26 Numerics library [numerics]

26.6 Random number generation [rand]

26.6.1 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 $2^{w}$ .

26.6.2 Header <random> synopsis [rand.synopsis]

#include <initializer_list> 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.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 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 bits, class URBG> RealType generate_canonical(URBG& g); // [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; }

26.6.3 Requirements [rand.req]

26.6.3.1 General requirements [rand.req.genl]

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

26.6.3.2 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 93 are valid and have the indicated semantics, and if S also meets all other requirements of this subclause [rand.req.seedseq].

In that Table and throughout this subclause:

(2.1)
T is the type named by S's associated result_type;
(2.2)
q is a value of S and r is a possibly const value of 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 initializer_list<T>.

Table 93: 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.	$O (r.size())$

26.6.3.3 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&>.

26.6.3.4 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 \geq 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 94 are valid and have the indicated semantics, and if E also meets all other requirements of this subclause [rand.req.eng].

In that Table 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 94: 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)247		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)248	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 29) and Cpp17CopyAssignable (Table 31) requirements.

These operations shall each be of complexity no worse than

O (size of state)

247)

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

⮥

248)

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().

⮥

26.6.3.5 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.

26.6.3.6 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)

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 95 are valid and have the indicated semantics, and if D and its associated types also meet all other requirements of this subclause [rand.req.dist].

In that Table 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 95: 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 29) and Cpp17CopyAssignable (Table 31) 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 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 this subclause [rand], declarations of D::param_type are in the form of typedefs for convenience of exposition only.

P shall meet the Cpp17CopyConstructible (Table 29), Cpp17CopyAssignable (Table 31), and Cpp17EqualityComparable (Table 25) 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;

26.6.4 Random number engine class templates [rand.eng]

26.6.4.1 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.

26.6.4.2 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}

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); // generating functions result_type operator()(); void discard(unsigned long long z); };

If the template parameter m is 0, the modulus m used throughout this subclause [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.

26.6.4.3 Class template mersenne_twister_engine [rand.eng.mers]

A mersenne_twister_engine random number engine249 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 $z_{4} = z_{3} x o r (z_{3} r s h i f t ℓ)$ .

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); // generating functions result_type operator()(); void discard(unsigned long long z); };

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}

249)

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.

⮥

26.6.4.4 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.

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 result_type default_seed = 19780503u; // constructors and seeding functions subtract_with_carry_engine() : subtract_with_carry_engine(default_seed) {} explicit subtract_with_carry_engine(result_type value); template<class Sseq> explicit subtract_with_carry_engine(Sseq& q); void seed(result_type value = default_seed); template<class Sseq> void seed(Sseq& q); // generating functions result_type operator()(); void discard(unsigned long long z); };

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<result_type, 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 taken modulo

2^{32}

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.

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

26.6.5.1 In general [rand.adapt.general]

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

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

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

Every function described in this subclause [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 this subclause [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 this subclause [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.

26.6.5.2 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 \geq r

values delivered by e.

The state transition is performed as follows: If

n \geq 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.

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); // generating functions result_type operator()(); void discard(unsigned long long z); // property functions const Engine& base() const noexcept { return e; }; private: Engine e; // exposition only int 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.

26.6.5.3 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 \neq 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}

; }

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); // generating functions result_type operator()(); void discard(unsigned long long z); // property functions const Engine& base() const noexcept { return e; }; 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.

26.6.5.4 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.

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); // generating functions result_type operator()(); void discard(unsigned long long z); // property functions const Engine& base() const noexcept { return e; }; 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], \dots, V[k-1]

and Y, in that order, with values returned by successive invocations of e().

26.6.6 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]

26.6.7 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.

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 could not be initialized.

Remarks: The semantics of the token parameter and the token value used by the default constructor are implementation-defined.250

🔗

double entropy() const noexcept;

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

Otherwise, returns an entropy estimate251 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 could not be obtained.

250)

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

⮥

251)

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}

⮥

26.6.8 Utilities [rand.util]

26.6.8.1 Class seed_seq [rand.util.seedseq]

class seed_seq { public: // types using result_type = uint_least32_t; // constructors seed_seq(); 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();

Postconditions: v.empty() is true.

Throws: Nothing.

🔗

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

Mandates: 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 x o r (x r s h i f t 27)

(10.1)
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
$t = (n \geq 623) ? 11 : (n \geq 68) ? 7 : (n \geq 39) ? 5 : (n \geq 7) ? 3 : (n - 1) / 2;$
(10.2)
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 $r_{2}$ .
(10.3)
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.

