator
	customization point for operator overloads More...
struct	custom_operator< _impl::convert_op, op_value< Input, _impl::native_tag >, op_value< scaled_integer< ResultRep, power< ResultExponent, ResultRadix > >, neg_inf_rounding_tag > >
	cnl::neg_inf_rounding_tag More...
struct	custom_operator< _impl::convert_op, op_value< Input, _impl::native_tag >, op_value< scaled_integer< ResultRep, power< ResultExponent, ResultRadix > >, tie_to_pos_inf_rounding_tag > >
	cnl::tie_to_pos_inf_rounding_tag More...
struct	custom_operator< _impl::convert_op, op_value< Input, nearest_rounding_tag >, op_value< Result, power< 0, Radix > > >
	cnl::nearest_rounding_tag More...
struct	fraction
	numeric type represented as the fraction, numerator / denominator More...
struct	from_rep
	generic function object that returns the number encapsulating a given value More...
struct	from_rep< Number, Rep >
	Specialization of from_rep for integer types. More...
struct	from_rep< scaled_integer< ArchetypeRep, power< Exponent, Radix > >, Rep >
	scaled_integer specialization of from_rep More...
struct	native_overflow_tag
	tag to match the overflow behavior of fundamental arithmetic types More...
struct	native_rounding_tag
	tag to match the rounding behavior of fundamental arithmetic types More...
struct	nearest_rounding_tag
	tag to specify round-to-nearest behavior in arithemtic operations More...
struct	neg_inf_rounding_tag
	tag to specify floor or round towards minus infinity rounding behavior in arithmetic operations More...
class	op_value
	operand or result of operation;used as Operands parameter of custom_operator More...
struct	power
	tag representing the scaling of an integer by a fixed factor More...
struct	rep_of
	meta-function object that returns the Rep type of a component More...
struct	rounding
	given a numeric type, defines member type as its rounding mode More...
struct	saturated_overflow_tag
	tag to match the overflow behavior of fundamental arithmetic types More...
struct	set_digits
	convert a type to a similar type with the given number of digits More...
struct	set_rep
	meta-function object that transforms a component from one Rep type to another More...
struct	set_rep< elastic_integer< Digits, Narrowest >, Rep >
	elastic_integer specialization of set_rep More...
struct	set_rounding
	given a numeric type, defines member type as the equivalent type with the given rounding mode User-specializations of this type are permitted. More...
struct	set_tag
	meta-function object that transforms a component from one Tag type to another More...
struct	set_tag< elastic_integer< Digits, Narrowest >, elastic_tag< NewDigits, NewNarrowest > >
	elastic_integer specialization of set_tag More...
struct	tag_of
	meta-function object that returns the Tag type of a component More...
struct	tie_to_pos_inf_rounding_tag
	tag to specify tie to positive inf rounding behavior in arithmetic operations More...
struct	to_rep
	Returns the value encapsulated in number. More...
struct	trapping_overflow_tag
	tag to specify trap-on-overflow behavior in arithemtic operations More...
struct	undefined_overflow_tag
	tag to specify undefined-behavior-on-overflow behavior in arithemtic operations More...

