29 Numerics library [numerics]

29.1 General [numerics.general]

This Clause describes components that C++ programs may use to perform seminumerical operations.

The following subclauses describe components for complex number types, random number generation, numeric (n-at-a-time) arrays, generalized numeric algorithms, and mathematical constants and functions for floating-point types, as summarized in Table 120 .

Table 120: Numerics library summary [tab:numerics.summary]

🔗		Subclause	Header
🔗	[numeric.requirements]	Requirements
🔗	[cfenv]	Floating-point environment	<cfenv>
🔗	[complex.numbers]	Complex numbers	<complex>
🔗	[rand]	Random number generation	<random>
🔗	[numarray]	Numeric arrays	<valarray>
🔗	[c.math]	Mathematical functions for floating-point types	<cmath>, <cstdlib>
🔗	[numbers]	Numbers	<numbers>
🔗	[linalg]	Linear algebra	<linalg>

29.2 Numeric type requirements [numeric.requirements]

The complex and valarray components are parameterized by the type of information they contain and manipulate.

A C++ program shall instantiate these components only with a numeric type.

A numeric type is a cv-unqualified object type T that meets the Cpp17DefaultConstructible, Cpp17CopyConstructible, Cpp17CopyAssignable, and Cpp17Destructible requirements ([utility.arg.requirements]).245

If any operation on T throws an exception the effects are undefined.

In addition, many member and related functions of valarray<T> can be successfully instantiated and will exhibit well-defined behavior if and only if T meets additional requirements specified for each such member or related function.

[Example 1:

It is valid to instantiate valarray<complex>, but operator>() will not be successfully instantiated for valarray<complex> operands, since complex does not have any ordering operators.

— end example]

245)245)

In other words, value types.

These include arithmetic types, pointers, the library class complex, and instantiations of valarray for value types.

29.3 The floating-point environment [cfenv]

29.3.1 Header <cfenv> synopsis [cfenv.syn]

🔗

#define FE_ALL_EXCEPT see below #define FE_DIVBYZERO see below // optional #define FE_INEXACT see below // optional #define FE_INVALID see below // optional #define FE_OVERFLOW see below // optional #define FE_UNDERFLOW see below // optional #define FE_DOWNWARD see below // optional #define FE_TONEAREST see below // optional #define FE_TOWARDZERO see below // optional #define FE_UPWARD see below // optional #define FE_DFL_ENV see below namespace std { // types using fenv_t = object type; using fexcept_t = object type; // functions int feclearexcept(int except); int fegetexceptflag(fexcept_t* pflag, int except); int feraiseexcept(int except); int fesetexceptflag(const fexcept_t* pflag, int except); int fetestexcept(int except); int fegetround(); int fesetround(int mode); int fegetenv(fenv_t* penv); int feholdexcept(fenv_t* penv); int fesetenv(const fenv_t* penv); int feupdateenv(const fenv_t* penv); }

The contents and meaning of the header <cfenv> are the same as the C standard library header <fenv.h>.

[Note 1:

This document does not require an implementation to support the FENV_ACCESS pragma; it is implementation-defined ([cpp.pragma]) whether the pragma is supported.

As a consequence, it is implementation-defined whether these functions can be used to test floating-point status flags, set floating-point control modes, or run under non-default mode settings.

If the pragma is used to enable control over the floating-point environment, this document does not specify the effect on floating-point evaluation in constant expressions.

— end note]

29.3.2 Threads [cfenv.thread]

The floating-point environment has thread storage duration .

The initial state for a thread's floating-point environment is the state of the floating-point environment of the thread that constructs the corresponding thread object ([thread.thread.class]) or jthread object ([thread.jthread.class]) at the time it constructed the object.

[Note 1:

That is, the child thread gets the floating-point state of the parent thread at the time of the child's creation.

— end note]

A separate floating-point environment is maintained for each thread.

Each function accesses the environment corresponding to its calling thread.

29.4 Complex numbers [complex.numbers]

29.4.1 General [complex.numbers.general]

The header <complex> defines a class template, and numerous functions for representing and manipulating complex numbers.

The effect of instantiating the template complex for any type that is not a cv-unqualified floating-point type ([basic.fundamental]) is unspecified.

Specializations of complex for cv-unqualified floating-point types are trivially copyable literal types ([basic.types.general]).

If the result of a function is not mathematically defined or not in the range of representable values for its type, the behavior is undefined.

If z is an lvalue of type cv complex<T> then:

(4.1)
the expression reinterpret_cast<cv T(&)[2]>(z) is well-formed,
(4.2)
reinterpret_cast<cv T(&)[2]>(z)[0] designates the real part of z, and
(4.3)
reinterpret_cast<cv T(&)[2]>(z)[1] designates the imaginary part of z.

Moreover, if a is an expression of type cv complex<T>* and the expression a[i] is well-defined for an integer expression i, then:

(4.4)
reinterpret_cast<cv T*>(a)[2*i] designates the real part of a[i], and
(4.5)
reinterpret_cast<cv T*>(a)[2*i + 1] designates the imaginary part of a[i].

29.4.2 Header <complex> synopsis [complex.syn]

🔗

namespace std { // [complex], class template complex template<class T> class complex; // [complex.ops], operators template<class T> constexpr complex<T> operator+(const complex<T>&, const complex<T>&); template<class T> constexpr complex<T> operator+(const complex<T>&, const T&); template<class T> constexpr complex<T> operator+(const T&, const complex<T>&); template<class T> constexpr complex<T> operator-(const complex<T>&, const complex<T>&); template<class T> constexpr complex<T> operator-(const complex<T>&, const T&); template<class T> constexpr complex<T> operator-(const T&, const complex<T>&); template<class T> constexpr complex<T> operator*(const complex<T>&, const complex<T>&); template<class T> constexpr complex<T> operator*(const complex<T>&, const T&); template<class T> constexpr complex<T> operator*(const T&, const complex<T>&); template<class T> constexpr complex<T> operator/(const complex<T>&, const complex<T>&); template<class T> constexpr complex<T> operator/(const complex<T>&, const T&); template<class T> constexpr complex<T> operator/(const T&, const complex<T>&); template<class T> constexpr complex<T> operator+(const complex<T>&); template<class T> constexpr complex<T> operator-(const complex<T>&); template<class T> constexpr bool operator==(const complex<T>&, const complex<T>&); template<class T> constexpr bool operator==(const complex<T>&, const T&); template<class T, class charT, class traits> basic_istream<charT, traits>& operator>>(basic_istream<charT, traits>&, complex<T>&); template<class T, class charT, class traits> basic_ostream<charT, traits>& operator<<(basic_ostream<charT, traits>&, const complex<T>&); // [complex.value.ops], values template<class T> constexpr T real(const complex<T>&); template<class T> constexpr T imag(const complex<T>&); template<class T> constexpr T abs(const complex<T>&); template<class T> constexpr T arg(const complex<T>&); template<class T> constexpr T norm(const complex<T>&); template<class T> constexpr complex<T> conj(const complex<T>&); template<class T> constexpr complex<T> proj(const complex<T>&); template<class T> constexpr complex<T> polar(const T&, const T& = T()); // [complex.transcendentals], transcendentals template<class T> constexpr complex<T> acos(const complex<T>&); template<class T> constexpr complex<T> asin(const complex<T>&); template<class T> constexpr complex<T> atan(const complex<T>&); template<class T> constexpr complex<T> acosh(const complex<T>&); template<class T> constexpr complex<T> asinh(const complex<T>&); template<class T> constexpr complex<T> atanh(const complex<T>&); template<class T> constexpr complex<T> cos (const complex<T>&); template<class T> constexpr complex<T> cosh (const complex<T>&); template<class T> constexpr complex<T> exp (const complex<T>&); template<class T> constexpr complex<T> log (const complex<T>&); template<class T> constexpr complex<T> log10(const complex<T>&); template<class T> constexpr complex<T> pow (const complex<T>&, const T&); template<class T> constexpr complex<T> pow (const complex<T>&, const complex<T>&); template<class T> constexpr complex<T> pow (const T&, const complex<T>&); template<class T> constexpr complex<T> sin (const complex<T>&); template<class T> constexpr complex<T> sinh (const complex<T>&); template<class T> constexpr complex<T> sqrt (const complex<T>&); template<class T> constexpr complex<T> tan (const complex<T>&); template<class T> constexpr complex<T> tanh (const complex<T>&); // [complex.tuple], tuple interface template<class T> struct tuple_size; template<size_t I, class T> struct tuple_element; template<class T> struct tuple_size<complex<T>>; template<size_t I, class T> struct tuple_element<I, complex<T>>; template<size_t I, class T> constexpr T& get(complex<T>&) noexcept; template<size_t I, class T> constexpr T&& get(complex<T>&&) noexcept; template<size_t I, class T> constexpr const T& get(const complex<T>&) noexcept; template<size_t I, class T> constexpr const T&& get(const complex<T>&&) noexcept; // [complex.literals], complex literals inline namespace literals { inline namespace complex_literals { constexpr complex<long double> operator""il(long double); constexpr complex<long double> operator""il(unsigned long long); constexpr complex<double> operator""i(long double); constexpr complex<double> operator""i(unsigned long long); constexpr complex<float> operator""if(long double); constexpr complex<float> operator""if(unsigned long long); } } }

29.4.3 Class template complex [complex]

🔗

namespace std { template<class T> class complex { public: using value_type = T; constexpr complex(const T& re = T(), const T& im = T()); constexpr complex(const complex&) = default; template<class X> constexpr explicit(see below) complex(const complex<X>&); constexpr T real() const; constexpr void real(T); constexpr T imag() const; constexpr void imag(T); constexpr complex& operator= (const T&); constexpr complex& operator+=(const T&); constexpr complex& operator-=(const T&); constexpr complex& operator*=(const T&); constexpr complex& operator/=(const T&); constexpr complex& operator=(const complex&); template<class X> constexpr complex& operator= (const complex<X>&); template<class X> constexpr complex& operator+=(const complex<X>&); template<class X> constexpr complex& operator-=(const complex<X>&); template<class X> constexpr complex& operator*=(const complex<X>&); template<class X> constexpr complex& operator/=(const complex<X>&); }; }

The class complex describes an object that can store the Cartesian components, real() and imag(), of a complex number.

29.4.4 Member functions [complex.members]

🔗

constexpr complex(const T& re = T(), const T& im = T());

Postconditions: real() == re && imag() == im is true.

🔗

template<class X> constexpr explicit(see below) complex(const complex<X>& other);

Effects: Initializes the real part with other.real() and the imaginary part with other.imag().

Remarks: The expression inside explicit evaluates to false if and only if the floating-point conversion rank of T is greater than or equal to the floating-point conversion rank of X.

🔗

constexpr T real() const;

Returns: The value of the real component.

🔗

constexpr void real(T val);

Effects: Assigns val to the real component.

🔗

constexpr T imag() const;

Returns: The value of the imaginary component.

🔗

constexpr void imag(T val);

Effects: Assigns val to the imaginary component.

29.4.5 Member operators [complex.member.ops]

🔗

constexpr complex& operator+=(const T& rhs);

Effects: Adds the scalar value rhs to the real part of the complex value *this and stores the result in the real part of *this, leaving the imaginary part unchanged.

Returns: *this.

🔗

constexpr complex& operator-=(const T& rhs);

Effects: Subtracts the scalar value rhs from the real part of the complex value *this and stores the result in the real part of *this, leaving the imaginary part unchanged.

Returns: *this.

🔗

constexpr complex& operator*=(const T& rhs);

Effects: Multiplies the scalar value rhs by the complex value *this and stores the result in *this.

Returns: *this.

🔗

constexpr complex& operator/=(const T& rhs);

Effects: Divides the scalar value rhs into the complex value *this and stores the result in *this.

Returns: *this.

🔗

template<class X> constexpr complex& operator=(const complex<X>& rhs);

Effects: Assigns the value rhs.real() to the real part and the value rhs.imag() to the imaginary part of the complex value *this.

