26 Numerics library [numerics]

26.1 General [numerics.general]

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

The following subclauses describe components for complex number types, random number generation, numeric (n-at-a-time) arrays, generalized numeric algorithms, and mathematical constants and functions for floating-point types, as summarized in Table 92 .

Table 92: Numerics library summary [tab:numerics.summary]

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

26.2 Numeric type requirements [numeric.requirements]

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

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

A numeric type is a cv-unqualified object type T that meets the Cpp17DefaultConstructible, Cpp17CopyConstructible, Cpp17CopyAssignable, and Cpp17Destructible requirements ([utility.arg.requirements]).240

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

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

[Example

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

— end example

]

240)

In other words, value types.

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

⮥

26.3 The floating-point environment [cfenv]

26.3.1 Header <cfenv> synopsis [cfenv.syn]

#define FE_ALL_EXCEPT see below
#define FE_DIVBYZERO see below    // optional
#define FE_INEXACT see below      // optional
#define FE_INVALID see below      // optional
#define FE_OVERFLOW see below     // optional
#define FE_UNDERFLOW see below    // optional

#define FE_DOWNWARD see below     // optional
#define FE_TONEAREST see below    // optional
#define FE_TOWARDZERO see below   // optional
#define FE_UPWARD see below       // optional

#define FE_DFL_ENV see below

namespace std {
  // types
  using fenv_t    = object type;
  using fexcept_t = integer type;

  // functions
  int feclearexcept(int except);
  int fegetexceptflag(fexcept_t* pflag, int except);
  int feraiseexcept(int except);
  int fesetexceptflag(const fexcept_t* pflag, int except);
  int fetestexcept(int except);

  int fegetround();
  int fesetround(int mode);

  int fegetenv(fenv_t* penv);
  int feholdexcept(fenv_t* penv);
  int fesetenv(const fenv_t* penv);
  int feupdateenv(const fenv_t* penv);
}

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

[Note

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

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

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

— end note

]

The floating-point environment has thread storage duration .

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

[Note

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

— end note

]

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

Each function accesses the environment corresponding to its calling thread.

26.4 Complex numbers [complex.numbers]

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

The effect of instantiating the template complex for any type other than float, double, or long double is unspecified.

The specializations complex<float>, complex<double>, and complex<long double> are literal types .

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

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

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

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

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

26.4.1 Header <complex> synopsis [complex.syn]

namespace std {
  // [complex], class template complex
  template<class T> class complex;

  // [complex.special], specializations
  template<> class complex<float>;
  template<> class complex<double>;
  template<> class complex<long double>;

  // [complex.ops], operators
  template<class T> constexpr complex<T> operator+(const complex<T>&, const complex<T>&);
  template<class T> constexpr complex<T> operator+(const complex<T>&, const T&);
  template<class T> constexpr complex<T> operator+(const T&, const complex<T>&);

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

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

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

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

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

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

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

  // [complex.value.ops], values
  template<class T> constexpr T real(const complex<T>&);
  template<class T> constexpr T imag(const complex<T>&);

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

  template<class T> constexpr complex<T> conj(const complex<T>&);
  template<class T> complex<T> proj(const complex<T>&);
  template<class T> complex<T> polar(const T&, const T& = T());

  // [complex.transcendentals], transcendentals
  template<class T> complex<T> acos(const complex<T>&);
  template<class T> complex<T> asin(const complex<T>&);
  template<class T> complex<T> atan(const complex<T>&);

  template<class T> complex<T> acosh(const complex<T>&);
  template<class T> complex<T> asinh(const complex<T>&);
  template<class T> complex<T> atanh(const complex<T>&);

  template<class T> complex<T> cos  (const complex<T>&);
  template<class T> complex<T> cosh (const complex<T>&);
  template<class T> complex<T> exp  (const complex<T>&);
  template<class T> complex<T> log  (const complex<T>&);
  template<class T> complex<T> log10(const complex<T>&);

  template<class T> complex<T> pow  (const complex<T>&, const T&);
  template<class T> complex<T> pow  (const complex<T>&, const complex<T>&);
  template<class T> complex<T> pow  (const T&, const complex<T>&);

  template<class T> complex<T> sin  (const complex<T>&);
  template<class T> complex<T> sinh (const complex<T>&);
  template<class T> complex<T> sqrt (const complex<T>&);
  template<class T> complex<T> tan  (const complex<T>&);
  template<class T> complex<T> tanh (const complex<T>&);