Typedefs
template<int Digits = digits_v<int>, integer Narrowest = int>
using	elastic_integer = _impl::wrapper< typename elastic_tag< Digits, Narrowest >::rep, elastic_tag< Digits, Narrowest > >
	An integer type with auto-widening operators. More...
template<int Digits, scaled_tag Scale = power<>, class Narrowest = signed>
using	elastic_scaled_integer = scaled_integer< elastic_integer< Digits, Narrowest >, Scale >
	literal real number approximation that uses fixed-point arithmetic and auto-widens to avoid overflow More...
template<unsigned NumDigits = digits_v<int>>
using	multiprecision = signed_multiprecision< NumDigits >
	alias to a signed, fixed-size Boost.Multiprecision type suitable for specializing scaled_integer More...
template<typename Rep = int, overflow_tag Tag = undefined_overflow_tag>
using	overflow_integer = _impl::wrapper< Rep, Tag >
	An integer for which out-of-range arithmetic results in overflow. More...
template<typename Rep = int, rounding_tag Tag = nearest_rounding_tag>
using	rounding_integer = _impl::wrapper< Rep, Tag >
	An integer which facilitates different rounding modes. More...
template<typename Number >
using	rounding_t = typename rounding< Number >::type
	helper alias of rounding More...
template<integer Rep = int, class Scale = power<>>
using	scaled_integer = _impl::wrapper< Rep, Scale >
	literal real number approximation that uses fixed-point arithmetic More...
template<class T , int Digits>
using	set_digits_t = typename set_digits< T, Digits >::type
	Alias to set_digits.
template<typename Number , rounding_tag RoundingTag>
using	set_rounding_t = typename set_rounding< Number, RoundingTag >::type
	helper alias of set_rounding More...
template<unsigned NumDigits = digits_v<int>>
using	signed_multiprecision = _sized_integer_impl::number< NumDigits+1, _bmp::signed_magnitude >
	alias to a signed, fixed-size Boost.Multiprecision type suitable for specializing scaled_integer More...
template<int Digits = digits_v<int>, rounding_tag RoundingTag = nearest_rounding_tag, overflow_tag OverflowTag = undefined_overflow_tag, class Narrowest = int>
using	static_integer = _impl::static_integer< Digits, RoundingTag, OverflowTag, Narrowest >
	a general-purpose integer type More...
template<int Digits, int Exponent = 0, rounding_tag RoundingTag = nearest_rounding_tag, overflow_tag OverflowTag = undefined_overflow_tag, integer Narrowest = int>
using	static_number = scaled_integer< _impl::static_integer< Digits, RoundingTag, OverflowTag, Narrowest >, power< Exponent > >
	a general-purpose fixed-point real number type More...
template<unsigned NumDigits = digits_v<unsigned>>
using	unsigned_multiprecision = _sized_integer_impl::number< NumDigits, _bmp::unsigned_magnitude >
	alias to an unsigned, fixed-size Boost.Multiprecision type suitable for specializing scaled_integer More...
template<int Digits = digits_v<int>, typename Narrowest = int>
using	wide_integer = _impl::wrapper< typename wide_tag< Digits, Narrowest >::rep, wide_tag< Digits, Narrowest > >
	An integer of limitless width.

Functions
template<typename Rep , class Scale >
constexpr auto	abs (scaled_integer< Rep, Scale > const &x) noexcept -> decltype(-x)
	absolute value More...
template<class Rep , int Exponent>
constexpr auto	exp2 (scaled_integer< Rep, power< Exponent >> x) -> scaled_integer< Rep, power< Exponent >>
	Calculates exp2(x), i.e. 2^xAccurate to 1LSB for up to 32 bit underlying representation. More...
template<typename Narrowest = int, ::cnl::intmax_t Value = 0>
constexpr auto	make_elastic_scaled_integer (constant< Value >) -> elastic_scaled_integer< std::max(digits_v< constant< Value >> - trailing_bits(Value), 1), power< trailing_bits(Value)>, Narrowest >
	generate an elastic_scaled_integer object of given value More...
template<typename Narrowest = void, typename Integral = int>
constexpr auto	make_elastic_scaled_integer (Integral const &value) -> elastic_scaled_integer< std::numeric_limits< Integral >::digits, power<>, typename std::conditional< std::is_same< void, Narrowest >::value, _impl::adopt_signedness_t< int, Integral >, Narrowest >::type >
	generate an elastic_scaled_integer object of given value More...
template<typename Value >
constexpr auto	make_scaled_integer (Value const &value)
	makes a scaled_integer object from a given value More...
template<rounding_tag RoundingTag = _impl::tag_of_t<rounding_integer<>>, overflow_tag OverflowTag = _impl::tag_of_t<overflow_integer<>>, integer Narrowest = int, class Input = int, ::cnl::intmax_t Value>
constexpr auto	make_static_number (constant< Value > const &) -> static_number< _impl::used_digits(Value) - trailing_bits(Value), trailing_bits(Value), RoundingTag, OverflowTag, Narrowest >
	constructs a static_number from a given constant value
template<rounding_tag RoundingTag = nearest_rounding_tag, overflow_tag OverflowTag = undefined_overflow_tag, integer Narrowest = int, class Input = int>
constexpr auto	make_static_number (Input const &input) -> static_number< std::numeric_limits< Input >::digits, 0, RoundingTag, OverflowTag, Narrowest >
	constructs a static_number from a given variable
template<int LhsExponent, int RhsExponent, int Radix>
constexpr auto	operator+ (power< LhsExponent, Radix >, power< RhsExponent, Radix >) -> power< std::min(LhsExponent, RhsExponent), Radix >
	value of template parameter, Exponent
template<class Dividend , class Divisor >
constexpr auto	quotient (Dividend const &dividend, Divisor const &divisor)
	calculates the quotient of two scaled_integer values More...
template<int Digits, class Narrowest >
constexpr auto	sqrt (elastic_integer< Digits, Narrowest > const &x)
	elastic_integer overload of cnl::sqrt More...
template<integer Integer>
constexpr auto	sqrt (Integer const &x)
	integer overload of cnl::sqrt More...
template<typename Rep , int Exponent, int Radix>
constexpr auto	sqrt (scaled_integer< Rep, power< Exponent, Radix >> const &x)
	scaled_integer overload of cnl::sqrt More...