Returns: *this.

🔗

template<class X> constexpr complex& operator+=(const complex<X>& rhs);

Effects: Adds the complex value rhs to the complex value *this and stores the sum in *this.

Returns: *this.

🔗

template<class X> constexpr complex& operator-=(const complex<X>& rhs);

Effects: Subtracts the complex value rhs from the complex value *this and stores the difference in *this.

Returns: *this.

🔗

template<class X> constexpr complex& operator*=(const complex<X>& rhs);

Effects: Multiplies the complex value rhs by the complex value *this and stores the product in *this.

Returns: *this.

🔗

template<class X> constexpr complex& operator/=(const complex<X>& rhs);

Effects: Divides the complex value rhs into the complex value *this and stores the quotient in *this.

Returns: *this.

29.4.6 Non-member operations [complex.ops]

🔗

template<class T> constexpr complex<T> operator+(const complex<T>& lhs);

Returns: complex<T>(lhs).

🔗

template<class T> constexpr complex<T> operator+(const complex<T>& lhs, const complex<T>& rhs);
template<class T> constexpr complex<T> operator+(const complex<T>& lhs, const T& rhs);
template<class T> constexpr complex<T> operator+(const T& lhs, const complex<T>& rhs);

Returns: complex<T>(lhs) += rhs.

🔗

template<class T> constexpr complex<T> operator-(const complex<T>& lhs);

Returns: complex<T>(-lhs.real(),-lhs.imag()).

🔗

template<class T> constexpr complex<T> operator-(const complex<T>& lhs, const complex<T>& rhs);
template<class T> constexpr complex<T> operator-(const complex<T>& lhs, const T& rhs);
template<class T> constexpr complex<T> operator-(const T& lhs, const complex<T>& rhs);

Returns: complex<T>(lhs) -= rhs.

🔗

template<class T> constexpr complex<T> operator*(const complex<T>& lhs, const complex<T>& rhs);
template<class T> constexpr complex<T> operator*(const complex<T>& lhs, const T& rhs);
template<class T> constexpr complex<T> operator*(const T& lhs, const complex<T>& rhs);

Returns: complex<T>(lhs) *= rhs.

🔗

template<class T> constexpr complex<T> operator/(const complex<T>& lhs, const complex<T>& rhs);
template<class T> constexpr complex<T> operator/(const complex<T>& lhs, const T& rhs);
template<class T> constexpr complex<T> operator/(const T& lhs, const complex<T>& rhs);

Returns: complex<T>(lhs) /= rhs.

🔗

template<class T> constexpr bool operator==(const complex<T>& lhs, const complex<T>& rhs);
template<class T> constexpr bool operator==(const complex<T>& lhs, const T& rhs);

Returns: lhs.real() == rhs.real() && lhs.imag() == rhs.imag().

Remarks: The imaginary part is assumed to be T(), or 0.0, for the T arguments.

🔗

template<class T, class charT, class traits>
  basic_istream<charT, traits>& operator>>(basic_istream<charT, traits>& is, complex<T>& x);

Preconditions: The input values are convertible to T.

Effects: Extracts a complex number x of the form: u, (u), or (u,v), where u is the real part and v is the imaginary part ([istream.formatted]).

If bad input is encountered, calls is.setstate(ios_base::failbit) (which may throw ios_base::failure ([iostate.flags])).

Returns: is.

Remarks: This extraction is performed as a series of simpler extractions.

Therefore, the skipping of whitespace is specified to be the same for each of the simpler extractions.

🔗

template<class T, class charT, class traits>
  basic_ostream<charT, traits>& operator<<(basic_ostream<charT, traits>& o, const complex<T>& x);

Effects: Inserts the complex number x onto the stream o as if it were implemented as follows: basic_ostringstream<charT, traits> s; s.flags(o.flags()); s.imbue(o.getloc()); s.precision(o.precision()); s << '(' << x.real() << ',' << x.imag() << ')'; return o << s.str();

[Note 1:

In a locale in which comma is used as a decimal point character, the use of comma as a field separator can be ambiguous.

Inserting showpoint into the output stream forces all outputs to show an explicit decimal point character; as a result, all inserted sequences of complex numbers can be extracted unambiguously.

— end note]

29.4.7 Value operations [complex.value.ops]

🔗

template<class T> constexpr T real(const complex<T>& x);

Returns: x.real().

🔗

template<class T> constexpr T imag(const complex<T>& x);

Returns: x.imag().

🔗

template<class T> constexpr T abs(const complex<T>& x);

Returns: The magnitude of x.

🔗

template<class T> constexpr T arg(const complex<T>& x);

Returns: The phase angle of x, or atan2(imag(x), real(x)).

🔗

template<class T> constexpr T norm(const complex<T>& x);

Returns: The squared magnitude of x.

🔗

template<class T> constexpr complex<T> conj(const complex<T>& x);

Returns: The complex conjugate of x.

🔗

template<class T> constexpr complex<T> proj(const complex<T>& x);

Returns: The projection of x onto the Riemann sphere.

Remarks: Behaves the same as the C function cproj.

29.4.8 Transcendentals [complex.transcendentals]

🔗

template<class T> constexpr complex<T> acos(const complex<T>& x);

Returns: The complex arc cosine of x.

Remarks: Behaves the same as the C function cacos.

29.4.9 Tuple interface [complex.tuple]

🔗

template<class T>
struct tuple_size<complex<T>> : integral_constant<size_t, 2> {};

template<size_t I, class T>
struct tuple_element<I, complex<T>> {
  using type = T;
};

Mandates: I < 2 is true.

🔗

template<size_t I, class T>
  constexpr T& get(complex<T>& z) noexcept;
template<size_t I, class T>
  constexpr T&& get(complex<T>&& z) noexcept;
template<size_t I, class T>
  constexpr const T& get(const complex<T>& z) noexcept;
template<size_t I, class T>
  constexpr const T&& get(const complex<T>&& z) noexcept;

Mandates: I < 2 is true.

Returns: A reference to the real part of z if I == 0 is true, otherwise a reference to the imaginary part of z.

29.4.10 Additional overloads [cmplx.over]

The following function templates have additional constexpr overloads: arg norm conj proj imag real

The additional constexpr overloads are sufficient to ensure:

(2.1)
If the argument has a floating-point type T, then it is effectively cast to complex<T>.
(2.2)
Otherwise, if the argument has integer type, then it is effectively cast to complex<double>.

Function template pow has additional constexpr overloads sufficient to ensure, for a call with one argument of type complex<T1> and the other argument of type T2 or complex<T2>, both arguments are effectively cast to complex<common_type_t<T1, T2>>.

If common_type_t<T1, T2> is not well-formed, then the program is ill-formed.

29.4.11 Suffixes for complex number literals [complex.literals]

This subclause describes literal suffixes for constructing complex number literals.

The suffixes i, il, and if create complex numbers of the types complex<double>, complex<long double>, and complex<float> respectively, with their imaginary part denoted by the given literal number and the real part being zero.

🔗

constexpr complex<long double> operator""il(long double d);
constexpr complex<long double> operator""il(unsigned long long d);

Returns: complex<long double>{0.0L, static_cast<long double>(d)}.

🔗

constexpr complex<double> operator""i(long double d);
constexpr complex<double> operator""i(unsigned long long d);

Returns: complex<double>{0.0, static_cast<double>(d)}.

🔗

constexpr complex<float> operator""if(long double d);
constexpr complex<float> operator""if(unsigned long long d);

Returns: complex<float>{0.0f, static_cast<float>(d)}.

29.5 Random number generation [rand]

29.5.1 General [rand.general]

Subclause [rand] defines a facility for generating (pseudo-)random numbers.

In addition to a few utilities, four categories of entities are described: uniform random bit generators, random number engines, random number engine adaptors, and random number distributions.

These categorizations are applicable to types that meet the corresponding requirements, to objects instantiated from such types, and to templates producing such types when instantiated.

[Note 1:

These entities are specified in such a way as to permit the binding of any uniform random bit generator object e as the argument to any random number distribution object d, thus producing a zero-argument function object such as given by bind(d,e).

— end note]

Each of the entities specified in [rand] has an associated arithmetic type ([basic.fundamental]) identified as result_type.

With T as the result_type thus associated with such an entity, that entity is characterized:

(3.1)
as boolean or equivalently as boolean-valued, if T is bool;
(3.2)
otherwise as integral or equivalently as integer-valued, if numeric_limits<T>::is_integer is true;
(3.3)
otherwise as floating-point or equivalently as real-valued.

If integer-valued, an entity may optionally be further characterized as signed or unsigned, according to numeric_limits<T>::is_signed.

Unless otherwise specified, all descriptions of calculations in [rand] use mathematical real numbers.

Throughout [rand], the operators bitand, bitor, and xor denote the respective conventional bitwise operations.

Further:

(5.1)
the operator rshift denotes a bitwise right shift with zero-valued bits appearing in the high bits of the result, and
(5.2)
the operator ${l s h i f t}_{w}$ denotes a bitwise left shift with zero-valued bits appearing in the low bits of the result, and whose result is always taken modulo $2^{w}$ .

29.5.2 Header <random> synopsis [rand.synopsis]

#include <initializer_list> // see [initializer.list.syn] namespace std { // [rand.req.urng], uniform random bit generator requirements template<class G> concept uniform_random_bit_generator = see below; // [rand.eng.lcong], class template linear_congruential_engine template<class UIntType, UIntType a, UIntType c, UIntType m> class linear_congruential_engine; // [rand.eng.mers], class template mersenne_twister_engine template<class UIntType, size_t w, size_t n, size_t m, size_t r, UIntType a, size_t u, UIntType d, size_t s, UIntType b, size_t t, UIntType c, size_t l, UIntType f> class mersenne_twister_engine; // [rand.eng.sub], class template subtract_with_carry_engine template<class UIntType, size_t w, size_t s, size_t r> class subtract_with_carry_engine; // [rand.adapt.disc], class template discard_block_engine template<class Engine, size_t p, size_t r> class discard_block_engine; // [rand.adapt.ibits], class template independent_bits_engine template<class Engine, size_t w, class UIntType> class independent_bits_engine; // [rand.adapt.shuf], class template shuffle_order_engine template<class Engine, size_t k> class shuffle_order_engine; // [rand.eng.philox], class template philox_engine template<class UIntType, size_t w, size_t n, size_t r, UIntType... consts> class philox_engine; // [rand.predef], engines and engine adaptors with predefined parameters using minstd_rand0 = see below; using minstd_rand = see below; using mt19937 = see below; using mt19937_64 = see below; using ranlux24_base = see below; using ranlux48_base = see below; using ranlux24 = see below; using ranlux48 = see below; using knuth_b = see below; using philox4x32 = see below; using philox4x64 = see below; using default_random_engine = see below; // [rand.device], class random_device class random_device; // [rand.util.seedseq], class seed_seq class seed_seq; // [rand.util.canonical], function template generate_canonical template<class RealType, size_t digits, class URBG> RealType generate_canonical(URBG& g); namespace ranges { // [alg.rand.generate], generate_random template<class R, class G> requires output_range<R, invoke_result_t<G&>> && uniform_random_bit_generator<remove_cvref_t<G>> constexpr borrowed_iterator_t<R> generate_random(R&& r, G&& g); template<class G, output_iterator<invoke_result_t<G&>> O, sentinel_for<O> S> requires uniform_random_bit_generator<remove_cvref_t<G>> constexpr O generate_random(O first, S last, G&& g); template<class R, class G, class D> requires output_range<R, invoke_result_t<D&, G&>> && invocable<D&, G&> && uniform_random_bit_generator<remove_cvref_t<G>> constexpr borrowed_iterator_t<R> generate_random(R&& r, G&& g, D&& d); template<class G, class D, output_iterator<invoke_result_t<D&, G&>> O, sentinel_for<O> S> requires invocable<D&, G&> && uniform_random_bit_generator<remove_cvref_t<G>> constexpr O generate_random(O first, S last, G&& g, D&& d); } // [rand.dist.uni.int], class template uniform_int_distribution template<class IntType = int> class uniform_int_distribution; // [rand.dist.uni.real], class template uniform_real_distribution template<class RealType = double> class uniform_real_distribution; // [rand.dist.bern.bernoulli], class bernoulli_distribution class bernoulli_distribution; // [rand.dist.bern.bin], class template binomial_distribution template<class IntType = int> class binomial_distribution; // [rand.dist.bern.geo], class template geometric_distribution template<class IntType = int> class geometric_distribution; // [rand.dist.bern.negbin], class template negative_binomial_distribution template<class IntType = int> class negative_binomial_distribution; // [rand.dist.pois.poisson], class template poisson_distribution template<class IntType = int> class poisson_distribution; // [rand.dist.pois.exp], class template exponential_distribution template<class RealType = double> class exponential_distribution; // [rand.dist.pois.gamma], class template gamma_distribution template<class RealType = double> class gamma_distribution; // [rand.dist.pois.weibull], class template weibull_distribution template<class RealType = double> class weibull_distribution; // [rand.dist.pois.extreme], class template extreme_value_distribution template<class RealType = double> class extreme_value_distribution; // [rand.dist.norm.normal], class template normal_distribution template<class RealType = double> class normal_distribution; // [rand.dist.norm.lognormal], class template lognormal_distribution template<class RealType = double> class lognormal_distribution; // [rand.dist.norm.chisq], class template chi_squared_distribution template<class RealType = double> class chi_squared_distribution; // [rand.dist.norm.cauchy], class template cauchy_distribution template<class RealType = double> class cauchy_distribution; // [rand.dist.norm.f], class template fisher_f_distribution template<class RealType = double> class fisher_f_distribution; // [rand.dist.norm.t], class template student_t_distribution template<class RealType = double> class student_t_distribution; // [rand.dist.samp.discrete], class template discrete_distribution template<class IntType = int> class discrete_distribution; // [rand.dist.samp.pconst], class template piecewise_constant_distribution template<class RealType = double> class piecewise_constant_distribution; // [rand.dist.samp.plinear], class template piecewise_linear_distribution template<class RealType = double> class piecewise_linear_distribution; }