  // [complex.literals], complex literals
  inline namespace literals {
  inline namespace complex_literals {
    constexpr complex<long double> operator""il(long double);
    constexpr complex<long double> operator""il(unsigned long long);
    constexpr complex<double> operator""i(long double);
    constexpr complex<double> operator""i(unsigned long long);
    constexpr complex<float> operator""if(long double);
    constexpr complex<float> operator""if(unsigned long long);
  }
  }
}

26.4.2 Class template complex [complex]

namespace std {
  template<class T> class complex {
  public:
    using value_type = T;

    constexpr complex(const T& re = T(), const T& im = T());
    constexpr complex(const complex&);
    template<class X> constexpr complex(const complex<X>&);

    constexpr T real() const;
    constexpr void real(T);
    constexpr T imag() const;
    constexpr void imag(T);

    constexpr complex& operator= (const T&);
    constexpr complex& operator+=(const T&);
    constexpr complex& operator-=(const T&);
    constexpr complex& operator*=(const T&);
    constexpr complex& operator/=(const T&);

    constexpr complex& operator=(const complex&);
    template<class X> constexpr complex& operator= (const complex<X>&);
    template<class X> constexpr complex& operator+=(const complex<X>&);
    template<class X> constexpr complex& operator-=(const complex<X>&);
    template<class X> constexpr complex& operator*=(const complex<X>&);
    template<class X> constexpr complex& operator/=(const complex<X>&);
  };
}

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

26.4.3 Specializations [complex.special]

namespace std {
  template<> class complex<float> {
  public:
    using value_type = float;

    constexpr complex(float re = 0.0f, float im = 0.0f);
    constexpr complex(const complex<float>&) = default;
    constexpr explicit complex(const complex<double>&);
    constexpr explicit complex(const complex<long double>&);

    constexpr float real() const;
    constexpr void real(float);
    constexpr float imag() const;
    constexpr void imag(float);

    constexpr complex& operator= (float);
    constexpr complex& operator+=(float);
    constexpr complex& operator-=(float);
    constexpr complex& operator*=(float);
    constexpr complex& operator/=(float);

    constexpr complex& operator=(const complex&);
    template<class X> constexpr complex& operator= (const complex<X>&);
    template<class X> constexpr complex& operator+=(const complex<X>&);
    template<class X> constexpr complex& operator-=(const complex<X>&);
    template<class X> constexpr complex& operator*=(const complex<X>&);
    template<class X> constexpr complex& operator/=(const complex<X>&);
  };

  template<> class complex<double> {
  public:
    using value_type = double;

    constexpr complex(double re = 0.0, double im = 0.0);
    constexpr complex(const complex<float>&);
    constexpr complex(const complex<double>&) = default;
    constexpr explicit complex(const complex<long double>&);

    constexpr double real() const;
    constexpr void real(double);
    constexpr double imag() const;
    constexpr void imag(double);

    constexpr complex& operator= (double);
    constexpr complex& operator+=(double);
    constexpr complex& operator-=(double);
    constexpr complex& operator*=(double);
    constexpr complex& operator/=(double);

    constexpr complex& operator=(const complex&);
    template<class X> constexpr complex& operator= (const complex<X>&);
    template<class X> constexpr complex& operator+=(const complex<X>&);
    template<class X> constexpr complex& operator-=(const complex<X>&);
    template<class X> constexpr complex& operator*=(const complex<X>&);
    template<class X> constexpr complex& operator/=(const complex<X>&);
  };

  template<> class complex<long double> {
  public:
    using value_type = long double;

    constexpr complex(long double re = 0.0L, long double im = 0.0L);
    constexpr complex(const complex<float>&);
    constexpr complex(const complex<double>&);
    constexpr complex(const complex<long double>&) = default;