26.6.8.2 Function template generate_canonical [rand.util.canonical]

🔗

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

Complexity: Exactly

k = max (1, ⌈ b / log_{2} R ⌉)

invocations of g, where b252 is the lesser of numeric_limits<RealType>::digits and bits, and R is the value of

g.max() - g.min() + 1

Effects: Invokes g() k times to obtain values

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

, respectively.

Calculates a quantity

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

using arithmetic of type RealType.

Returns:

S / R^{k}

[Note 1:

0 \leq S / R^{k} < 1

— end note]

Throws: What and when g throws.

[Note 2:

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]

252)

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

⮥

26.6.9 Random number distribution class templates [rand.dist]

26.6.9.1 In general [rand.dist.general]

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

Descriptions are provided in this subclause [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.

26.6.9.2 Uniform distributions [rand.dist.uni]

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

A uniform_int_distribution random number distribution produces random integers i,

a \leq i \leq b

, distributed according to the constant discrete probability function

P (i | a, b) = 1 / (b - a + 1) .

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(); // 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; };

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

Preconditions:

a \leq b

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

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result_type a() const;

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

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result_type b() const;

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

26.6.9.2.2 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

p (x | a, b) = 1 / (b - a) .

[Note 1:

This implies that

p (x | a, b)

is undefined when a == b.

— end note]

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(); // 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; };

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

Preconditions:

a \leq b

and

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

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

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result_type a() const;

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

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result_type b() const;

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

26.6.9.3 Bernoulli distributions [rand.dist.bern]

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

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

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

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(); // 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; };

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explicit bernoulli_distribution(double p);

Preconditions:

0 \leq p \leq 1

Remarks: p corresponds to the parameter of the distribution.

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double p() const;

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

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

A binomial_distribution random number distribution produces integer values

i \geq 0

distributed according to the discrete probability function

P (i | t, p) = (\frac{t}{i}) \cdot p^{i} \cdot (1 - p)^{t - i} .

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(); // 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; };

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

Preconditions:

0 \leq p \leq 1

and

0 \leq t

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

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IntType t() const;

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

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double p() const;

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

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

A geometric_distribution random number distribution produces integer values

i \geq 0

distributed according to the discrete probability function

P (i | p) = p \cdot (1 - p)^{i} .

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(); // 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; };

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explicit geometric_distribution(double p);

Preconditions:

0 < p < 1

Remarks: p corresponds to the parameter of the distribution.

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double p() const;

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

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

A negative_binomial_distribution random number distribution produces random integers

i \geq 0

distributed according to the discrete probability function

P (i | k, p) = (\frac{k + i - 1}{i}) \cdot p^{k} \cdot (1 - p)^{i} .

[Note 1:

This implies that

P (i | k, p)

is undefined when p == 1.

— end note]

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(); // 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; };

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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.

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IntType k() const;

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

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double p() const;

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

26.6.9.4 Poisson distributions [rand.dist.pois]

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

A poisson_distribution random number distribution produces integer values

i \geq 0

distributed according to the discrete probability function

P (i | μ) = \frac{e^{- μ} μ^{i}}{i!} .

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

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(); // 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; };

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explicit poisson_distribution(double mean);

Preconditions:

0 < mean

Remarks: mean corresponds to the parameter of the distribution.

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double mean() const;

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

26.6.9.4.2 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

p (x | λ) = λ e^{- λ x} .

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(); // 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; };

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explicit exponential_distribution(RealType lambda);

Preconditions:

0 < lambda

Remarks: lambda corresponds to the parameter of the distribution.

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RealType lambda() const;

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

26.6.9.4.3 Class template gamma_distribution [rand.dist.pois.gamma]

A gamma_distribution random number distribution produces random numbers

x > 0

distributed according to the probability density function

p (x | α, β) = \frac{e^{- x / β}}{β^{α} \cdot Γ (α)} \cdot x^{α - 1} .

template<class RealType = double> class gamma_distribution { public: // types using result_type = RealType; using param_type = unspecified; // constructors and reset functions gamma_distribution() : gamma_distribution(1.0) {} explicit gamma_distribution(RealType alpha, RealType beta = 1.0); explicit gamma_distribution(const param_type& parm); void reset(); // 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 alpha() const; RealType beta() const; param_type param() const; void param(const param_type& parm); result_type min() const; result_type max() const; };

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explicit gamma_distribution(RealType alpha, RealType beta = 1.0);

Preconditions:

0 < alpha

and

0 < beta

Remarks: alpha and beta correspond to the parameters of the distribution.