29.5.3 Requirements [rand.req]

29.5.3.1 General requirements [rand.req.genl]

Throughout [rand], the effect of instantiating a template:

(1.1)
that has a template type parameter named Sseq is undefined unless the corresponding template argument is cv-unqualified and meets the requirements of seed sequence .
(1.2)
that has a template type parameter named URBG is undefined unless the corresponding template argument is cv-unqualified and meets the requirements of uniform random bit generator .
(1.3)
that has a template type parameter named Engine is undefined unless the corresponding template argument is cv-unqualified and meets the requirements of random number engine .
(1.4)
that has a template type parameter named RealType is undefined unless the corresponding template argument is cv-unqualified and is one of float, double, or long double.
(1.5)
that has a template type parameter named IntType is undefined unless the corresponding template argument is cv-unqualified and is one of short, int, long, long long, unsigned short, unsigned int, unsigned long, or unsigned long long.
(1.6)
that has a template type parameter named UIntType is undefined unless the corresponding template argument is cv-unqualified and is one of unsigned short, unsigned int, unsigned long, or unsigned long long.

Throughout [rand], phrases of the form “x is an iterator of a specific kind” shall be interpreted as equivalent to the more formal requirement that “x is a value of a type meeting the requirements of the specified iterator type”.

Throughout [rand], any constructor that can be called with a single argument and that meets a requirement specified in this subclause shall be declared explicit.

29.5.3.2 Seed sequence requirements [rand.req.seedseq]

A seed sequence is an object that consumes a sequence of integer-valued data and produces a requested number of unsigned integer values i,

0 \leq i < 2^{32}

, based on the consumed data.

[Note 1:

Such an object provides a mechanism to avoid replication of streams of random variates.

This can be useful, for example, in applications requiring large numbers of random number engines.

— end note]

A class S meets the requirements of a seed sequence if the expressions shown in Table 121 are valid and have the indicated semantics, and if S also meets all other requirements of [rand.req.seedseq].

In Table 121 and throughout this subclause:

(2.1)
T is the type named by S's associated result_type;
(2.2)
q is a value of type S and r is a value of type S or const S;
(2.3)
ib and ie are input iterators with an unsigned integer value_type of at least 32 bits;
(2.4)
rb and re are mutable random access iterators with an unsigned integer value_type of at least 32 bits;
(2.5)
ob is an output iterator; and
(2.6)
il is a value of type initializer_list<T>.

Table 121: Seed sequence requirements [tab:rand.req.seedseq]

🔗	Expression	Return type	Pre/post-condition	Complexity
🔗	S::result_type	T	T is an unsigned integer type of at least 32 bits.	compile-time
🔗	S()		Creates a seed sequence with the same initial state as all other default-constructed seed sequences of type S.	constant
🔗	S(ib,ie)		Creates a seed sequence having internal state that depends on some or all of the bits of the supplied sequence [ib,ie).	$O (ie - ib)$
🔗	S(il)		Same as S(il.begin(), il.end()).	same as S(il.begin(), il.end())
🔗	q.generate(rb,re)	void	Does nothing if rb == re. Otherwise, fills the supplied sequence [rb,re) with 32-bit quantities that depend on the sequence supplied to the constructor and possibly also depend on the history of generate's previous invocations.	$O (re - rb)$
🔗	r.size()	size_t	The number of 32-bit units that would be copied by a call to r.param.	constant
🔗	r.param(ob)	void	Copies to the given destination a sequence of 32-bit units that can be provided to the constructor of a second object of type S, and that would reproduce in that second object a state indistinguishable from the state of the first object.	$O (r.size())$

29.5.3.3 Uniform random bit generator requirements [rand.req.urng]

A uniform random bit generator g of type G is a function object returning unsigned integer values such that each value in the range of possible results has (ideally) equal probability of being returned.

[Note 1:

The degree to which g's results approximate the ideal is often determined statistically.

— end note]

template<class G> concept uniform_random_bit_generator = invocable<G&> && unsigned_integral<invoke_result_t<G&>> && requires { { G::min() } -> same_as<invoke_result_t<G&>>; { G::max() } -> same_as<invoke_result_t<G&>>; requires bool_constant<(G::min() < G::max())>::value; };

Let g be an object of type G.

G models uniform_random_bit_generator only if

(2.1)
G::min() <= g(),
(2.2)
g() <= G::max(), and
(2.3)
g() has amortized constant complexity.

A class G meets the uniform random bit generator requirements if G models uniform_random_bit_generator, invoke_result_t<G&> is an unsigned integer type ([basic.fundamental]), and G provides a nested typedef-name result_type that denotes the same type as invoke_result_t<G&>.

29.5.3.4 Random number engine requirements [rand.req.eng]

A random number engine (commonly shortened to engine) e of type E is a uniform random bit generator that additionally meets the requirements (e.g., for seeding and for input/output) specified in this subclause.

At any given time, e has a state e

_{i}

for some integer i ≥ 0.

Upon construction, e has an initial state e

_{0}

An engine's state may be established via a constructor, a seed function, assignment, or a suitable operator>>.

E's specification shall define:

(3.1)
the size of E's state in multiples of the size of result_type, given as an integral constant expression;
(3.2)
the transition algorithm TA by which e's state e $_{i}$ is advanced to its successor state e $_{i + 1}$ ; and
(3.3)
the generation algorithm GA by which an engine's state is mapped to a value of type result_type.

A class E that meets the requirements of a uniform random bit generator also meets the requirements of a random number engine if the expressions shown in Table 122 are valid and have the indicated semantics, and if E also meets all other requirements of [rand.req.eng].

In Table 122 and throughout this subclause:

(4.1)
T is the type named by E's associated result_type;
(4.2)
e is a value of E, v is an lvalue of E, x and y are (possibly const) values of E;
(4.3)
s is a value of T;
(4.4)
q is an lvalue meeting the requirements of a seed sequence;
(4.5)
z is a value of type unsigned long long;
(4.6)
os is an lvalue of the type of some class template specialization basic_ostream<charT, traits>; and
(4.7)
is is an lvalue of the type of some class template specialization basic_istream<charT, traits>;

where charT and traits are constrained according to [strings] and [input.output].

Table 122: Random number engine requirements [tab:rand.req.eng]

🔗	Expression	Return type	Pre/post-condition	Complexity
🔗	E()		Creates an engine with the same initial state as all other default-constructed engines of type E.	$O (size of state)$
🔗	E(x)		Creates an engine that compares equal to x.	$O (size of state)$
🔗	E(s)		Creates an engine with initial state determined by s.	$O (size of state)$
🔗	E(q)246		Creates an engine with an initial state that depends on a sequence produced by one call to q.generate.	same as complexity of q.generate called on a sequence whose length is size of state
🔗	e.seed()	void	Postconditions: e == E().	same as E()
🔗	e.seed(s)	void	Postconditions: e == E(s).	same as E(s)
🔗	e.seed(q)	void	Postconditions: e == E(q).	same as E(q)
🔗	e()	T	Advances e's state e $_{i}$ to e $_{i + 1}$ $= T A ($ e $_{i}$ ) and returns GA(e $_{i}$ ).	per [rand.req.urng]
🔗	e.discard(z)247	void	Advances e's state e $_{i}$ to $e_{i + z}$ by any means equivalent to z consecutive calls e().	no worse than the complexity of z consecutive calls e()
🔗	x == y	bool	This operator is an equivalence relation. With $S_{x}$ and $S_{y}$ as the infinite sequences of values that would be generated by repeated future calls to x() and y(), respectively, returns true if $S_{x} = S_{y}$ ; else returns false.	$O (size of state)$
🔗	x != y	bool	!(x == y).	$O (size of state)$
🔗	os << x	reference to the type of os	With os.fmtflags set to ios_base::dec\|ios_base::left and the fill character set to the space character, writes to os the textual representation of x's current state. In the output, adjacent numbers are separated by one or more space characters. Postconditions: The os.fmtflags and fill character are unchanged.	$O (size of state)$
🔗	is >> v	reference to the type of is	With is.fmtflags set to ios_base::dec, sets v's state as determined by reading its textual representation from is. If bad input is encountered, ensures that v's state is unchanged by the operation and calls is.setstate(ios_base::failbit) (which may throw ios_base::failure ([iostate.flags])). If a textual representation written via os << x was subsequently read via is >> v, then x == v provided that there have been no intervening invocations of x or of v. Preconditions: is provides a textual representation that was previously written using an output stream whose imbued locale was the same as that of is, and whose type's template specialization arguments charT and traits were respectively the same as those of is. Postconditions: The is.fmtflags are unchanged.	$O (size of state)$

E shall meet the Cpp17CopyConstructible (Table 32) and Cpp17CopyAssignable (Table 34) requirements.

These operations shall each be of complexity no worse than

O (size of state)

246)246)

This constructor (as well as the subsequent corresponding seed() function) can be particularly useful to applications requiring a large number of independent random sequences.

247)247)

This operation is common in user code, and can often be implemented in an engine-specific manner so as to provide significant performance improvements over an equivalent naive loop that makes z consecutive calls e().

29.5.3.5 Random number engine adaptor requirements [rand.req.adapt]

A random number engine adaptor (commonly shortened to adaptor) a of type A is a random number engine that takes values produced by some other random number engine, and applies an algorithm to those values in order to deliver a sequence of values with different randomness properties.

An engine b of type B adapted in this way is termed a base engine in this context.

The expression a.base() shall be valid and shall return a const reference to a's base engine.

The requirements of a random number engine type shall be interpreted as follows with respect to a random number engine adaptor type.

🔗

A::A();

Effects: The base engine is initialized as if by its default constructor.

🔗

bool operator==(const A& a1, const A& a2);

Returns: true if a1's base engine is equal to a2's base engine.

Otherwise returns false.

🔗

A::A(result_type s);

Effects: The base engine is initialized with s.