    constexpr long double real() const;
    constexpr void real(long double);
    constexpr long double imag() const;
    constexpr void imag(long double);

    constexpr complex& operator= (long double);
    constexpr complex& operator+=(long double);
    constexpr complex& operator-=(long double);
    constexpr complex& operator*=(long double);
    constexpr complex& operator/=(long double);

    constexpr complex& operator=(const complex&);
    template<class X> constexpr complex& operator= (const complex<X>&);
    template<class X> constexpr complex& operator+=(const complex<X>&);
    template<class X> constexpr complex& operator-=(const complex<X>&);
    template<class X> constexpr complex& operator*=(const complex<X>&);
    template<class X> constexpr complex& operator/=(const complex<X>&);
  };
}

26.4.4 Member functions [complex.members]

🔗

template<class T> constexpr complex(const T& re = T(), const T& im = T());

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

🔗

constexpr T real() const;

Returns: The value of the real component.

🔗

constexpr void real(T val);

Effects: Assigns val to the real component.

🔗

constexpr T imag() const;

Returns: The value of the imaginary component.

🔗

constexpr void imag(T val);

Effects: Assigns val to the imaginary component.

26.4.5 Member operators [complex.member.ops]

🔗

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

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

Returns: *this.

🔗

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

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

Returns: *this.

🔗

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

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

Returns: *this.

🔗

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

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

Returns: *this.

🔗

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

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

Returns: *this.

🔗

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

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

Returns: *this.

🔗

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

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

Returns: *this.

🔗

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

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

Returns: *this.

26.4.6 Non-member operations [complex.ops]

🔗

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

Returns: complex<T>(lhs).

🔗

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

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

🔗

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

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

🔗

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

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

🔗

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

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

🔗

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

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

🔗

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

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

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

🔗

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

Preconditions: The input values are convertible to T.

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

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

Returns: is.

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

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

🔗

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

Effects: Inserts the complex number x onto the stream o as if it were implemented as follows:

basic_ostringstream<charT, traits> s;
s.flags(o.flags());
s.imbue(o.getloc());
s.precision(o.precision());
s << '(' << x.real() << "," << x.imag() << ')';
return o << s.str();

[Note

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

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

— end note

]

26.4.7 Value operations [complex.value.ops]

🔗

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

Returns: x.real().

🔗

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

Returns: x.imag().

🔗

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

Returns: The magnitude of x.

🔗

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

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

🔗

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

Returns: The squared magnitude of x.

🔗

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

Returns: The complex conjugate of x.

🔗

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

Returns: The projection of x onto the Riemann sphere.

Remarks: Behaves the same as the C function cproj.

26.4.8 Transcendentals [complex.transcendentals]

🔗

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

Returns: The complex arc cosine of x.

Remarks: Behaves the same as the C function cacos.

26.4.9 Additional overloads [cmplx.over]

The following function templates shall have additional overloads:

arg                   norm
conj                  proj
imag                  real

where norm, conj, imag, and real are constexpr overloads.

The additional overloads shall be sufficient to ensure:

(2.1)
If the argument has type long double, then it is effectively cast to complex<long double>.
(2.2)
Otherwise, if the argument has type double or an integer type, then it is effectively cast to complex<double>.
(2.3)
Otherwise, if the argument has type float, then it is effectively cast to complex<float>.

Function template pow shall have additional overloads sufficient to ensure, for a call with at least one argument of type complex<T>:

(3.1)
If either argument has type complex<long double> or type long double, then both arguments are effectively cast to complex<long double>.
(3.2)
Otherwise, if either argument has type complex<double>, double, or an integer type, then both arguments are effectively cast to complex<double>.
(3.3)
Otherwise, if either argument has type complex<float> or float, then both arguments are effectively cast to complex<float>.

26.4.10 Suffixes for complex number literals [complex.literals]

This subclause describes literal suffixes for constructing complex number literals.

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

🔗

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

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

🔗

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

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

🔗

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

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

26.5 Bit manipulation [bit]

26.5.1 General [bit.general]

The header <bit> provides components to access, manipulate and process both individual bits and bit sequences.