Variables
template<typename T >
constexpr int	digits_v = 0
	provide number of binary digits of the given type More...

Detailed Description

compositional numeric library

Typedef Documentation

elastic_integer

template<int Digits = digits_v<int>, integer Narrowest = int>

using cnl::elastic_integer = typedef _impl::wrapper< typename elastic_tag<Digits, Narrowest>::rep, elastic_tag<Digits, Narrowest> >

An integer type with auto-widening operators.

Template Parameters

Digits	a count of the number of digits needed to express the number
Narrowest	the most narrow integer type to use for storage

Arithmetic operations return numbers with the appropriate number of digits necessary to store the correct value, without risk of overflow. For instance, when two elastic_integer values are multiplied together, the resultant type has Digits set to the sum of the operands' Digits.

Note

The lowest value of a signed elastic_integer type is the negative of its maximum value. E.g. the range of elastic_integer<7> is [-127..127], and not [-128..127]. This prevents range errors or wasteful stretching or Digits that would otherwise occur in relation to - and / operators.

The value is stored in an object of Narrowest or set_digits_t<Narrowest, Digits>.

See also

elastic_scaled_integer

elastic_scaled_integer

template<int Digits, scaled_tag Scale = power<>, class Narrowest = signed>

using cnl::elastic_scaled_integer = typedef scaled_integer<elastic_integer<Digits, Narrowest>, Scale>

literal real number approximation that uses fixed-point arithmetic and auto-widens to avoid overflow

Template Parameters

Digits	the total number of integer and fractional digits stored
Scale	the scale of the integer; defaults to power
Narrowest	the most narrow integer type to use to represent values

See also

elastic_integer

multiprecision

template<unsigned NumDigits = digits_v<int>>

using cnl::multiprecision = typedef signed_multiprecision<NumDigits>

alias to a signed, fixed-size Boost.Multiprecision type suitable for specializing scaled_integer

Template Parameters

NumDigits

number of digits constituting the integer value (excluding sign bit)

See also

signed_multiprecision, unsigned_multiprecision

overflow_integer

template<typename Rep = int, overflow_tag Tag = undefined_overflow_tag>

using cnl::overflow_integer = typedef _impl::wrapper<Rep, Tag>

An integer for which out-of-range arithmetic results in overflow.

Template Parameters

Rep	the underlying type used to represent the value; defaults to int
Tag	tag specifying the overflow-handling strategy; defaults to undefined_overflow_tag

See also

native_overflow_tag, saturated_overflow_tag, trapping_overflow_tag, undefined_overflow_tag

rounding_integer

template<typename Rep = int, rounding_tag Tag = nearest_rounding_tag>

using cnl::rounding_integer = typedef _impl::wrapper<Rep, Tag>

An integer which facilitates different rounding modes.

Template Parameters

Rep	the underlying type used to represent the value; defaults to int
Tag	tag specifying the rounding mode; defaults to nearest_rounding_tag

See also

native_rounding_tag, nearest_rounding_tag, neg_inf_rounding_tag, tie_to_pos_inf_rounding_tag

rounding_t

template<typename Number >

using cnl::rounding_t = typedef typename rounding<Number>::type

helper alias of rounding

See also

rounding, set_rounding_t, native_rounding_tag, nearest_rounding_tag

scaled_integer

template<integer Rep = int, class Scale = power<>>

using cnl::scaled_integer = typedef _impl::wrapper<Rep, Scale>

literal real number approximation that uses fixed-point arithmetic

Template Parameters

Rep	the underlying type used to represent the value; defaults to int
Scale	the scale of the value represented with Rep; defaults to power

Uses an integer to approximate a real number. Scales the integer by a factor specified by Scale to produce the scaled number. By default, represents an int with no fractional digits using binary arithmetic.