🔗

template<class Sseq> A::A(Sseq& q);

Effects: The base engine is initialized with q.

🔗

void seed();

Effects: With b as the base engine, invokes b.seed().

🔗

void seed(result_type s);

Effects: With b as the base engine, invokes b.seed(s).

🔗

template<class Sseq> void seed(Sseq& q);

Effects: With b as the base engine, invokes b.seed(q).

A shall also meet the following additional requirements:

(10.1)
The complexity of each function shall not exceed the complexity of the corresponding function applied to the base engine.
(10.2)
The state of A shall include the state of its base engine.

The size of A's state shall be no less than the size of the base engine.
(10.3)
Copying A's state (e.g., during copy construction or copy assignment) shall include copying the state of the base engine of A.
(10.4)
The textual representation of A shall include the textual representation of its base engine.

29.5.3.6 Random number distribution requirements [rand.req.dist]

A random number distribution (commonly shortened to distribution) d of type D is a function object returning values that are distributed according to an associated mathematical probability density function p(z) or according to an associated discrete probability function

P (z_{i})

A distribution's specification identifies its associated probability function p(z) or

P (z_{i})

An associated probability function is typically expressed using certain externally-supplied quantities known as the parameters of the distribution.

Such distribution parameters are identified in this context by writing, for example, p(z | a,b) or

P (z_{i} | a, b)

, to name specific parameters, or by writing, for example, p(z |{p}) or

P (z_{i} | {p})

, to denote a distribution's parameters p taken as a whole.

A class D meets the requirements of a random number distribution if the expressions shown in Table 123 are valid and have the indicated semantics, and if D and its associated types also meet all other requirements of [rand.req.dist].

In Table 123 and throughout this subclause,

(3.1)
T is the type named by D's associated result_type;
(3.2)
P is the type named by D's associated param_type;
(3.3)
d is a value of D, and x and y are (possibly const) values of D;
(3.4)
glb and lub are values of T respectively corresponding to the greatest lower bound and the least upper bound on the values potentially returned by d's operator(), as determined by the current values of d's parameters;
(3.5)
p is a (possibly const) value of P;
(3.6)
g, g1, and g2 are lvalues of a type meeting the requirements of a uniform random bit generator;
(3.7)
os is an lvalue of the type of some class template specialization basic_ostream<charT, traits>; and
(3.8)
is is an lvalue of the type of some class template specialization basic_istream<charT, traits>;

where charT and traits are constrained according to [strings] and [input.output].

Table 123: Random number distribution requirements [tab:rand.req.dist]

🔗	Expression	Return type	Pre/post-condition	Complexity
🔗	D::result_type	T	T is an arithmetic type .	compile-time
🔗	D::param_type	P		compile-time
🔗	D()		Creates a distribution whose behavior is indistinguishable from that of any other newly default-constructed distribution of type D.	constant
🔗	D(p)		Creates a distribution whose behavior is indistinguishable from that of a distribution newly constructed directly from the values used to construct p.	same as p's construction
🔗	d.reset()	void	Subsequent uses of d do not depend on values produced by any engine prior to invoking reset.	constant
🔗	x.param()	P	Returns a value p such that D(p).param() == p.	no worse than the complexity of D(p)
🔗	d.param(p)	void	Postconditions: d.param() == p.	no worse than the complexity of D(p)
🔗	d(g)	T	With $p = d.param()$ , the sequence of numbers returned by successive invocations with the same object g is randomly distributed according to the associated p(z \|{p}) or $P (z_{i} \| {p})$ function.	amortized constant number of invocations of g
🔗	d(g,p)	T	The sequence of numbers returned by successive invocations with the same objects g and p is randomly distributed according to the associated p(z \|{p}) or $P (z_{i} \| {p})$ function.	amortized constant number of invocations of g
🔗	x.min()	T	Returns glb.	constant
🔗	x.max()	T	Returns lub.	constant
🔗	x == y	bool	This operator is an equivalence relation. Returns true if x.param() == y.param() and $S_{1} = S_{2}$ , where $S_{1}$ and $S_{2}$ are the infinite sequences of values that would be generated, respectively, by repeated future calls to x(g1) and y(g2) whenever g1 == g2. Otherwise returns false.	constant
🔗	x != y	bool	!(x == y).	same as x == y.
🔗	os << x	reference to the type of os	Writes to os a textual representation for the parameters and the additional internal data of x. Postconditions: The os.fmtflags and fill character are unchanged.
🔗	is >> d	reference to the type of is	Restores from is the parameters and additional internal data of the lvalue d. If bad input is encountered, ensures that d is unchanged by the operation and calls is.setstate(ios_base::failbit) (which may throw ios_base::failure ([iostate.flags])). Preconditions: is provides a textual representation that was previously written using an os whose imbued locale and whose type's template specialization arguments charT and traits were the same as those of is. Postconditions: The is.fmtflags are unchanged.

D shall meet the Cpp17CopyConstructible (Table 32) and Cpp17CopyAssignable (Table 34) requirements.

The sequence of numbers produced by repeated invocations of d(g) shall be independent of any invocation of os << d or of any const member function of D between any of the invocations of d(g).

If a textual representation is written using os << x and that representation is restored into the same or a different object y of the same type using is >> y, repeated invocations of y(g) shall produce the same sequence of numbers as would repeated invocations of x(g).

It is unspecified whether D::param_type is declared as a (nested) class or via a typedef.

In [rand], declarations of D::param_type are in the form of typedefs for convenience of exposition only.

P shall meet the Cpp17CopyConstructible (Table 32), Cpp17CopyAssignable (Table 34), and Cpp17EqualityComparable (Table 28) requirements.

For each of the constructors of D taking arguments corresponding to parameters of the distribution, P shall have a corresponding constructor subject to the same requirements and taking arguments identical in number, type, and default values.

Moreover, for each of the member functions of D that return values corresponding to parameters of the distribution, P shall have a corresponding member function with the identical name, type, and semantics.

P shall have a declaration of the form using distribution_type = D;

29.5.4 Random number engine class templates [rand.eng]

29.5.4.1 General [rand.eng.general]

Each type instantiated from a class template specified in [rand.eng] meets the requirements of a random number engine type.

Except where specified otherwise, the complexity of each function specified in [rand.eng] is constant.

Except where specified otherwise, no function described in [rand.eng] throws an exception.

Every function described in [rand.eng] that has a function parameter q of type Sseq& for a template type parameter named Sseq that is different from type seed_seq throws what and when the invocation of q.generate throws.

Descriptions are provided in [rand.eng] only for engine operations that are not described in [rand.req.eng] or for operations where there is additional semantic information.

In particular, declarations for copy constructors, for copy assignment operators, for streaming operators, and for equality and inequality operators are not shown in the synopses.

Each template specified in [rand.eng] requires one or more relationships, involving the value(s) of its non-type template parameter(s), to hold.

A program instantiating any of these templates is ill-formed if any such required relationship fails to hold.

For every random number engine and for every random number engine adaptor X defined in [rand.eng] and in [rand.adapt]:

(7.1)
if the constructor template<class Sseq> explicit X(Sseq& q); is called with a type Sseq that does not qualify as a seed sequence, then this constructor shall not participate in overload resolution;
(7.2)
if the member function template<class Sseq> void seed(Sseq& q); is called with a type Sseq that does not qualify as a seed sequence, then this function shall not participate in overload resolution.

The extent to which an implementation determines that a type cannot be a seed sequence is unspecified, except that as a minimum a type shall not qualify as a seed sequence if it is implicitly convertible to X::result_type.

29.5.4.2 Class template linear_congruential_engine [rand.eng.lcong]

A linear_congruential_engine random number engine produces unsigned integer random numbers.

The state x

_{i}

of a linear_congruential_engine object x is of size 1 and consists of a single integer.

The transition algorithm is a modular linear function of the form

T A (x_{i}) = (a \cdot x_{i} + c) mod m

; the generation algorithm is

G A (x_{i}) = x_{i + 1}

🔗

namespace std { template<class UIntType, UIntType a, UIntType c, UIntType m> class linear_congruential_engine { public: // types using result_type = UIntType; // engine characteristics static constexpr result_type multiplier = a; static constexpr result_type increment = c; static constexpr result_type modulus = m; static constexpr result_type min() { return c == 0u ? 1u: 0u; } static constexpr result_type max() { return m - 1u; } static constexpr result_type default_seed = 1u; // constructors and seeding functions linear_congruential_engine() : linear_congruential_engine(default_seed) {} explicit linear_congruential_engine(result_type s); template<class Sseq> explicit linear_congruential_engine(Sseq& q); void seed(result_type s = default_seed); template<class Sseq> void seed(Sseq& q); // equality operators friend bool operator==(const linear_congruential_engine& x, const linear_congruential_engine& y); // generating functions result_type operator()(); void discard(unsigned long long z); // inserters and extractors template<class charT, class traits> friend basic_ostream<charT, traits>& operator<<(basic_ostream<charT, traits>& os, const linear_congruential_engine& x); template<class charT, class traits> friend basic_istream<charT, traits>& operator>>(basic_istream<charT, traits>& is, linear_congruential_engine& x); }; }

If the template parameter m is 0, the modulus m used throughout [rand.eng.lcong] is numeric_limits<result_type>::max() plus 1.

[Note 1:

m need not be representable as a value of type result_type.

— end note]

If the template parameter m is not 0, the following relations shall hold: a < m and c < m.

The textual representation consists of the value of x

_{i}

🔗

explicit linear_congruential_engine(result_type s);

Effects: If

c mod m

is 0 and

s mod m

is 0, sets the engine's state to 1, otherwise sets the engine's state to

s mod m

🔗

template<class Sseq> explicit linear_congruential_engine(Sseq& q);

Effects: With

k = ⌈ \frac{log_{2} m}{32} ⌉

and a an array (or equivalent) of length

k + 3

, invokes q.generate(

a + 0

a + k + 3

) and then computes

S = (\sum_{j = 0}^{k - 1} a_{j + 3} \cdot 2^{32 j}) mod m

c mod m

is 0 and S is 0, sets the engine's state to 1, else sets the engine's state to S.

29.5.4.3 Class template mersenne_twister_engine [rand.eng.mers]

A mersenne_twister_engine random number engine248 produces unsigned integer random numbers in the closed interval

[0, 2^{w} - 1]

The state x

_{i}

of a mersenne_twister_engine object x is of size n and consists of a sequence X of n values of the type delivered by x; all subscripts applied to X are to be taken modulo n.

The transition algorithm employs a twisted generalized feedback shift register defined by shift values n and m, a twist value r, and a conditional xor-mask a.

To improve the uniformity of the result, the bits of the raw shift register are additionally tempered (i.e., scrambled) according to a bit-scrambling matrix defined by values u, d, s, b, t, c, and ℓ.

The state transition is performed as follows:

(2.1)
Concatenate the upper $w - r$ bits of $X_{i - n}$ with the lower r bits of $X_{i + 1 - n}$ to obtain an unsigned integer value Y.
(2.2)
With $α = a \cdot (Y b i t a n d 1)$ , set $X_{i}$ to $X_{i + m - n} x o r (Y r s h i f t 1) x o r α$ .

The sequence X is initialized with the help of an initialization multiplier f.

The generation algorithm determines the unsigned integer values

z_{1}, z_{2}, z_{3}, z_{4}

as follows, then delivers

z_{4}

as its result:

(3.1)
Let $z_{1} = X_{i} x o r ((X_{i} r s h i f t u) b i t a n d d)$ .
(3.2)
Let $z_{2} = z_{1} x o r ((z_{1} {l s h i f t}_{w} s) b i t a n d b)$ .
(3.3)
Let $z_{3} = z_{2} x o r ((z_{2} {l s h i f t}_{w} t) b i t a n d c)$ .
(3.4)
Let $z_{4} = z_{3} x o r (z_{3} r s h i f t ℓ)$ .