26.5.2 Header <bit> synopsis [bit.syn]

namespace std {
  // [bit.cast], bit_cast
  template<class To, class From>
    constexpr To bit_cast(const From& from) noexcept;

  // [bit.pow.two], integral powers of 2
  template<class T>
    constexpr bool has_single_bit(T x) noexcept;
  template<class T>
    constexpr T bit_ceil(T x);
  template<class T>
    constexpr T bit_floor(T x) noexcept;
  template<class T>
    constexpr T bit_width(T x) noexcept;

  // [bit.rotate], rotating
  template<class T>
    [[nodiscard]] constexpr T rotl(T x, int s) noexcept;
  template<class T>
    [[nodiscard]] constexpr T rotr(T x, int s) noexcept;

  // [bit.count], counting
  template<class T>
    constexpr int countl_zero(T x) noexcept;
  template<class T>
    constexpr int countl_one(T x) noexcept;
  template<class T>
    constexpr int countr_zero(T x) noexcept;
  template<class T>
    constexpr int countr_one(T x) noexcept;
  template<class T>
    constexpr int popcount(T x) noexcept;

  // [bit.endian], endian
  enum class endian {
    little = see below,
    big    = see below,
    native = see below
  };
}

26.5.3 Function template bit_cast [bit.cast]

🔗

template<class To, class From>
  constexpr To bit_cast(const From& from) noexcept;

Constraints:

(1.1)
sizeof(To) == sizeof(From) is true;
(1.2)
is_trivially_copyable_v<To> is true; and
(1.3)
is_trivially_copyable_v<From> is true.

Returns: An object of type To.

Implicitly creates objects nested within the result ([intro.object]).

Each bit of the value representation of the result is equal to the corresponding bit in the object representation of from.

Padding bits of the result are unspecified.

For the result and each object created within it, if there is no value of the object's type corresponding to the value representation produced, the behavior is undefined.

If there are multiple such values, which value is produced is unspecified.

Remarks: This function is constexpr if and only if To, From, and the types of all subobjects of To and From are types T such that:

(3.1)
is_union_v<T> is false;
(3.2)
is_pointer_v<T> is false;
(3.3)
is_member_pointer_v<T> is false;
(3.4)
is_volatile_v<T> is false; and
(3.5)
T has no non-static data members of reference type.

26.5.4 Integral powers of 2 [bit.pow.two]

🔗

template<class T>
  constexpr bool has_single_bit(T x) noexcept;

Constraints: T is an unsigned integer type ([basic.fundamental]).

Returns: true if x is an integral power of two; false otherwise.

🔗

template<class T>
  constexpr T bit_ceil(T x);

Let N be the smallest power of 2 greater than or equal to x.

Constraints: T is an unsigned integer type ([basic.fundamental]).

Preconditions: N is representable as a value of type T.

Returns: N.

Throws: Nothing.

Remarks: A function call expression that violates the precondition in the Preconditions: element is not a core constant expression ([expr.const]).

🔗

template<class T>
  constexpr T bit_floor(T x) noexcept;

Constraints: T is an unsigned integer type ([basic.fundamental]).

Returns: If x == 0, 0; otherwise the maximal value y such that has_single_bit(y) is true and y <= x.

🔗

template<class T>
  constexpr T bit_width(T x) noexcept;

Constraints: T is an unsigned integer type ([basic.fundamental]).

Returns: If x == 0, 0; otherwise one plus the base-2 logarithm of x, with any fractional part discarded.

26.5.5 Rotating [bit.rotate]

In the following descriptions, let N denote numeric_limits<T>::digits.

🔗

template<class T>
  [[nodiscard]] constexpr T rotl(T x, int s) noexcept;

Constraints: T is an unsigned integer type ([basic.fundamental]).

Let r be s % N.

Returns: If r is 0, x; if r is positive, (x << r) | (x >> (N - r)); if r is negative, rotr(x, -r).