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RealType alpha() const;

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

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RealType beta() const;

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

26.6.9.4.4 Class template weibull_distribution [rand.dist.pois.weibull]

A weibull_distribution random number distribution produces random numbers

x \geq 0

distributed according to the probability density function

p (x | a, b) = \frac{a}{b} \cdot {(\frac{x}{b})}^{a - 1} \cdot exp (- {(\frac{x}{b})}^{a}) .

template<class RealType = double> class weibull_distribution { public: // types using result_type = RealType; using param_type = unspecified; // constructor and reset functions weibull_distribution() : weibull_distribution(1.0) {} explicit weibull_distribution(RealType a, RealType b = 1.0); explicit weibull_distribution(const param_type& parm); void reset(); // 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 a() const; RealType b() const; param_type param() const; void param(const param_type& parm); result_type min() const; result_type max() const; };

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

Preconditions:

0 < a

and

0 < b

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

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RealType a() const;

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

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RealType b() const;

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

26.6.9.4.5 Class template extreme_value_distribution [rand.dist.pois.extreme]

An extreme_value_distribution random number distribution produces random numbers x distributed according to the probability density function253

p (x | a, b) = \frac{1}{b} \cdot exp (\frac{a - x}{b} - exp (\frac{a - x}{b})) .

template<class RealType = double> class extreme_value_distribution { public: // types using result_type = RealType; using param_type = unspecified; // constructor and reset functions extreme_value_distribution() : extreme_value_distribution(0.0) {} explicit extreme_value_distribution(RealType a, RealType b = 1.0); explicit extreme_value_distribution(const param_type& parm); void reset(); // 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 a() const; RealType b() const; param_type param() const; void param(const param_type& parm); result_type min() const; result_type max() const; };

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

Preconditions:

0 < b

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

🔗

RealType a() const;

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

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RealType b() const;

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

253)

The distribution corresponding to this probability density function is also known (with a possible change of variable) as the Gumbel Type I, the log-Weibull, or the Fisher-Tippett Type I distribution.

⮥

26.6.9.5 Normal distributions [rand.dist.norm]

26.6.9.5.1 Class template normal_distribution [rand.dist.norm.normal]

A normal_distribution random number distribution produces random numbers x distributed according to the probability density function

p (x | μ, σ) = \frac{1}{σ \sqrt{2 π}} \cdot exp (- \frac{(x - μ)^{2}}{2 σ^{2}}) .

The distribution parameters μ and σ are also known as this distribution's mean and standard deviation.

template<class RealType = double> class normal_distribution { public: // types using result_type = RealType; using param_type = unspecified; // constructors and reset functions normal_distribution() : normal_distribution(0.0) {} explicit normal_distribution(RealType mean, RealType stddev = 1.0); explicit normal_distribution(const param_type& parm); void reset(); // 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 mean() const; RealType stddev() const; param_type param() const; void param(const param_type& parm); result_type min() const; result_type max() const; };

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explicit normal_distribution(RealType mean, RealType stddev = 1.0);

Preconditions:

0 < stddev

Remarks: mean and stddev correspond to the respective parameters of the distribution.

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RealType mean() const;

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

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RealType stddev() const;

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

26.6.9.5.2 Class template lognormal_distribution [rand.dist.norm.lognormal]

A lognormal_distribution random number distribution produces random numbers

x > 0

distributed according to the probability density function

p (x | m, s) = \frac{1}{s x \sqrt{2 π}} \cdot exp (- \frac{(ln x - m)^{2}}{2 s^{2}}) .

template<class RealType = double> class lognormal_distribution { public: // types using result_type = RealType; using param_type = unspecified; // constructor and reset functions lognormal_distribution() : lognormal_distribution(0.0) {} explicit lognormal_distribution(RealType m, RealType s = 1.0); explicit lognormal_distribution(const param_type& parm); void reset(); // 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 m() const; RealType s() const; param_type param() const; void param(const param_type& parm); result_type min() const; result_type max() const; };

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explicit lognormal_distribution(RealType m, RealType s = 1.0);

Preconditions:

0 < s

Remarks: m and s correspond to the respective parameters of the distribution.