🔗

namespace std { template<class UIntType, size_t w, size_t n, size_t m, size_t r, UIntType a, size_t u, UIntType d, size_t s, UIntType b, size_t t, UIntType c, size_t l, UIntType f> class mersenne_twister_engine { public: // types using result_type = UIntType; // engine characteristics static constexpr size_t word_size = w; static constexpr size_t state_size = n; static constexpr size_t shift_size = m; static constexpr size_t mask_bits = r; static constexpr UIntType xor_mask = a; static constexpr size_t tempering_u = u; static constexpr UIntType tempering_d = d; static constexpr size_t tempering_s = s; static constexpr UIntType tempering_b = b; static constexpr size_t tempering_t = t; static constexpr UIntType tempering_c = c; static constexpr size_t tempering_l = l; static constexpr UIntType initialization_multiplier = f; static constexpr result_type min() { return 0; } static constexpr result_type max() { return

2^{w} - 1

; } static constexpr result_type default_seed = 5489u; // constructors and seeding functions mersenne_twister_engine() : mersenne_twister_engine(default_seed) {} explicit mersenne_twister_engine(result_type value); template<class Sseq> explicit mersenne_twister_engine(Sseq& q); void seed(result_type value = default_seed); template<class Sseq> void seed(Sseq& q); // equality operators friend bool operator==(const mersenne_twister_engine& x, const mersenne_twister_engine& y); // generating functions result_type operator()(); void discard(unsigned long long z); // inserters and extractors template<class charT, class traits> friend basic_ostream<charT, traits>& operator<<(basic_ostream<charT, traits>& os, const mersenne_twister_engine& x); template<class charT, class traits> friend basic_istream<charT, traits>& operator>>(basic_istream<charT, traits>& is, mersenne_twister_engine& x); }; }

The following relations shall hold: 0 < m, m <= n, 2u < w, r <= w, u <= w, s <= w, t <= w, l <= w, w <= numeric_limits<UIntType>::digits, a <= (1u<<w) - 1u, b <= (1u<<w) - 1u, c <= (1u<<w) - 1u, d <= (1u<<w) - 1u, and f <= (1u<<w) - 1u.

The textual representation of x

_{i}

consists of the values of

X_{i - n}, \dots, X_{i - 1}

, in that order.

🔗

explicit mersenne_twister_engine(result_type value);

Effects: Sets

X_{- n}

value mod 2^{w}

Then, iteratively for

i = 1 - n, \dots, - 1

, sets

X_{i}

[f \cdot (X_{i - 1} x o r (X_{i - 1} r s h i f t (w - 2))) + i mod n] mod 2^{w} .

Complexity:

O (n)

🔗

template<class Sseq> explicit mersenne_twister_engine(Sseq& q);

Effects: With

k = ⌈ w / 32 ⌉

and a an array (or equivalent) of length

n \cdot k

, invokes q.generate(

a + 0

a + n \cdot k

) and then, iteratively for

i = - n, \dots, - 1

, sets

X_{i}

(\sum_{j = 0}^{k - 1} a_{k (i + n) + j} \cdot 2^{32 j}) mod 2^{w}

Finally, if the most significant

w - r

bits of

X_{- n}

are zero, and if each of the other resulting

X_{i}

is 0, changes

X_{- n}

2^{w - 1}

248)248)

The name of this engine refers, in part, to a property of its period: For properly-selected values of the parameters, the period is closely related to a large Mersenne prime number.

29.5.4.4 Class template subtract_with_carry_engine [rand.eng.sub]

A subtract_with_carry_engine random number engine produces unsigned integer random numbers.

The state x

_{i}

of a subtract_with_carry_engine object x is of size

O (r)

, and consists of a sequence X of r integer values

0 \leq X_{i} < m = 2^{w}

; all subscripts applied to X are to be taken modulo r.

The state x

_{i}

additionally consists of an integer c (known as the carry) whose value is either 0 or 1.

The state transition is performed as follows:

(3.1)
Let $Y = X_{i - s} - X_{i - r} - c$ .
(3.2)
Set $X_{i}$ to $y = Y mod m$ .

Set c to 1 if $Y < 0$ , otherwise set c to 0.

[Note 1:

This algorithm corresponds to a modular linear function of the form

T A (x_{i}) = (a \cdot x_{i}) mod b

, where b is of the form

m^{r} - m^{s} + 1

and

a = b - (b - 1) / m

— end note]

The generation algorithm is given by

G A (x_{i}) = y

, where y is the value produced as a result of advancing the engine's state as described above.

🔗

namespace std { template<class UIntType, size_t w, size_t s, size_t r> class subtract_with_carry_engine { public: // types using result_type = UIntType; // engine characteristics static constexpr size_t word_size = w; static constexpr size_t short_lag = s; static constexpr size_t long_lag = r; static constexpr result_type min() { return 0; } static constexpr result_type max() { return

m - 1

; } static constexpr uint_least32_t default_seed = 19780503u; // constructors and seeding functions subtract_with_carry_engine() : subtract_with_carry_engine(0u) {} explicit subtract_with_carry_engine(result_type value); template<class Sseq> explicit subtract_with_carry_engine(Sseq& q); void seed(result_type value = 0u); template<class Sseq> void seed(Sseq& q); // equality operators friend bool operator==(const subtract_with_carry_engine& x, const subtract_with_carry_engine& y); // generating functions result_type operator()(); void discard(unsigned long long z); // inserters and extractors template<class charT, class traits> friend basic_ostream<charT, traits>& operator<<(basic_ostream<charT, traits>& os, const subtract_with_carry_engine& x); template<class charT, class traits> friend basic_istream<charT, traits>& operator>>(basic_istream<charT, traits>& is, subtract_with_carry_engine& x); }; }

The following relations shall hold: 0u < s, s < r, 0 < w, and w <= numeric_limits<UIntType>::digits.

The textual representation consists of the values of

X_{i - r}, \dots, X_{i - 1}

, in that order, followed by c.

🔗

explicit subtract_with_carry_engine(result_type value);

Effects: Sets the values of

X_{- r}, \dots, X_{- 1}

, in that order, as specified below.

X_{- 1}

is then 0, sets c to 1; otherwise sets c to 0.

To set the values

X_{k}

, first construct e, a linear_congruential_engine object, as if by the following definition: linear_congruential_engine<uint_least32_t, 40014u,0u,2147483563u> e(value == 0u ? default_seed : value);

Then, to set each

X_{k}

, obtain new values

z_{0}, \dots, z_{n - 1}

from

n = ⌈ w / 32 ⌉

successive invocations of e.

Set

X_{k}

(\sum_{j = 0}^{n - 1} z_{j} \cdot 2^{32 j}) mod m

Complexity: Exactly

n \cdot r

invocations of e.

🔗

template<class Sseq> explicit subtract_with_carry_engine(Sseq& q);

Effects: With

k = ⌈ w / 32 ⌉

and a an array (or equivalent) of length

r \cdot k

, invokes q.generate(

a + 0

a + r \cdot k

) and then, iteratively for

i = - r, \dots, - 1

, sets

X_{i}

(\sum_{j = 0}^{k - 1} a_{k (i + r) + j} \cdot 2^{32 j}) mod m

X_{- 1}

is then 0, sets c to 1; otherwise sets c to 0.

29.5.4.5 Class template philox_engine [rand.eng.philox]

A philox_engine random number engine produces unsigned integer random numbers in the closed interval [0, m], where

m = 2^{w} - 1

and the template parameter w defines the range of the produced numbers.

The state of a philox_engine object consists of a sequence X of n unsigned integer values of width w, a sequence K of

n / 2

values of result_type, a sequence Y of n values of result_type, and a scalar i, where

(1.1)
X is the interpretation of the unsigned integer counter value $Z : = \sum_{j = 0}^{n - 1} X_{j} \cdot 2^{w j}$ of $n \cdot w$ bits,
(1.2)
K are keys, which are generated once from the seed (see constructors below) and remain constant unless the seed function ([rand.req.eng]) is invoked,
(1.3)
Y stores a batch of output values, and
(1.4)
i is an index for an element of the sequence Y.

The generation algorithm returns

Y_{i}

, the value stored in the

i^{t h}

element of Y after applying the transition algorithm.

The state transition is performed as if by the following algorithm: i = i + 1 if (i == n) { Y = Philox(K, X) // see below Z = Z + 1 i = 0 }

The Philox function maps the length-

n / 2

sequence K and the length-n sequence X into a length-n output sequence Y.

Philox applies an r-round substitution-permutation network to the values in X.

A single round of the generation algorithm performs the following steps:

(4.1)
The output sequence $X^{'}$ of the previous round (X in case of the first round) is permuted to obtain the intermediate state V: $V_{j} = X_{f_{n} (j)}^{'}$ where $j = 0, \dots, n - 1$ and $f_{n} (j)$ is defined in Table 124.
Table 124: Values for the word permutation $f_{n} (j)$ [tab:rand.eng.philox.f]
🔗
$f_{n} (j)$
j
🔗
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
🔗
n
2
0
1
🔗
4
0
3
2
1
🔗
8
2
1
4
7
6
5
0
3
🔗
16
0
9
2
13
6
11
4
15
10
7
12
3
14
5
8
1
🔗

[Note 1:
For $n = 2$ the sequence is not permuted.
— end note]
(4.2)
The following computations are applied to the elements of the V sequence: $X_{2 k + 0} = mullo (V_{2 k + 1}, M_{k}, w)$ $X_{2 k + 1} = mulhi (V_{2 k + 1}, M_{k}, w) x o r {k e y}_{k}^{q} x o r V_{2 k}$ where:
- (4.2.1)
  mullo(a, b, w) is the low half of the modular multiplication of a and b: $(a \cdot b) mod 2^{w}$ ,
- (4.2.2)
  mulhi(a, b, w) is the high half of the modular multiplication of a and b: $(⌊ (a \cdot b) / 2^{w} ⌋)$ ,
- (4.2.3)
  $k = 0, \dots, n / 2 - 1$ is the index in the sequences,
- (4.2.4)
  $q = 0, \dots, r - 1$ is the index of the round,
- (4.2.5)
  ${k e y}_{k}^{q}$ is the $k^{th}$ round key for round q, ${k e y}_{k}^{q} : = (K_{k} + q \cdot C_{k}) mod 2^{w}$ ,
- (4.2.6)
  $K_{k}$ are the elements of the key sequence K,
- (4.2.7)
  $M_{k}$ is multipliers[k], and
- (4.2.8)
  $C_{k}$ is round_consts[k].

After r applications of the single-round function, Philox returns the sequence

Y = X^{'}

🔗

namespace std { template<class UIntType, size_t w, size_t n, size_t r, UIntType... consts> class philox_engine { static constexpr size_t array-size = n / 2; // exposition only public: // types using result_type = UIntType; // engine characteristics static constexpr size_t word_size = w; static constexpr size_t word_count = n; static constexpr size_t round_count = r; static constexpr array<result_type, array-size> multipliers; static constexpr array<result_type, array-size> round_consts; static constexpr result_type min() { return 0; } static constexpr result_type max() { return m - 1; } static constexpr result_type default_seed = 20111115u; // constructors and seeding functions philox_engine() : philox_engine(default_seed) {} explicit philox_engine(result_type value); template<class Sseq> explicit philox_engine(Sseq& q); void seed(result_type value = default_seed); template<class Sseq> void seed(Sseq& q); void set_counter(const array<result_type, n>& counter); // equality operators friend bool operator==(const philox_engine& x, const philox_engine& y); // generating functions result_type operator()(); void discard(unsigned long long z); // inserters and extractors template<class charT, class traits> friend basic_ostream<charT, traits>& operator<<(basic_ostream<charT, traits>& os, const philox_engine& x); template<class charT, class traits> friend basic_istream<charT, traits>& operator>>(basic_istream<charT, traits>& is, philox_engine& x); }; }

Mandates:

(6.1)
sizeof...(consts) == n is true, and
(6.2)
n == 2 || n == 4 || n == 8 || n == 16 is true, and
(6.3)
0 < r is true, and
(6.4)
0 < w && w <= numeric_limits<UIntType>::digits is true.