🔗

template<class T>
  [[nodiscard]] constexpr T rotr(T x, int s) noexcept;

Constraints: T is an unsigned integer type ([basic.fundamental]).

Let r be s % N.

Returns: If r is 0, x; if r is positive, (x >> r) | (x << (N - r)); if r is negative, rotl(x, -r).

26.5.6 Counting [bit.count]

In the following descriptions, let N denote numeric_limits<T>::digits.

🔗

template<class T>
  constexpr int countl_zero(T x) noexcept;

Constraints: T is an unsigned integer type ([basic.fundamental]).

Returns: The number of consecutive 0 bits in the value of x, starting from the most significant bit.

[Note

Returns N if x == 0.

— end note

]

🔗

template<class T>
  constexpr int countl_one(T x) noexcept;

Constraints: T is an unsigned integer type ([basic.fundamental]).

Returns: The number of consecutive 1 bits in the value of x, starting from the most significant bit.

[Note

Returns N if x == numeric_limits<T>::max().

— end note

]

🔗

template<class T>
  constexpr int countr_zero(T x) noexcept;

Constraints: T is an unsigned integer type ([basic.fundamental]).

Returns: The number of consecutive 0 bits in the value of x, starting from the least significant bit.

[Note

Returns N if x == 0.

— end note

]

🔗

template<class T>
  constexpr int countr_one(T x) noexcept;

Constraints: T is an unsigned integer type ([basic.fundamental]).

Returns: The number of consecutive 1 bits in the value of x, starting from the least significant bit.

[Note

Returns N if x == numeric_limits<T>::max().

— end note

]

🔗

template<class T>
  constexpr int popcount(T x) noexcept;

Constraints: T is an unsigned integer type ([basic.fundamental]).

Returns: The number of 1 bits in the value of x.

26.5.7 Endian [bit.endian]

Two common methods of byte ordering in multibyte scalar types are big-endian and little-endian in the execution environment.

Big-endian is a format for storage of binary data in which the most significant byte is placed first, with the rest in descending order.

Little-endian is a format for storage of binary data in which the least significant byte is placed first, with the rest in ascending order.

This subclause describes the endianness of the scalar types of the execution environment.

🔗

enum class endian {
  little = see below,
  big    = see below,
  native = see below
};

If all scalar types have size 1 byte, then all of endian::little, endian::big, and endian::native have the same value.

Otherwise, endian::little is not equal to endian::big.

If all scalar types are big-endian, endian::native is equal to endian::big.

If all scalar types are little-endian, endian::native is equal to endian::little.

Otherwise, endian::native is not equal to either endian::big or endian::little.

26.6 Random number generation [rand]

This subclause 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

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 via this subclause has an associated arithmetic type 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 this subclause use mathematical real numbers.

Throughout this subclause, 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 $2^{w}$ .

26.6.1 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;
}

26.6.2 Requirements [rand.req]

26.6.2.1 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.

26.6.2.2 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

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.	$O (r.size())$

26.6.2.3 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

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&>.

26.6.2.4 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)241		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) 242	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)

241)

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

⮥

242)

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().

⮥

26.6.2.5 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.

26.6.2.6 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;

26.6.3 Random number engine class templates [rand.eng]

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

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

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

Every function described in this subclause [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 this subclause [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 this subclause [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 this subclause ([rand.eng]) and in subclause [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.

26.6.3.1 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

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.

26.6.3.2 Class template mersenne_twister_engine [rand.eng.mers]

A mersenne_twister_engine random number engine243 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}

243)

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.

⮥

26.6.3.3 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

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.

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

26.6.4.1 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.

26.6.4.2 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.

26.6.4.3 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

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.

26.6.4.4 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().

26.6.5 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

The implementation may 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 may select different underlying engine types, code that uses this typedef need not generate identical sequences across implementations.

— end note

]

26.6.6 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);

Remarks: The semantics of the token parameter and the token value used by the default constructor are implementation-defined.244

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

🔗

double entropy() const noexcept;

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

Otherwise, returns an entropy estimate245 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.

244)

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

⮥

245)

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}

⮥

26.6.7 Utilities [rand.util]

26.6.7.1 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.