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RealType m() const;

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

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RealType s() const;

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

26.6.9.5.3 Class template chi_squared_distribution [rand.dist.norm.chisq]

A chi_squared_distribution random number distribution produces random numbers

x > 0

distributed according to the probability density function

p (x | n) = \frac{x^{(n / 2) - 1} \cdot e^{- x / 2}}{Γ (n / 2) \cdot 2^{n / 2}} .

template<class RealType = double> class chi_squared_distribution { public: // types using result_type = RealType; using param_type = unspecified; // constructor and reset functions chi_squared_distribution() : chi_squared_distribution(1.0) {} explicit chi_squared_distribution(RealType n); explicit chi_squared_distribution(const param_type& parm); void reset(); // 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 n() const; param_type param() const; void param(const param_type& parm); result_type min() const; result_type max() const; };

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explicit chi_squared_distribution(RealType n);

Preconditions:

0 < n

Remarks: n corresponds to the parameter of the distribution.

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RealType n() const;

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

26.6.9.5.4 Class template cauchy_distribution [rand.dist.norm.cauchy]

A cauchy_distribution random number distribution produces random numbers x distributed according to the probability density function

p (x | a, b) = {(π b (1 + {(\frac{x - a}{b})}^{2}))}^{- 1} .

template<class RealType = double> class cauchy_distribution { public: // types using result_type = RealType; using param_type = unspecified; // constructor and reset functions cauchy_distribution() : cauchy_distribution(0.0) {} explicit cauchy_distribution(RealType a, RealType b = 1.0); explicit cauchy_distribution(const param_type& parm); void reset(); // 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 a() const; RealType b() const; param_type param() const; void param(const param_type& parm); result_type min() const; result_type max() const; };

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

Preconditions:

0 < b

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

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RealType a() const;

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

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RealType b() const;

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

26.6.9.5.5 Class template fisher_f_distribution [rand.dist.norm.f]

A fisher_f_distribution random number distribution produces random numbers

x \geq 0

distributed according to the probability density function

p (x | m, n) = \frac{Γ ((m + n) / 2)}{Γ (m / 2) Γ (n / 2)} \cdot {(\frac{m}{n})}^{m / 2} \cdot x^{(m / 2) - 1} \cdot {(1 + \frac{m x}{n})}^{- (m + n) / 2} .

template<class RealType = double> class fisher_f_distribution { public: // types using result_type = RealType; using param_type = unspecified; // constructor and reset functions fisher_f_distribution() : fisher_f_distribution(1.0) {} explicit fisher_f_distribution(RealType m, RealType n = 1.0); explicit fisher_f_distribution(const param_type& parm); void reset(); // 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 m() const; RealType n() const; param_type param() const; void param(const param_type& parm); result_type min() const; result_type max() const; };

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explicit fisher_f_distribution(RealType m, RealType n = 1);

Preconditions:

0 < m

and

0 < n

Remarks: m and n correspond to the respective parameters of the distribution.

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RealType m() const;

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

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RealType n() const;

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

26.6.9.5.6 Class template student_t_distribution [rand.dist.norm.t]

A student_t_distribution random number distribution produces random numbers x distributed according to the probability density function

p (x | n) = \frac{1}{\sqrt{n π}} \cdot \frac{Γ ((n + 1) / 2)}{Γ (n / 2)} \cdot {(1 + \frac{x^{2}}{n})}^{- (n + 1) / 2} .

template<class RealType = double> class student_t_distribution { public: // types using result_type = RealType; using param_type = unspecified; // constructor and reset functions student_t_distribution() : student_t_distribution(1.0) {} explicit student_t_distribution(RealType n); explicit student_t_distribution(const param_type& parm); void reset(); // 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 n() const; param_type param() const; void param(const param_type& parm); result_type min() const; result_type max() const; };

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explicit student_t_distribution(RealType n);

Preconditions:

0 < n

Remarks: n corresponds to the parameter of the distribution.

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RealType n() const;

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

26.6.9.6 Sampling distributions [rand.dist.samp]

26.6.9.6.1 Class template discrete_distribution [rand.dist.samp.discrete]

A discrete_distribution random number distribution produces random integers i,

0 \leq i < n

, distributed according to the discrete probability function

P (i | p_{0}, \dots, p_{n - 1}) = p_{i} .

Unless specified otherwise, the distribution parameters are calculated as:

p_{k} = w_{k} / S

for

k = 0, \dots, n - 1

, in which the values

w_{k}

, commonly known as the weights, shall be non-negative, non-NaN, and non-infinity.