The template parameter pack consts represents the

M_{k}

and

C_{k}

constants which are grouped as follows:

[M_{0}, C_{0}, M_{1}, C_{1}, M_{2}, C_{2}, \dots, M_{n / 2 - 1}, C_{n / 2 - 1}]

The textual representation consists of the values of

K_{0}, \dots, K_{n / 2 - 1}, X_{0}, \dots, X_{n - 1}, i

, in that order.

[Note 2:

The stream extraction operator can reconstruct Y from K and X, as needed.

— end note]

🔗

explicit philox_engine(result_type value);

Effects: Sets the

K_{0}

element of sequence K to

value mod 2^{w}

All elements of sequences X and K (except

K_{0}

) are set to 0.

The value of i is set to

n - 1

🔗

template<class Sseq> explicit philox_engine(Sseq& q);

Effects: With

p = ⌈ w / 32 ⌉

and an array (or equivalent) a of length

(n / 2) \cdot p

, invokes q.generate(a + 0, a + n / 2 * p) and then iteratively for

k = 0, \dots, n / 2 - 1

, sets

K_{k}

(\sum_{j = 0}^{p - 1} a_{k p + j} \cdot 2^{32 j}) mod 2^{w}

All elements of sequence X are set to 0.

The value of i is set to

n - 1

🔗

void set_counter(const array<result_type, n>& c);

Effects: For

j = 0, \dots, n - 1

sets

X_{j}

C_{n - 1 - j} mod 2^{w}

The value of i is set to

n - 1

[Note 3:

The counter is the value Z introduced at the beginning of this subclause.

— end note]

29.5.5 Random number engine adaptor class templates [rand.adapt]

29.5.5.1 General [rand.adapt.general]

Each type instantiated from a class template specified in [rand.adapt] meets the requirements of a random number engine adaptor type.

Except where specified otherwise, the complexity of each function specified in [rand.adapt] is constant.

Except where specified otherwise, no function described in [rand.adapt] throws an exception.

Every function described in [rand.adapt] that has a function parameter q of type Sseq& for a template type parameter named Sseq that is different from type seed_seq throws what and when the invocation of q.generate throws.

Descriptions are provided in [rand.adapt] only for adaptor operations that are not described in subclause [rand.req.adapt] or for operations where there is additional semantic information.

In particular, declarations for copy constructors, for copy assignment operators, for streaming operators, and for equality and inequality operators are not shown in the synopses.

Each template specified in [rand.adapt] requires one or more relationships, involving the value(s) of its non-type template parameter(s), to hold.

A program instantiating any of these templates is ill-formed if any such required relationship fails to hold.

29.5.5.2 Class template discard_block_engine [rand.adapt.disc]

A discard_block_engine random number engine adaptor produces random numbers selected from those produced by some base engine e.

The state x

_{i}

of a discard_block_engine engine adaptor object x consists of the state e

_{i}

of its base engine e and an additional integer n.

The size of the state is the size of e's state plus 1.

The transition algorithm discards all but

r > 0

values from each block of p ≥ r values delivered by e.

The state transition is performed as follows: If n ≥ r, advance the state of e from e

_{i}

to e

_{i + p - r}

and set n to 0.

In any case, then increment n and advance e's then-current state e

_{j}

to e

_{j + 1}

The generation algorithm yields the value returned by the last invocation of e() while advancing e's state as described above.

🔗

namespace std { template<class Engine, size_t p, size_t r> class discard_block_engine { public: // types using result_type = typename Engine::result_type; // engine characteristics static constexpr size_t block_size = p; static constexpr size_t used_block = r; static constexpr result_type min() { return Engine::min(); } static constexpr result_type max() { return Engine::max(); } // constructors and seeding functions discard_block_engine(); explicit discard_block_engine(const Engine& e); explicit discard_block_engine(Engine&& e); explicit discard_block_engine(result_type s); template<class Sseq> explicit discard_block_engine(Sseq& q); void seed(); void seed(result_type s); template<class Sseq> void seed(Sseq& q); // equality operators friend bool operator==(const discard_block_engine& x, const discard_block_engine& y); // generating functions result_type operator()(); void discard(unsigned long long z); // property functions const Engine& base() const noexcept { return e; } // inserters and extractors template<class charT, class traits> friend basic_ostream<charT, traits>& operator<<(basic_ostream<charT, traits>& os, const discard_block_engine& x); template<class charT, class traits> friend basic_istream<charT, traits>& operator>>(basic_istream<charT, traits>& is, discard_block_engine& x); private: Engine e; // exposition only size_t n; // exposition only }; }

The following relations shall hold: 0 < r and r <= p.

The textual representation consists of the textual representation of e followed by the value of n.

In addition to its behavior pursuant to subclause [rand.req.adapt], each constructor that is not a copy constructor sets n to 0.

29.5.5.3 Class template independent_bits_engine [rand.adapt.ibits]

An independent_bits_engine random number engine adaptor combines random numbers that are produced by some base engine e, so as to produce random numbers with a specified number of bits w.

The state x

_{i}

of an independent_bits_engine engine adaptor object x consists of the state e

_{i}

of its base engine e; the size of the state is the size of e's state.

The transition and generation algorithms are described in terms of the following integral constants:

(2.1)
Let $R = e.max() - e.min() + 1$ and $m = ⌊ log_{2} R ⌋$ .
(2.2)
With n as determined below, let $w_{0} = ⌊ w / n ⌋$ , $n_{0} = n - w mod n$ , $y_{0} = 2^{w_{0}} ⌊ R / 2^{w_{0}} ⌋$ , and $y_{1} = 2^{w_{0} + 1} ⌊ R / 2^{w_{0} + 1} ⌋$ .
(2.3)
Let $n = ⌈ w / m ⌉$ if and only if the relation $R - y_{0} \leq ⌊ y_{0} / n ⌋$ holds as a result.

Otherwise let $n = 1 + ⌈ w / m ⌉$ .

[Note 1:

The relation

w = n_{0} w_{0} + (n - n_{0}) (w_{0} + 1)

always holds.

— end note]

The transition algorithm is carried out by invoking e() as often as needed to obtain

n_{0}

values less than

y_{0} + e.min()

and

n - n_{0}

values less than

y_{1} + e.min()

The generation algorithm uses the values produced while advancing the state as described above to yield a quantity S obtained as if by the following algorithm: S = 0; for (k = 0;

k \neq n_{0}

; k += 1) { do u = e() - e.min(); while (

u \geq y_{0}

); S =

2^{w_{0}} \cdot S + u mod 2^{w_{0}}

; } for (k =

n_{0}

; k ≠ n; k += 1) { do u = e() - e.min(); while (

u \geq y_{1}

); S =

2^{w_{0} + 1} \cdot S + u mod 2^{w_{0} + 1}

; }

🔗

namespace std { template<class Engine, size_t w, class UIntType> class independent_bits_engine { public: // types using result_type = UIntType; // engine characteristics static constexpr result_type min() { return 0; } static constexpr result_type max() { return

2^{w} - 1

; } // constructors and seeding functions independent_bits_engine(); explicit independent_bits_engine(const Engine& e); explicit independent_bits_engine(Engine&& e); explicit independent_bits_engine(result_type s); template<class Sseq> explicit independent_bits_engine(Sseq& q); void seed(); void seed(result_type s); template<class Sseq> void seed(Sseq& q); // equality operators friend bool operator==(const independent_bits_engine& x, const independent_bits_engine& y); // generating functions result_type operator()(); void discard(unsigned long long z); // property functions const Engine& base() const noexcept { return e; } // inserters and extractors template<class charT, class traits> friend basic_ostream<charT, traits>& operator<<(basic_ostream<charT, traits>& os, const independent_bits_engine& x); template<class charT, class traits> friend basic_istream<charT, traits>& operator>>(basic_istream<charT, traits>& is, independent_bits_engine& x); private: Engine e; // exposition only }; }

The following relations shall hold: 0 < w and w <= numeric_limits<result_type>::digits.

The textual representation consists of the textual representation of e.

29.5.5.4 Class template shuffle_order_engine [rand.adapt.shuf]

A shuffle_order_engine random number engine adaptor produces the same random numbers that are produced by some base engine e, but delivers them in a different sequence.

The state x

_{i}

of a shuffle_order_engine engine adaptor object x consists of the state e

_{i}

of its base engine e, an additional value Y of the type delivered by e, and an additional sequence V of k values also of the type delivered by e.

The size of the state is the size of e's state plus

k + 1

The transition algorithm permutes the values produced by e.

The state transition is performed as follows:

(2.1)
Calculate an integer $j = ⌊ \frac{k \cdot (Y - e_{min})}{e_{max} - e_{min} + 1} ⌋$ .
(2.2)
Set Y to $V_{j}$ and then set $V_{j}$ to e().

The generation algorithm yields the last value of Y produced while advancing e's state as described above.

🔗

namespace std { template<class Engine, size_t k> class shuffle_order_engine { public: // types using result_type = typename Engine::result_type; // engine characteristics static constexpr size_t table_size = k; static constexpr result_type min() { return Engine::min(); } static constexpr result_type max() { return Engine::max(); } // constructors and seeding functions shuffle_order_engine(); explicit shuffle_order_engine(const Engine& e); explicit shuffle_order_engine(Engine&& e); explicit shuffle_order_engine(result_type s); template<class Sseq> explicit shuffle_order_engine(Sseq& q); void seed(); void seed(result_type s); template<class Sseq> void seed(Sseq& q); // equality operators friend bool operator==(const shuffle_order_engine& x, const shuffle_order_engine& y); // generating functions result_type operator()(); void discard(unsigned long long z); // property functions const Engine& base() const noexcept { return e; } // inserters and extractors template<class charT, class traits> friend basic_ostream<charT, traits>& operator<<(basic_ostream<charT, traits>& os, const shuffle_order_engine& x); template<class charT, class traits> friend basic_istream<charT, traits>& operator>>(basic_istream<charT, traits>& is, shuffle_order_engine& x); private: Engine e; // exposition only result_type V[k]; // exposition only result_type Y; // exposition only }; }

The following relation shall hold: 0 < k.

The textual representation consists of the textual representation of e, followed by the k values of V, followed by the value of Y.

In addition to its behavior pursuant to subclause [rand.req.adapt], each constructor that is not a copy constructor initializes V[0], …, V[k-1] and Y, in that order, with values returned by successive invocations of e().

29.5.6 Engines and engine adaptors with predefined parameters [rand.predef]

🔗

using minstd_rand0 =
      linear_congruential_engine<uint_fast32_t, 16'807, 0, 2'147'483'647>;

Required behavior: The

10000^{th}

consecutive invocation of a default-constructed object of type minstd_rand0 produces the value 1043618065.

🔗

using minstd_rand =
      linear_congruential_engine<uint_fast32_t, 48'271, 0, 2'147'483'647>;

Required behavior: The

10000^{th}

consecutive invocation of a default-constructed object of type minstd_rand produces the value 399268537.

🔗

using mt19937 =
      mersenne_twister_engine<uint_fast32_t, 32, 624, 397, 31,
       0x9908'b0df, 11, 0xffff'ffff, 7, 0x9d2c'5680, 15, 0xefc6'0000, 18, 1'812'433'253>;

Required behavior: The

10000^{th}

consecutive invocation of a default-constructed object of type mt19937 produces the value 4123659995.

🔗

using mt19937_64 =
      mersenne_twister_engine<uint_fast64_t, 64, 312, 156, 31,
       0xb502'6f5a'a966'19e9, 29, 0x5555'5555'5555'5555, 17,
       0x71d6'7fff'eda6'0000, 37, 0xfff7'eee0'0000'0000, 43, 6'364'136'223'846'793'005>;

Required behavior: The

10000^{th}

consecutive invocation of a default-constructed object of type mt19937_64 produces the value 9981545732273789042.

