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 .
A distribution's specification identifies its associated probability function p(z) or .
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, or , to name specific parameters, or by writing, for example, p(z|{p}) or , 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 [strings] and [input.output].
Table 105 — Random number distribution requirements
Expression
Return type
Pre/post-condition
Complexity
D​::​result_­type
T
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 , the sequence of numbers returned by successive invocations with the same object g is randomly distributed according to the associated p(z|{p}) or 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 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 , where and 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;