Moreover, the following relation shall hold:

0 < S = w_{0} + \dots + w_{n - 1}

template<class IntType = int> class discrete_distribution { public: // types using result_type = IntType; using param_type = unspecified; // constructor and reset functions discrete_distribution(); template<class InputIterator> discrete_distribution(InputIterator firstW, InputIterator lastW); discrete_distribution(initializer_list<double> wl); template<class UnaryOperation> discrete_distribution(size_t nw, double xmin, double xmax, UnaryOperation fw); explicit discrete_distribution(const param_type& parm); void reset(); // generating functions template<class URBG> result_type operator()(URBG& g); template<class URBG> result_type operator()(URBG& g, const param_type& parm); // property functions vector<double> probabilities() const; param_type param() const; void param(const param_type& parm); result_type min() const; result_type max() const; };

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discrete_distribution();

Effects: Constructs a discrete_distribution object with

n = 1

and

p_{0} = 1

[Note 1:

Such an object will always deliver the value 0.

— end note]

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template<class InputIterator>
  discrete_distribution(InputIterator firstW, InputIterator lastW);

Mandates: is_convertible_v<iterator_traits<InputIterator>::value_type, double> is true.

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

If firstW == lastW, let

n = 1

and

w_{0} = 1

Otherwise,

[firstW, lastW)

forms a sequence w of length

n > 0

Effects: Constructs a discrete_distribution object with probabilities given by the formula above.

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discrete_distribution(initializer_list<double> wl);

Effects: Same as discrete_distribution(wl.begin(), wl.end()).

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template<class UnaryOperation>
  discrete_distribution(size_t nw, double xmin, double xmax, UnaryOperation fw);

Mandates: is_invocable_r_v<double, UnaryOperation&, double> is true.

Preconditions: If

nw = 0

, let

n = 1

, otherwise let

n = nw

The relation

0 < δ = (xmax - xmin) / n

holds.

Effects: Constructs a discrete_distribution object with probabilities given by the formula above, using the following values: If

nw = 0

, let

w_{0} = 1

Otherwise, let

w_{k} = fw (xmin + k \cdot δ + δ / 2)

for

k = 0, \dots, n - 1

Complexity: The number of invocations of fw does not exceed n.

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vector<double> probabilities() const;

Returns: A vector<double> whose size member returns n and whose operator[] member returns

p_{k}

when invoked with argument k for

k = 0, \dots, n - 1

26.6.9.6.2 Class template piecewise_constant_distribution [rand.dist.samp.pconst]

A piecewise_constant_distribution random number distribution produces random numbers x,

b_{0} \leq x < b_{n}

, uniformly distributed over each subinterval

[b_{i}, b_{i + 1})

according to the probability density function

p (x | b_{0}, \dots, b_{n}, ρ_{0}, \dots, ρ_{n - 1}) = ρ_{i}, for b_{i} \leq x < b_{i + 1} .

The

n + 1

distribution parameters

b_{i}

, also known as this distribution's interval boundaries, shall satisfy the relation

b_{i} < b_{i + 1}

for

i = 0, \dots, n - 1

Unless specified otherwise, the remaining n distribution parameters are calculated as:

ρ_{k} = \frac{w_{k}}{S \cdot (b_{k + 1} - b_{k})} for k = 0, \dots, n - 1,

in which the values

w_{k}

, commonly known as the weights, shall be non-negative, non-NaN, and non-infinity.

Moreover, the following relation shall hold:

0 < S = w_{0} + \dots + w_{n - 1}

template<class RealType = double> class piecewise_constant_distribution { public: // types using result_type = RealType; using param_type = unspecified; // constructor and reset functions piecewise_constant_distribution(); template<class InputIteratorB, class InputIteratorW> piecewise_constant_distribution(InputIteratorB firstB, InputIteratorB lastB, InputIteratorW firstW); template<class UnaryOperation> piecewise_constant_distribution(initializer_list<RealType> bl, UnaryOperation fw); template<class UnaryOperation> piecewise_constant_distribution(size_t nw, RealType xmin, RealType xmax, UnaryOperation fw); explicit piecewise_constant_distribution(const param_type& parm); void reset(); // generating functions template<class URBG> result_type operator()(URBG& g); template<class URBG> result_type operator()(URBG& g, const param_type& parm); // property functions vector<result_type> intervals() const; vector<result_type> densities() const; param_type param() const; void param(const param_type& parm); result_type min() const; result_type max() const; };

🔗

piecewise_constant_distribution();

Effects: Constructs a piecewise_constant_distribution object with

n = 1

ρ_{0} = 1

b_{0} = 0

, and

b_{1} = 1

🔗

template<class InputIteratorB, class InputIteratorW>
 piecewise_constant_distribution(InputIteratorB firstB, InputIteratorB lastB,
                                 InputIteratorW firstW);

Mandates: Both of

(4.1)
is_convertible_v<iterator_traits<InputIteratorB>::value_type, double>
(4.2)
is_convertible_v<iterator_traits<InputIteratorW>::value_type, double>

are true.

