# 29 Numerics library [numerics]

## 29.6 Random number generation [rand]

### 29.6.1 Requirements [rand.req]

#### 29.6.1.6 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(zi). A distribution's specification identifies its associated probability function p(z) or P(zi).

An associated probability function is typically expressed using certain externally-supplied quantities known as the parameters of the distribution. Such distribution parameters are identified in this context by writing, for example, p(z|a,b) or P(zi|a,b), to name specific parameters, or by writing, for example, p(z|{p}) or P(zi|{p}), to denote a distribution's parameters p taken as a whole.

A class D satisfies the requirements of a random number distribution if the expressions shown in Table 105 are valid and have the indicated semantics, and if D and its associated types also satisfy all other requirements of this section [rand.req.dist]. In that Table and throughout this section,

1. a)T is the type named by D's associated result_­type;

2. b)P is the type named by D's associated param_­type;

3. c)d is a value of D, and x and y are (possibly const) values of D;

4. d)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;

5. e)p is a (possibly const) value of P;

6. f)g, g1, and g2 are lvalues of a type satisfying the requirements of a uniform random bit generator;

7. g)os is an lvalue of the type of some class template specialization basic_­ostream<charT, traits>; and

8. h)is is an lvalue of the type of some class template specialization basic_­istream<charT, traits>;

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

Table 105 — Random number distribution requirements
 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(zi|{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(zi|{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 S1=S2, where S1 and S2 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​::​failbit) (which may throw ios​::​failure ([iostate.flags])). Requires: 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 satisfy the requirements of CopyConstructible and CopyAssignable types.

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 satisfy the requirements of CopyConstructible, CopyAssignable, and EqualityComparable types.

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;`