26.6.7.2 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 b246 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

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

— end note

]

Throws: What and when g throws.

[Note

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

]

246)

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

⮥

26.6.8 Random number distribution class templates [rand.dist]

26.6.8.1 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.

26.6.8.2 Uniform distributions [rand.dist.uni]

26.6.8.2.1 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;
  };

🔗

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.

🔗

result_type a() const;

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

🔗

result_type b() const;

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

26.6.8.2.2 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

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;
  };

🔗

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.

🔗

result_type a() const;

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

🔗

result_type b() const;

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

26.6.8.3 Bernoulli distributions [rand.dist.bern]

26.6.8.3.1 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;
};

🔗

explicit bernoulli_distribution(double p);

Preconditions:

0 \leq p \leq 1

Remarks: p corresponds to the parameter of the distribution.

🔗

double p() const;

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

26.6.8.3.2 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;
  };

🔗

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.

🔗

IntType t() const;

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

🔗

double p() const;

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

26.6.8.3.3 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;
  };

🔗

explicit geometric_distribution(double p);

Preconditions:

0 < p < 1

Remarks: p corresponds to the parameter of the distribution.

🔗

double p() const;

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

26.6.8.3.4 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

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;
  };

🔗

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

Preconditions:

0 < p \leq 1

and

0 < k

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

🔗

IntType k() const;

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

🔗

double p() const;

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

26.6.8.4 Poisson distributions [rand.dist.pois]

26.6.8.4.1 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;
  };

🔗

explicit poisson_distribution(double mean);

Preconditions:

0 < mean

Remarks: mean corresponds to the parameter of the distribution.

🔗

double mean() const;

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

26.6.8.4.2 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;
  };

🔗

explicit exponential_distribution(RealType lambda);

Preconditions:

0 < lambda

Remarks: lambda corresponds to the parameter of the distribution.

🔗

RealType lambda() const;

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

26.6.8.4.3 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;
  };

🔗

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.

🔗

RealType alpha() const;

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

🔗

RealType beta() const;

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

26.6.8.4.4 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;
  };

🔗

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.

🔗

RealType a() const;

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

🔗

RealType b() const;

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

26.6.8.4.5 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 function247

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;
  };

🔗

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.

🔗

RealType b() const;

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

247)

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.

⮥

26.6.8.5 Normal distributions [rand.dist.norm]

26.6.8.5.1 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;
  };

🔗

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

Preconditions:

0 < stddev

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

🔗

RealType mean() const;

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

🔗

RealType stddev() const;

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

26.6.8.5.2 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;
  };

🔗

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

Preconditions:

0 < s

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

🔗

RealType m() const;

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

🔗

RealType s() const;

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

26.6.8.5.3 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;
  };

🔗

explicit chi_squared_distribution(RealType n);

Preconditions:

0 < n

Remarks: n corresponds to the parameter of the distribution.

🔗

RealType n() const;

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

26.6.8.5.4 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;
  };

🔗

explicit cauchy_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.

🔗

RealType b() const;

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

26.6.8.5.5 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;
  };

🔗

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.

🔗

RealType m() const;

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

🔗

RealType n() const;

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

26.6.8.5.6 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;
  };

🔗

explicit student_t_distribution(RealType n);

Preconditions:

0 < n

Remarks: n corresponds to the parameter of the distribution.

🔗

RealType n() const;

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

26.6.8.6 Sampling distributions [rand.dist.samp]

26.6.8.6.1 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;
  };

🔗

discrete_distribution();

Effects: Constructs a discrete_distribution object with

n = 1

and

p_{0} = 1

[Note

Such an object will always deliver the value 0.

— end note

]

🔗

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.

🔗

discrete_distribution(initializer_list<double> wl);

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

🔗

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.