Preconditions: InputIteratorB and InputIteratorW each meet the Cpp17InputIterator requirements ([input.iterators]).

If firstB == lastB or ++firstB == lastB, let

n = 1

w_{0} = 1

b_{0} = 0

, and

b_{1} = 1

Otherwise,

[firstB, lastB)

forms a sequence b of length

n + 1

, the length of the sequence w starting from firstW is at least n, and any

w_{k}

for

k \geq n

are ignored by the distribution.

Effects: Constructs a piecewise_constant_distribution object with parameters as specified above.

🔗

template<class UnaryOperation>
 piecewise_constant_distribution(initializer_list<RealType> bl, UnaryOperation fw);

Mandates: is_invocable_r_v<double, UnaryOperation&, double> is true.

Effects: Constructs a piecewise_constant_distribution object with parameters taken or calculated from the following values: If

bl.size() < 2

, let

n = 1

w_{0} = 1

b_{0} = 0

, and

b_{1} = 1

Otherwise, let

[bl.begin(), bl.end())

form a sequence

b_{0}, \dots, b_{n}

, and let

w_{k} = fw ((b_{k + 1} + b_{k}) / 2)

for

k = 0, \dots, n - 1

Complexity: The number of invocations of fw does not exceed n.

🔗

template<class UnaryOperation>
 piecewise_constant_distribution(size_t nw, RealType xmin, RealType xmax, UnaryOperation fw);

Mandates: is_invocable_r_v<double, UnaryOperation&, double> is true.

Preconditions: If

nw = 0

, let

n = 1

, otherwise let

n = nw

The relation

0 < δ = (xmax - xmin) / n

holds.

Effects: Constructs a piecewise_constant_distribution object with parameters taken or calculated from the following values: Let

b_{k} = xmin + k \cdot δ

for

k = 0, \dots, n

, and

w_{k} = fw (b_{k} + δ / 2)

for

k = 0, \dots, n - 1

Complexity: The number of invocations of fw does not exceed n.

🔗

vector<result_type> intervals() const;

Returns: A vector<result_type> whose size member returns

n + 1

and whose operator[] member returns

b_{k}

when invoked with argument k for

k = 0, \dots, n

🔗

vector<result_type> densities() const;

Returns: A vector<result_type> whose size member returns n and whose operator[] member returns

ρ_{k}

when invoked with argument k for

k = 0, \dots, n - 1

26.6.9.6.3 Class template piecewise_linear_distribution [rand.dist.samp.plinear]

A piecewise_linear_distribution random number distribution produces random numbers x,

b_{0} \leq x < b_{n}

, distributed over each subinterval

[b_{i}, b_{i + 1})

according to the probability density function

p (x | b_{0}, \dots, b_{n}, ρ_{0}, \dots, ρ_{n}) = ρ_{i} \cdot \frac{b_{i + 1} - x}{b_{i + 1} - b_{i}} + ρ_{i + 1} \cdot \frac{x - b_{i}}{b_{i + 1} - b_{i}}, for b_{i} \leq x < b_{i + 1} .

The

n + 1

distribution parameters

b_{i}

, also known as this distribution's interval boundaries, shall satisfy the relation

b_{i} < b_{i + 1}

for

i = 0, \dots, n - 1

Unless specified otherwise, the remaining

n + 1

distribution parameters are calculated as

ρ_{k} = w_{k} / S

for

k = 0, \dots, n

, in which the values

w_{k}

, commonly known as the weights at boundaries, shall be non-negative, non-NaN, and non-infinity.