Note

Currently, only binary support and partial decimal support is tested.

The type makes no effort to avoid errors resulting from overflow or underflow. Consider specializing with:

• rounding_integer to reduce precision loss;
• elastic_integer to avoid out-of-range errors;
• overflow_integer to detect out-of-range errors; and
• static_integer to combine all of the above.

Examples

To define a fixed-point value 1 byte in size with a sign bit, 4 integer bits and 3 fractional bits:

    constexpr auto n = scaled_integer<std::int8_t, power<-3>>{-2.75};
    static_assert(n == -2.75, "fixed-point type was unable to store the value");

set_rounding_t

template<typename Number , rounding_tag RoundingTag>

Tag cnl::set_rounding_t< _impl::rep_of_t< Number >, Tag >

Initial value:

{
    }

helper alias of set_rounding

See also

set_rounding, rounding_t, native_rounding_tag, nearest_rounding_tag

signed_multiprecision

template<unsigned NumDigits = digits_v<int>>

using cnl::signed_multiprecision = typedef _sized_integer_impl::number<NumDigits + 1, _bmp::signed_magnitude>

alias to a signed, fixed-size Boost.Multiprecision type suitable for specializing scaled_integer

Template Parameters

NumDigits

number of digits constituting the integer value (excluding sign bit)

See also

unsigned_multiprecision, multiprecision

static_integer

template<int Digits = digits_v<int>, rounding_tag RoundingTag = nearest_rounding_tag, overflow_tag OverflowTag = undefined_overflow_tag, class Narrowest = int>

using cnl::static_integer = typedef _impl::static_integer<Digits, RoundingTag, OverflowTag, Narrowest>

a general-purpose integer type

Template Parameters

Digits	number of binary digits
OverflowTag	behavior exhibited on out-of-range conditions
RoundingTag	behavior exhibited on precision loss
Narrowest	narrowest integer with which to represent the value

See also

static_number

static_number

template<int Digits, int Exponent = 0, rounding_tag RoundingTag = nearest_rounding_tag, overflow_tag OverflowTag = undefined_overflow_tag, integer Narrowest = int>

using cnl::static_number = typedef scaled_integer< _impl::static_integer<Digits, RoundingTag, OverflowTag, Narrowest>, power<Exponent> >

a general-purpose fixed-point real number type

Template Parameters

Digits	number of binary digits
Exponent	the exponent used to scale the integer value
OverflowTag	behavior exhibited on out-of-range conditions
RoundingTag	behavior exhibited on precision loss
Narrowest	narrowest integer with which to represent the value

See also

static_integer

unsigned_multiprecision

template<unsigned NumDigits = digits_v<unsigned>>

using cnl::unsigned_multiprecision = typedef _sized_integer_impl::number<NumDigits, _bmp::unsigned_magnitude>

alias to an unsigned, fixed-size Boost.Multiprecision type suitable for specializing scaled_integer

Template Parameters

NumDigits

number of digits constituting the integer value (excluding sign bit)

See also

signed_multiprecision, multiprecision

Function Documentation

abs()

template<typename Rep , class Scale >

constexpr auto cnl::abs ( scaled_integer< Rep, Scale > const &  x ) -> decltype(-x)

constexprnoexcept

absolute value

Parameters

x	input parameter

Returns

|x|

See also

std::abs

exp2()

template<class Rep , int Exponent>

constexpr auto cnl::exp2 ( scaled_integer< Rep, power< Exponent >>  x ) -> scaled_integer<Rep, power<Exponent>>

constexpr

Calculates exp2(x), i.e. 2^xAccurate to 1LSB for up to 32 bit underlying representation.