🔗

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

26.6.8.6.2 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

26.6.8.6.3 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

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

[Note

The header <cstdlib> ([cstdlib.syn]) 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

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

]

26.7 Numeric arrays [numarray]

26.7.1 Header <valarray> synopsis [valarray.syn]

#include <initializer_list>

namespace std {
  template<class T> class valarray;         // An array of type T
  class slice;                              // a BLAS-like slice out of an array
  template<class T> class slice_array;
  class gslice;                             // a generalized slice out of an array
  template<class T> class gslice_array;
  template<class T> class mask_array;       // a masked array
  template<class T> class indirect_array;   // an indirected array

  template<class T> void swap(valarray<T>&, valarray<T>&) noexcept;

  template<class T> valarray<T> operator* (const valarray<T>&, const valarray<T>&);
  template<class T> valarray<T> operator* (const valarray<T>&,
                                           const typename valarray<T>::value_type&);
  template<class T> valarray<T> operator* (const typename valarray<T>::value_type&,
                                           const valarray<T>&);

  template<class T> valarray<T> operator/ (const valarray<T>&, const valarray<T>&);
  template<class T> valarray<T> operator/ (const valarray<T>&,
                                           const typename valarray<T>::value_type&);
  template<class T> valarray<T> operator/ (const typename valarray<T>::value_type&,
                                           const valarray<T>&);

  template<class T> valarray<T> operator% (const valarray<T>&, const valarray<T>&);
  template<class T> valarray<T> operator% (const valarray<T>&,
                                           const typename valarray<T>::value_type&);
  template<class T> valarray<T> operator% (const typename valarray<T>::value_type&,
                                           const valarray<T>&);

  template<class T> valarray<T> operator+ (const valarray<T>&, const valarray<T>&);
  template<class T> valarray<T> operator+ (const valarray<T>&,
                                           const typename valarray<T>::value_type&);
  template<class T> valarray<T> operator+ (const typename valarray<T>::value_type&,
                                           const valarray<T>&);

  template<class T> valarray<T> operator- (const valarray<T>&, const valarray<T>&);
  template<class T> valarray<T> operator- (const valarray<T>&,
                                           const typename valarray<T>::value_type&);
  template<class T> valarray<T> operator- (const typename valarray<T>::value_type&,
                                           const valarray<T>&);

  template<class T> valarray<T> operator^ (const valarray<T>&, const valarray<T>&);
  template<class T> valarray<T> operator^ (const valarray<T>&,
                                           const typename valarray<T>::value_type&);
  template<class T> valarray<T> operator^ (const typename valarray<T>::value_type&,
                                           const valarray<T>&);

  template<class T> valarray<T> operator& (const valarray<T>&, const valarray<T>&);
  template<class T> valarray<T> operator& (const valarray<T>&,
                                           const typename valarray<T>::value_type&);
  template<class T> valarray<T> operator& (const typename valarray<T>::value_type&,
                                           const valarray<T>&);

  template<class T> valarray<T> operator| (const valarray<T>&, const valarray<T>&);
  template<class T> valarray<T> operator| (const valarray<T>&,
                                           const typename valarray<T>::value_type&);
  template<class T> valarray<T> operator| (const typename valarray<T>::value_type&,
                                           const valarray<T>&);

  template<class T> valarray<T> operator<<(const valarray<T>&, const valarray<T>&);
  template<class T> valarray<T> operator<<(const valarray<T>&,
                                           const typename valarray<T>::value_type&);
  template<class T> valarray<T> operator<<(const typename valarray<T>::value_type&,
                                           const valarray<T>&);

  template<class T> valarray<T> operator>>(const valarray<T>&, const valarray<T>&);
  template<class T> valarray<T> operator>>(const valarray<T>&,
                                           const typename valarray<T>::value_type&);
  template<class T> valarray<T> operator>>(const typename valarray<T>::value_type&,
                                           const valarray<T>&);

  template<class T> valarray<bool> operator&&(const valarray<T>&, const valarray<T>&);
  template<class T> valarray<bool> operator&&(const valarray<T>&,
                                              const typename valarray<T>::value_type&);
  template<class T> valarray<bool> operator&&(const typename valarray<T>::value_type&,
                                              const valarray<T>&);