Moreover, the following relation shall hold:

0 < S = \frac{1}{2} \cdot n - 1 \sum k = 0 (w_{k} + w_{k + 1}) \cdot (b_{k + 1} - b_{k}) .

template<class RealType = double> class piecewise_linear_distribution { public: // types using result_type = RealType; using param_type = unspecified; // constructor and reset functions piecewise_linear_distribution(); template<class InputIteratorB, class InputIteratorW> piecewise_linear_distribution(InputIteratorB firstB, InputIteratorB lastB, InputIteratorW firstW); template<class UnaryOperation> piecewise_linear_distribution(initializer_list<RealType> bl, UnaryOperation fw); template<class UnaryOperation> piecewise_linear_distribution(size_t nw, RealType xmin, RealType xmax, UnaryOperation fw); explicit piecewise_linear_distribution(const param_type& parm); void reset(); // generating functions template<class URBG> result_type operator()(URBG& g); template<class URBG> result_type operator()(URBG& g, const param_type& parm); // property functions vector<result_type> intervals() const; vector<result_type> densities() const; param_type param() const; void param(const param_type& parm); result_type min() const; result_type max() const; };

🔗

piecewise_linear_distribution();

Effects: Constructs a piecewise_linear_distribution object with

n = 1

ρ_{0} = ρ_{1} = 1

b_{0} = 0

, and

b_{1} = 1

🔗

template<class InputIteratorB, class InputIteratorW>
 piecewise_linear_distribution(InputIteratorB firstB, InputIteratorB lastB,
                               InputIteratorW firstW);

Mandates: is_invocable_r_v<double, UnaryOperation&, double> is true.

Preconditions: InputIteratorB and InputIteratorW each meet the Cpp17InputIterator requirements ([input.iterators]).

If firstB == lastB or ++firstB == lastB, let

n = 1

ρ_{0} = ρ_{1} = 1

b_{0} = 0

, and

b_{1} = 1

Otherwise,

[firstB, lastB)

forms a sequence b of length

n + 1

, the length of the sequence w starting from firstW is at least

n + 1

, and any

w_{k}

for

k \geq n + 1

are ignored by the distribution.

Effects: Constructs a piecewise_linear_distribution object with parameters as specified above.

🔗

template<class UnaryOperation>
 piecewise_linear_distribution(initializer_list<RealType> bl, UnaryOperation fw);

Mandates: is_invocable_r_v<double, UnaryOperation&, double> is true.

Effects: Constructs a piecewise_linear_distribution object with parameters taken or calculated from the following values: If

bl.size() < 2

, let

n = 1

ρ_{0} = ρ_{1} = 1

b_{0} = 0

, and

b_{1} = 1

Otherwise, let

[bl.begin(), bl.end())

form a sequence

b_{0}, \dots, b_{n}

, and let

w_{k} = fw (b_{k})

for

k = 0, \dots, n

Complexity: The number of invocations of fw does not exceed

n + 1

🔗

template<class UnaryOperation>
 piecewise_linear_distribution(size_t nw, RealType xmin, RealType xmax, UnaryOperation fw);

Mandates: is_invocable_r_v<double, UnaryOperation&, double> is true.

Preconditions: If

nw = 0

, let

n = 1

, otherwise let

n = nw

The relation

0 < δ = (xmax - xmin) / n

holds.

Effects: Constructs a piecewise_linear_distribution object with parameters taken or calculated from the following values: Let

b_{k} = xmin + k \cdot δ

for

k = 0, \dots, n

, and

w_{k} = fw (b_{k})

for

k = 0, \dots, n

Complexity: The number of invocations of fw does not exceed

n + 1

🔗

vector<result_type> intervals() const;

Returns: A vector<result_type> whose size member returns

n + 1

and whose operator[] member returns

b_{k}

when invoked with argument k for

k = 0, \dots, n

🔗

vector<result_type> densities() const;

Returns: A vector<result_type> whose size member returns n and whose operator[] member returns

ρ_{k}

when invoked with argument k for

k = 0, \dots, n

26.6.10 Low-quality random number generation [c.math.rand]

[Note 1:

The header <cstdlib> declares the functions described in this subclause.

— end note]

🔗

int rand();
void srand(unsigned int seed);

Effects: The rand and srand functions have the semantics specified in the C standard library.

Remarks: The implementation may specify that particular library functions may call rand.

It is implementation-defined whether the rand function may introduce data races ([res.on.data.races]).

[Note 2:

The other random number generation facilities in this document ([rand]) are often preferable to rand, because rand's underlying algorithm is unspecified.

Use of rand therefore continues to be non-portable, with unpredictable and oft-questionable quality and performance.