🔗

using ranlux24_base =
      subtract_with_carry_engine<uint_fast32_t, 24, 10, 24>;

Required behavior: The

10000^{th}

consecutive invocation of a default-constructed object of type ranlux24_base produces the value 7937952.

🔗

using ranlux48_base =
      subtract_with_carry_engine<uint_fast64_t, 48, 5, 12>;

Required behavior: The

10000^{th}

consecutive invocation of a default-constructed object of type ranlux48_base produces the value 61839128582725.

🔗

using ranlux24 = discard_block_engine<ranlux24_base, 223, 23>;

Required behavior: The

10000^{th}

consecutive invocation of a default-constructed object of type ranlux24 produces the value 9901578.

🔗

using ranlux48 = discard_block_engine<ranlux48_base, 389, 11>;

Required behavior: The

10000^{th}

consecutive invocation of a default-constructed object of type ranlux48 produces the value 249142670248501.

🔗

using knuth_b = shuffle_order_engine<minstd_rand0,256>;

Required behavior: The

10000^{th}

consecutive invocation of a default-constructed object of type knuth_b produces the value 1112339016.

🔗

using default_random_engine = implementation-defined;

Remarks: The choice of engine type named by this typedef is implementation-defined.

[Note 1:

The implementation can select this type on the basis of performance, size, quality, or any combination of such factors, so as to provide at least acceptable engine behavior for relatively casual, inexpert, and/or lightweight use.

Because different implementations can select different underlying engine types, code that uses this typedef need not generate identical sequences across implementations.

— end note]

🔗

using philox4x32 =
      philox_engine<uint_fast32_t, 32, 4, 10,
       0xD2511F53, 0x9E3779B9, 0xCD9E8D57, 0xBB67AE85>;

Required behavior: The

10000^{th}

consecutive invocation a default-constructed object of type philox4x32 produces the value 1955073260.

🔗

using philox4x64 =
      philox_engine<uint_fast64_t, 64, 4, 10,
       0xD2E7470EE14C6C93, 0x9E3779B97F4A7C15, 0xCA5A826395121157, 0xBB67AE8584CAA73B>;

Required behavior: The

10000^{th}

consecutive invocation a default-constructed object of type philox4x64 produces the value 3409172418970261260.

29.5.7 Class random_device [rand.device]

A random_device uniform random bit generator produces nondeterministic random numbers.

If implementation limitations prevent generating nondeterministic random numbers, the implementation may employ a random number engine.

🔗

namespace std { class random_device { public: // types using result_type = unsigned int; // generator characteristics static constexpr result_type min() { return numeric_limits<result_type>::min(); } static constexpr result_type max() { return numeric_limits<result_type>::max(); } // constructors random_device() : random_device(implementation-defined) {} explicit random_device(const string& token); // generating functions result_type operator()(); // property functions double entropy() const noexcept; // no copy functions random_device(const random_device&) = delete; void operator=(const random_device&) = delete; }; }

🔗

explicit random_device(const string& token);

Throws: A value of an implementation-defined type derived from exception if the random_device cannot be initialized.

Remarks: The semantics of the token parameter and the token value used by the default constructor are implementation-defined.249

🔗

double entropy() const noexcept;

Returns: If the implementation employs a random number engine, returns 0.0.

Otherwise, returns an entropy estimate250 for the random numbers returned by operator(), in the range min() to

log_{2} (max() + 1)

🔗

result_type operator()();

Returns: A nondeterministic random value, uniformly distributed between min() and max() (inclusive).

It is implementation-defined how these values are generated.

Throws: A value of an implementation-defined type derived from exception if a random number cannot be obtained.

249)249)

The parameter is intended to allow an implementation to differentiate between different sources of randomness.

250)250)

If a device has n states whose respective probabilities are

P_{0}, \dots, P_{n - 1}

, the device entropy S is defined as

S = - \sum_{i = 0}^{n - 1} P_{i} \cdot log P_{i}

29.5.8 Utilities [rand.util]

29.5.8.1 Class seed_seq [rand.util.seedseq]

🔗

namespace std { class seed_seq { public: // types using result_type = uint_least32_t; // constructors seed_seq() noexcept; template<class T> seed_seq(initializer_list<T> il); template<class InputIterator> seed_seq(InputIterator begin, InputIterator end); // generating functions template<class RandomAccessIterator> void generate(RandomAccessIterator begin, RandomAccessIterator end); // property functions size_t size() const noexcept; template<class OutputIterator> void param(OutputIterator dest) const; // no copy functions seed_seq(const seed_seq&) = delete; void operator=(const seed_seq&) = delete; private: vector<result_type> v; // exposition only }; }

🔗

seed_seq() noexcept;

Postconditions: v.empty() is true.

🔗

template<class T>
  seed_seq(initializer_list<T> il);

Constraints: T is an integer type.

Effects: Same as seed_seq(il.begin(), il.end()).

🔗

template<class InputIterator>
  seed_seq(InputIterator begin, InputIterator end);

Mandates: iterator_traits<InputIterator>::value_type is an integer type.

Preconditions: InputIterator meets the Cpp17InputIterator requirements ([input.iterators]).

Effects: Initializes v by the following algorithm: for (InputIterator s = begin; s != end; ++s) v.push_back((*s)

mod 2^{32}

);

🔗

template<class RandomAccessIterator>
  void generate(RandomAccessIterator begin, RandomAccessIterator end);

Mandates: iterator_traits<RandomAccessIterator>::value_type is an unsigned integer type capable of accommodating 32-bit quantities.

Preconditions: RandomAccessIterator meets the Cpp17RandomAccessIterator requirements ([random.access.iterators]) and the requirements of a mutable iterator.

Effects: Does nothing if begin == end.

Otherwise, with

s = v.size()

and

n = end - begin

, fills the supplied range [begin,end) according to the following algorithm in which each operation is to be carried out modulo

2^{32}

, each indexing operator applied to begin is to be taken modulo n, and T(x) is defined as x xor(x rshift27):

(9.1)
By way of initialization, set each element of the range to the value 0x8b8b8b8b.

Additionally, for use in subsequent steps, let $p = (n - t) / 2$ and let $q = p + t$ , where
$t = (n \geq 623) ? 11 : (n \geq 68) ? 7 : (n \geq 39) ? 5 : (n \geq 7) ? 3 : (n - 1) / 2;$
(9.2)
With m as the larger of $s + 1$ and n, transform the elements of the range: iteratively for $k = 0, \dots, m - 1$ , calculate values
$\begin{matrix} r_{1} & = & 1664525 \cdot T (begin[k] x o r begin[k + p] x o r begin[k - 1]) r_{2} & = & r_{1} + ⎧ ⎪ ⎨ ⎪ ⎩ \begin{matrix} s & , k = 0 k mod n + v[k - 1] & , 0 < k \leq s k mod n & , s < k \end{matrix} \end{matrix}$ and, in order, increment begin[ $k + p$ ] by $r_{1}$ , increment begin[ $k + q$ ] by $r_{2}$ , and set begin[k] to $r_{2}$ .
(9.3)
Transform the elements of the range again, beginning where the previous step ended: iteratively for $k = m, \dots, m + n - 1$ , calculate values
$\begin{matrix} r_{3} & = & 1566083941 \cdot T (begin[k] + begin[k + p] + begin[k - 1]) r_{4} & = & r_{3} - (k mod n) \end{matrix}$ and, in order, update begin[ $k + p$ ] by xoring it with $r_{3}$ , update begin[ $k + q$ ] by xoring it with $r_{4}$ , and set begin[k] to $r_{4}$ .

Throws: What and when RandomAccessIterator operations of begin and end throw.

🔗

size_t size() const noexcept;

Returns: The number of 32-bit units that would be returned by a call to param().

Complexity: Constant time.

🔗

template<class OutputIterator>
  void param(OutputIterator dest) const;

Mandates: Values of type result_type are writable ([iterator.requirements.general]) to dest.

Preconditions: OutputIterator meets the Cpp17OutputIterator requirements ([output.iterators]).

Effects: Copies the sequence of prepared 32-bit units to the given destination, as if by executing the following statement: copy(v.begin(), v.end(), dest);

Throws: What and when OutputIterator operations of dest throw.

29.5.8.2 Function template generate_canonical [rand.util.canonical]

🔗

template<class RealType, size_t digits, class URBG>
  RealType generate_canonical(URBG& g);

Let

(1.1)
r be numeric_limits<RealType>::radix,
(1.2)
R be $g.max() - g.min() + 1$ ,
(1.3)
d be the smaller of digits and numeric_limits<RealType>::digits,251
(1.4)
k be the smallest integer such that $R^{k} \geq r^{d}$ , and
(1.5)
x be $⌊ R^{k} / r^{d} ⌋$ .

An attempt is k invocations of g() to obtain values

g_{0}, \dots, g_{k - 1}

, respectively, and the calculation of a quantity S given by Formula 29.1:

\begin{matrix} S = k - 1 \sum i = 0 (g_{i} - g.min()) \cdot R^{i} \\ (29.1) \end{matrix}

Effects: Attempts are made until

S < x r^{d}

[Note 1:

When R is a power of r, precisely one attempt is made.

— end note]

Returns:

⌊ S / x ⌋ / r^{d}

[Note 2:

The return value c satisfies

0 \leq c < 1

— end note]

Throws: What and when g throws.

Complexity: Exactly k invocations of g per attempt.

[Note 3:

If the values

g_{i}

produced by g are uniformly distributed, the instantiation's results are distributed as uniformly as possible.

Obtaining a value in this way can be a useful step in the process of transforming a value generated by a uniform random bit generator into a value that can be delivered by a random number distribution.

— end note]

[Note 4:

When R is a power of r, an implementation can avoid using an arithmetic type that is wider than the output when computing S.

— end note]

251)251)

d is introduced to avoid any attempt to produce more bits of randomness than can be held in RealType.

29.5.9 Random number distribution class templates [rand.dist]

29.5.9.1 General [rand.dist.general]

Each type instantiated from a class template specified in [rand.dist] meets the requirements of a random number distribution type.

Descriptions are provided in [rand.dist] only for distribution operations that are not described in [rand.req.dist] or for operations where there is additional semantic information.

In particular, declarations for copy constructors, for copy assignment operators, for streaming operators, and for equality and inequality operators are not shown in the synopses.

The algorithms for producing each of the specified distributions are implementation-defined.

The value of each probability density function p(z) and of each discrete probability function

P (z_{i})

specified in this subclause is 0 everywhere outside its stated domain.

29.5.9.2 Uniform distributions [rand.dist.uni]

29.5.9.2.1 Class template uniform_int_distribution [rand.dist.uni.int]

A uniform_int_distribution random number distribution produces random integers i, a ≤ i ≤ b, distributed according to the constant discrete probability function in Formula 29.2 .

\begin{matrix} P (i | a, b) = 1 / (b - a + 1) \\ (29.2) \end{matrix}

🔗

namespace std { template<class IntType = int> class uniform_int_distribution { public: // types using result_type = IntType; using param_type = unspecified; // constructors and reset functions uniform_int_distribution() : uniform_int_distribution(0) {} explicit uniform_int_distribution(IntType a, IntType b = numeric_limits<IntType>::max()); explicit uniform_int_distribution(const param_type& parm); void reset(); // equality operators friend bool operator==(const uniform_int_distribution& x, const uniform_int_distribution& y); // generating functions template<class URBG> result_type operator()(URBG& g); template<class URBG> result_type operator()(URBG& g, const param_type& parm); // property functions result_type a() const; result_type b() const; param_type param() const; void param(const param_type& parm); result_type min() const; result_type max() const; // inserters and extractors template<class charT, class traits> friend basic_ostream<charT, traits>& operator<<(basic_ostream<charT, traits>& os, const uniform_int_distribution& x); template<class charT, class traits> friend basic_istream<charT, traits>& operator>>(basic_istream<charT, traits>& is, uniform_int_distribution& x); }; }

🔗

explicit uniform_int_distribution(IntType a, IntType b = numeric_limits<IntType>::max());

Preconditions: a ≤ b.