Template Parameters

x	the input value as a scaled_integer

Returns

the result of the exponential, in the same representation as x

make_elastic_scaled_integer() [1/2]

template<typename Narrowest = int, ::cnl::intmax_t Value = 0>

constexpr auto cnl::make_elastic_scaled_integer ( constant< Value >  ) -> elastic_scaled_integer< std::max(digits_v<constant<Value>> - trailing_bits(Value), 1), power<trailing_bits(Value)>, Narrowest>

constexpr

generate an elastic_scaled_integer object of given value

Template Parameters

Narrowest	the narrowest type to use as storage in the resultant elastic_scaled_integer object
Integral	the type of Value
Value	the integer number to be represented

Returns

the given value to be represented using an elastic_scaled_integer type

Note

The return type is guaranteed to be no larger than is necessary to represent the value.

Example

To define an int-sized object using make_elastic_scaled_integer and constant

    // std::uint8_t specifies the type of const_integer - not elastic
    constexpr auto n = make_elastic_scaled_integer(constant<0xAA>{});
 
    static_assert(n == 0xAA, "n now has the value, 1024");
    static_assert(
            std::is_same<decltype(n), elastic_scaled_integer<7, power<1>, int> const>::value,
            "by default make_elastic_scaled_integer uses the most efficient type it can");

make_elastic_scaled_integer() [2/2]

template<typename Narrowest = void, typename Integral = int>

constexpr auto cnl::make_elastic_scaled_integer ( Integral const &  value ) -> elastic_scaled_integer< std::numeric_limits<Integral>::digits, power<>, typename std::conditional< std::is_same<void, Narrowest>::value, _impl::adopt_signedness_t<int, Integral>, Narrowest>::type>

constexpr

generate an elastic_scaled_integer object of given value

Template Parameters

Narrowest	the most narrow storage type of the resultant cnl::elastic_scaled_integer object
Integral	the type of value

Parameters

value

the value with which to initialize the elastic_scaled_integer object

Note

The return type is guaranteed to be no larger than is necessary to represent the value.

Example

To define a byte-sized object using make_elastic_scaled_integer and _c:

    constexpr auto n = make_elastic_scaled_integer<char>(constant<1536>{});
 
    static_assert(n == 1536, "n now has the value, 1536");
    static_assert(
            std::is_same<decltype(n), elastic_scaled_integer<2, power<9>, char> const>::value,
            "by default make_elastic_scaled_integer uses the most efficient type it can");

make_scaled_integer()

template<typename Value >

constexpr auto cnl::make_scaled_integer ( Value const &  value )

constexpr

makes a scaled_integer object from a given value

Template Parameters

Value

the type of the value that is to be made into a scaled_integer value

Parameters

value

the value from which to make the scaled_integer object

Referenced by quotient().

quotient()

template<class Dividend , class Divisor >

constexpr auto cnl::quotient ( Dividend const &  dividend, Divisor const &  divisor  )

constexpr

calculates the quotient of two scaled_integer values

Template Parameters

Dividend	the dividend (top number) of the division
Divisor	the divisor (bottom number) of the division

Parameters

dividend,divisor

dividend and divisor

Returns

quotient: dividend / divisor

Note

This function provides 'quasi-exact' division as described in P1368.

References make_scaled_integer().

sqrt() [1/3]

template<int Digits, class Narrowest >

auto cnl::sqrt ( elastic_integer< Digits, Narrowest > const &  x )

constexpr

elastic_integer overload of cnl::sqrt

Returns

square root of x

Note

This function has O(n) complexity where n is the number of significant digits.

Precondition

x must be non-negative

References sqrt().

sqrt() [2/3]

template<integer Integer>

constexpr auto cnl::sqrt ( Integer const &  x )

constexpr

integer overload of cnl::sqrt

Returns

square root of x

Note

This function has O(n) complexity where n is the number of significant digits.

The algorithm is taken from here.

Precondition

x must be non-negative

Referenced by sqrt().

sqrt() [3/3]

template<typename Rep , int Exponent, int Radix>

auto cnl::sqrt ( scaled_integer< Rep, power< Exponent, Radix >> const &  x )

constexpr

scaled_integer overload of cnl::sqrt

Returns

square root of x

Note

This function has O(n) complexity where n is the number of significant digits.

Precondition

x must be non-negative

Exponent must be even

References sqrt().

Variable Documentation

digits_v

template<typename T >

constexpr int cnl::digits_v< _bmp::number< _bmp::cpp_int_backend< NumBits, NumBits, _bmp::unsigned_magnitude > > > = 0

constexpr

provide number of binary digits of the given type

Template Parameters

T	the source type
Digits	the desired number of digits

Note

does not include a sign bit

See also

set_digits