— end note]

26 Numerics library [numerics]

26.6 Random number generation [rand]

26.6.1 General [rand.general]

26.6.2 Header <random> synopsis [rand.synopsis]

26.6.3 Requirements [rand.req]

26.6.3.1 General requirements [rand.req.genl]

26.6.3.2 Seed sequence requirements [rand.req.seedseq]

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

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

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

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

26.6.4 Random number engine class templates [rand.eng]

26.6.4.1 General [rand.eng.general]

26.6.4.2 Class template linear_­congruential_­engine [rand.eng.lcong]

26.6.4.3 Class template mersenne_­twister_­engine [rand.eng.mers]

26.6.4.4 Class template subtract_­with_­carry_­engine [rand.eng.sub]

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

26.6.5.1 In general [rand.adapt.general]

26.6.5.2 Class template discard_­block_­engine [rand.adapt.disc]

26.6.5.3 Class template independent_­bits_­engine [rand.adapt.ibits]

26.6.5.4 Class template shuffle_­order_­engine [rand.adapt.shuf]

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

26.6.7 Class random_­device [rand.device]

26.6.8 Utilities [rand.util]

26.6.8.1 Class seed_­seq [rand.util.seedseq]

26.6.8.2 Function template generate_­canonical [rand.util.canonical]

26.6.9 Random number distribution class templates [rand.dist]

26.6.9.1 In general [rand.dist.general]

26.6.9.2 Uniform distributions [rand.dist.uni]

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

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

26.6.9.3 Bernoulli distributions [rand.dist.bern]

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

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

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

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

26.6.9.4 Poisson distributions [rand.dist.pois]

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

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

26.6.9.4.3 Class template gamma_­distribution [rand.dist.pois.gamma]

26.6.9.4.4 Class template weibull_­distribution [rand.dist.pois.weibull]

26.6.9.4.5 Class template extreme_­value_­distribution [rand.dist.pois.extreme]

26.6.9.5 Normal distributions [rand.dist.norm]

26.6.9.5.1 Class template normal_­distribution [rand.dist.norm.normal]

26.6.9.5.2 Class template lognormal_­distribution [rand.dist.norm.lognormal]

26.6.9.5.3 Class template chi_­squared_­distribution [rand.dist.norm.chisq]

26.6.9.5.4 Class template cauchy_­distribution [rand.dist.norm.cauchy]

26.6.9.5.5 Class template fisher_­f_­distribution [rand.dist.norm.f]

26.6.9.5.6 Class template student_­t_­distribution [rand.dist.norm.t]

26.6.9.6 Sampling distributions [rand.dist.samp]

26.6.9.6.1 Class template discrete_­distribution [rand.dist.samp.discrete]

26.6.9.6.2 Class template piecewise_­constant_­distribution [rand.dist.samp.pconst]

26.6.9.6.3 Class template piecewise_­linear_­distribution [rand.dist.samp.plinear]

26.6.10 Low-quality random number generation [c.math.rand]

26.6.4.2 Class template linear_congruential_engine [rand.eng.lcong]

26.6.4.3 Class template mersenne_twister_engine [rand.eng.mers]

26.6.4.4 Class template subtract_with_carry_engine [rand.eng.sub]

26.6.5.2 Class template discard_block_engine [rand.adapt.disc]

26.6.5.3 Class template independent_bits_engine [rand.adapt.ibits]

26.6.5.4 Class template shuffle_order_engine [rand.adapt.shuf]

26.6.7 Class random_device [rand.device]

26.6.8.1 Class seed_seq [rand.util.seedseq]

26.6.8.2 Function template generate_canonical [rand.util.canonical]

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

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

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

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

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

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

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

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

26.6.9.4.3 Class template gamma_distribution [rand.dist.pois.gamma]

26.6.9.4.4 Class template weibull_distribution [rand.dist.pois.weibull]

26.6.9.4.5 Class template extreme_value_distribution [rand.dist.pois.extreme]

26.6.9.5.1 Class template normal_distribution [rand.dist.norm.normal]

26.6.9.5.2 Class template lognormal_distribution [rand.dist.norm.lognormal]

26.6.9.5.3 Class template chi_squared_distribution [rand.dist.norm.chisq]

26.6.9.5.4 Class template cauchy_distribution [rand.dist.norm.cauchy]

26.6.9.5.5 Class template fisher_f_distribution [rand.dist.norm.f]

26.6.9.5.6 Class template student_t_distribution [rand.dist.norm.t]

26.6.9.6.1 Class template discrete_distribution [rand.dist.samp.discrete]

26.6.9.6.2 Class template piecewise_constant_distribution [rand.dist.samp.pconst]

26.6.9.6.3 Class template piecewise_linear_distribution [rand.dist.samp.plinear]