Remarks: a and b correspond to the respective parameters of the distribution.

🔗

result_type a() const;

Returns: The value of the a parameter with which the object was constructed.

🔗

result_type b() const;

Returns: The value of the b parameter with which the object was constructed.

29.5.9.2.2 Class template uniform_real_distribution [rand.dist.uni.real]

A uniform_real_distribution random number distribution produces random numbers x,

a \leq x < b

, distributed according to the constant probability density function in Formula 29.3 .

\begin{matrix} p (x | a, b) = 1 / (b - a) \\ (29.3) \end{matrix}

[Note 1:

This implies that p(x | a,b) is undefined when a == b.

— end note]

🔗

namespace std { template<class RealType = double> class uniform_real_distribution { public: // types using result_type = RealType; using param_type = unspecified; // constructors and reset functions uniform_real_distribution() : uniform_real_distribution(0.0) {} explicit uniform_real_distribution(RealType a, RealType b = 1.0); explicit uniform_real_distribution(const param_type& parm); void reset(); // equality operators friend bool operator==(const uniform_real_distribution& x, const uniform_real_distribution& y); // generating functions template<class URBG> result_type operator()(URBG& g); template<class URBG> result_type operator()(URBG& g, const param_type& parm); // property functions result_type a() const; result_type b() const; param_type param() const; void param(const param_type& parm); result_type min() const; result_type max() const; // inserters and extractors template<class charT, class traits> friend basic_ostream<charT, traits>& operator<<(basic_ostream<charT, traits>& os, const uniform_real_distribution& x); template<class charT, class traits> friend basic_istream<charT, traits>& operator>>(basic_istream<charT, traits>& is, uniform_real_distribution& x); }; }

🔗

explicit uniform_real_distribution(RealType a, RealType b = 1.0);

Preconditions: a ≤ b and

b - a \leq numeric_limits<RealType>::max()

Remarks: a and b correspond to the respective parameters of the distribution.

🔗

result_type a() const;

Returns: The value of the a parameter with which the object was constructed.

🔗

result_type b() const;

Returns: The value of the b parameter with which the object was constructed.

29.5.9.3 Bernoulli distributions [rand.dist.bern]

29.5.9.3.1 Class bernoulli_distribution [rand.dist.bern.bernoulli]

A bernoulli_distribution random number distribution produces bool values b distributed according to the discrete probability function in Formula 29.4 .

\begin{matrix} P (b | p) = {\begin{matrix} p & if b = true 1 - p & if b = false \end{matrix} \\ (29.4) \end{matrix}

🔗

namespace std { class bernoulli_distribution { public: // types using result_type = bool; using param_type = unspecified; // constructors and reset functions bernoulli_distribution() : bernoulli_distribution(0.5) {} explicit bernoulli_distribution(double p); explicit bernoulli_distribution(const param_type& parm); void reset(); // equality operators friend bool operator==(const bernoulli_distribution& x, const bernoulli_distribution& y); // generating functions template<class URBG> result_type operator()(URBG& g); template<class URBG> result_type operator()(URBG& g, const param_type& parm); // property functions double p() const; param_type param() const; void param(const param_type& parm); result_type min() const; result_type max() const; // inserters and extractors template<class charT, class traits> friend basic_ostream<charT, traits>& operator<<(basic_ostream<charT, traits>& os, const bernoulli_distribution& x); template<class charT, class traits> friend basic_istream<charT, traits>& operator>>(basic_istream<charT, traits>& is, bernoulli_distribution& x); }; }

🔗

explicit bernoulli_distribution(double p);

Preconditions: 0 ≤ p ≤ 1.

Remarks: p corresponds to the parameter of the distribution.

🔗

double p() const;

Returns: The value of the p parameter with which the object was constructed.

29.5.9.3.2 Class template binomial_distribution [rand.dist.bern.bin]

A binomial_distribution random number distribution produces integer values i ≥ 0 distributed according to the discrete probability function in Formula 29.5 .

\begin{matrix} P (i | t, p) = (\frac{t}{i}) \cdot p^{i} \cdot (1 - p)^{t - i} \\ (29.5) \end{matrix}

🔗

namespace std { template<class IntType = int> class binomial_distribution { public: // types using result_type = IntType; using param_type = unspecified; // constructors and reset functions binomial_distribution() : binomial_distribution(1) {} explicit binomial_distribution(IntType t, double p = 0.5); explicit binomial_distribution(const param_type& parm); void reset(); // equality operators friend bool operator==(const binomial_distribution& x, const binomial_distribution& y); // generating functions template<class URBG> result_type operator()(URBG& g); template<class URBG> result_type operator()(URBG& g, const param_type& parm); // property functions IntType t() const; double p() const; param_type param() const; void param(const param_type& parm); result_type min() const; result_type max() const; // inserters and extractors template<class charT, class traits> friend basic_ostream<charT, traits>& operator<<(basic_ostream<charT, traits>& os, const binomial_distribution& x); template<class charT, class traits> friend basic_istream<charT, traits>& operator>>(basic_istream<charT, traits>& is, binomial_distribution& x); }; }

🔗

explicit binomial_distribution(IntType t, double p = 0.5);

Preconditions: 0 ≤ p ≤ 1 and 0 ≤ t.

Remarks: t and p correspond to the respective parameters of the distribution.

🔗

IntType t() const;

Returns: The value of the t parameter with which the object was constructed.

🔗

double p() const;

Returns: The value of the p parameter with which the object was constructed.

29.5.9.3.3 Class template geometric_distribution [rand.dist.bern.geo]

A geometric_distribution random number distribution produces integer values i ≥ 0 distributed according to the discrete probability function in Formula 29.6 .

\begin{matrix} P (i | p) = p \cdot (1 - p)^{i} \\ (29.6) \end{matrix}

🔗

namespace std { template<class IntType = int> class geometric_distribution { public: // types using result_type = IntType; using param_type = unspecified; // constructors and reset functions geometric_distribution() : geometric_distribution(0.5) {} explicit geometric_distribution(double p); explicit geometric_distribution(const param_type& parm); void reset(); // equality operators friend bool operator==(const geometric_distribution& x, const geometric_distribution& y); // generating functions template<class URBG> result_type operator()(URBG& g); template<class URBG> result_type operator()(URBG& g, const param_type& parm); // property functions double p() const; param_type param() const; void param(const param_type& parm); result_type min() const; result_type max() const; // inserters and extractors template<class charT, class traits> friend basic_ostream<charT, traits>& operator<<(basic_ostream<charT, traits>& os, const geometric_distribution& x); template<class charT, class traits> friend basic_istream<charT, traits>& operator>>(basic_istream<charT, traits>& is, geometric_distribution& x); }; }

🔗

explicit geometric_distribution(double p);

Preconditions:

0 < p < 1

Remarks: p corresponds to the parameter of the distribution.

🔗

double p() const;

Returns: The value of the p parameter with which the object was constructed.

29.5.9.3.4 Class template negative_binomial_distribution [rand.dist.bern.negbin]

A negative_binomial_distribution random number distribution produces random integers i ≥ 0 distributed according to the discrete probability function in Formula 29.7 .

\begin{matrix} P (i | k, p) = (\frac{k + i - 1}{i}) \cdot p^{k} \cdot (1 - p)^{i} \\ (29.7) \end{matrix}

[Note 1:

This implies that P(i | k,p) is undefined when p == 1.

— end note]

🔗

namespace std { template<class IntType = int> class negative_binomial_distribution { public: // types using result_type = IntType; using param_type = unspecified; // constructor and reset functions negative_binomial_distribution() : negative_binomial_distribution(1) {} explicit negative_binomial_distribution(IntType k, double p = 0.5); explicit negative_binomial_distribution(const param_type& parm); void reset(); // equality operators friend bool operator==(const negative_binomial_distribution& x, const negative_binomial_distribution& y); // generating functions template<class URBG> result_type operator()(URBG& g); template<class URBG> result_type operator()(URBG& g, const param_type& parm); // property functions IntType k() const; double p() const; param_type param() const; void param(const param_type& parm); result_type min() const; result_type max() const; // inserters and extractors template<class charT, class traits> friend basic_ostream<charT, traits>& operator<<(basic_ostream<charT, traits>& os, const negative_binomial_distribution& x); template<class charT, class traits> friend basic_istream<charT, traits>& operator>>(basic_istream<charT, traits>& is, negative_binomial_distribution& x); }; }

🔗

explicit negative_binomial_distribution(IntType k, double p = 0.5);

Preconditions:

0 < p \leq 1

and

0 < k

Remarks: k and p correspond to the respective parameters of the distribution.

🔗

IntType k() const;

Returns: The value of the k parameter with which the object was constructed.

🔗

double p() const;

Returns: The value of the p parameter with which the object was constructed.

29.5.9.4 Poisson distributions [rand.dist.pois]

29.5.9.4.1 Class template poisson_distribution [rand.dist.pois.poisson]

A poisson_distribution random number distribution produces integer values i ≥ 0 distributed according to the discrete probability function in Formula 29.8 .

\begin{matrix} P (i | μ) = \frac{e^{- μ} μ^{i}}{i!} \\ (29.8) \end{matrix}

The distribution parameter μ is also known as this distribution's mean.

🔗

namespace std { template<class IntType = int> class poisson_distribution { public: // types using result_type = IntType; using param_type = unspecified; // constructors and reset functions poisson_distribution() : poisson_distribution(1.0) {} explicit poisson_distribution(double mean); explicit poisson_distribution(const param_type& parm); void reset(); // equality operators friend bool operator==(const poisson_distribution& x, const poisson_distribution& y); // generating functions template<class URBG> result_type operator()(URBG& g); template<class URBG> result_type operator()(URBG& g, const param_type& parm); // property functions double mean() const; param_type param() const; void param(const param_type& parm); result_type min() const; result_type max() const; // inserters and extractors template<class charT, class traits> friend basic_ostream<charT, traits>& operator<<(basic_ostream<charT, traits>& os, const poisson_distribution& x); template<class charT, class traits> friend basic_istream<charT, traits>& operator>>(basic_istream<charT, traits>& is, poisson_distribution& x); }; }

🔗

explicit poisson_distribution(double mean);

Preconditions:

0 < mean

Remarks: mean corresponds to the parameter of the distribution.

🔗

double mean() const;

Returns: The value of the mean parameter with which the object was constructed.

29.5.9.4.2 Class template exponential_distribution [rand.dist.pois.exp]

An exponential_distribution random number distribution produces random numbers

x > 0

distributed according to the probability density function in Formula 29.9 .

\begin{matrix} p (x | λ) = λ e^{- λ x} \\ (29.9) \end{matrix}

🔗

namespace std { template<class RealType = double> class exponential_distribution { public: // types using result_type = RealType; using param_type = unspecified; // constructors and reset functions exponential_distribution() : exponential_distribution(1.0) {} explicit exponential_distribution(RealType lambda); explicit exponential_distribution(const param_type& parm); void reset(); // equality operators friend bool operator==(const exponential_distribution& x, const exponential_distribution& y); // generating functions template<class URBG> result_type operator()(URBG& g); template<class URBG> result_type operator()(URBG& g, const param_type& parm); // property functions RealType lambda() const; param_type param() const; void param(const param_type& parm); result_type min() const; result_type max() const; // inserters and extractors template<class charT, class traits> friend basic_ostream<charT, traits>& operator<<(basic_ostream<charT, traits>& os, const exponential_distribution& x); template<class charT, class traits> friend basic_istream<charT, traits>& operator>>(basic_istream<charT, traits>& is, exponential_distribution& x); }; }

🔗

explicit exponential_distribution(RealType lambda);

Preconditions:

0 <

🔗	$f_{n} (j)$		j
🔗			0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15
🔗	n	2	0	1
🔗		4	0	3	2	1
🔗		8	2	1	4	7	6	5	0	3
🔗		16	0	9	2	13	6	11	4	15	10	7	12	3	14	5	8	1
🔗