  template<class T> valarray<bool> operator||(const valarray<T>&, const valarray<T>&);
  template<class T> valarray<bool> operator||(const valarray<T>&,
                                              const typename valarray<T>::value_type&);
  template<class T> valarray<bool> operator||(const typename valarray<T>::value_type&,
                                              const valarray<T>&);

  template<class T> valarray<bool> operator==(const valarray<T>&, const valarray<T>&);
  template<class T> valarray<bool> operator==(const valarray<T>&,
                                              const typename valarray<T>::value_type&);
  template<class T> valarray<bool> operator==(const typename valarray<T>::value_type&,
                                              const valarray<T>&);
  template<class T> valarray<bool> operator!=(const valarray<T>&, const valarray<T>&);
  template<class T> valarray<bool> operator!=(const valarray<T>&,
                                              const typename valarray<T>::value_type&);
  template<class T> valarray<bool> operator!=(const typename valarray<T>::value_type&,
                                              const valarray<T>&);

  template<class T> valarray<bool> operator< (const valarray<T>&, const valarray<T>&);
  template<class T> valarray<bool> operator< (const valarray<T>&,
                                              const typename valarray<T>::value_type&);
  template<class T> valarray<bool> operator< (const typename valarray<T>::value_type&,
                                              const valarray<T>&);
  template<class T> valarray<bool> operator> (const valarray<T>&, const valarray<T>&);
  template<class T> valarray<bool> operator> (const valarray<T>&,
                                              const typename valarray<T>::value_type&);
  template<class T> valarray<bool> operator> (const typename valarray<T>::value_type&,
                                              const valarray<T>&);
  template<class T> valarray<bool> operator<=(const valarray<T>&, const valarray<T>&);
  template<class T> valarray<bool> operator<=(const valarray<T>&,
                                              const typename valarray<T>::value_type&);
  template<class T> valarray<bool> operator<=(const typename valarray<T>::value_type&,
                                              const valarray<T>&);
  template<class T> valarray<bool> operator>=(const valarray<T>&, const valarray<T>&);
  template<class T> valarray<bool> operator>=(const valarray<T>&,
                                              const typename valarray<T>::value_type&);
  template<class T> valarray<bool> operator>=(const typename valarray<T>::value_type&,
                                              const valarray<T>&);

  template<class T> valarray<T> abs  (const valarray<T>&);
  template<class T> valarray<T> acos (const valarray<T>&);
  template<class T> valarray<T> asin (const valarray<T>&);
  template<class T> valarray<T> atan (const valarray<T>&);

  template<class T> valarray<T> atan2(const valarray<T>&, const valarray<T>&);
  template<class T> valarray<T> atan2(const valarray<T>&,
                                      const typename valarray<T>::value_type&);
  template<class T> valarray<T> atan2(const typename valarray<T>::value_type&,
                                      const valarray<T>&);

  template<class T> valarray<T> cos  (const valarray<T>&);
  template<class T> valarray<T> cosh (const valarray<T>&);
  template<class T> valarray<T> exp  (const valarray<T>&);
  template<class T> valarray<T> log  (const valarray<T>&);
  template<class T> valarray<T> log10(const valarray<T>&);

  template<class T> valarray<T> pow(const valarray<T>&, const valarray<T>&);
  template<class T> valarray<T> pow(const valarray<T>&, const typename valarray<T>::value_type&);
  template<class T> valarray<T> pow(const typename valarray<T>::value_type&, const valarray<T>&);

  template<class T> valarray<T> sin  (const valarray<T>&);
  template<class T> valarray<T> sinh (const valarray<T>&);
  template<class T> valarray<T> sqrt (const valarray<T>&);
  template<class T> valarray<T> tan  (const valarray<T>&);
  template<class T> valarray<T> tanh (const valarray<T>&);

  template<class T> unspecified1 begin(valarray<T>& v);
  template<class T> unspecified2 begin(const valarray<T>& v);
  template<class T> unspecified1 end(valarray<T>& v);
  template<class T> unspecified2 end(const valarray<T>& v);
}

The header <valarray> defines five class templates (valarray, slice_array, gslice_array, mask_array, and indirect_array), two classes (slice and gslice), and a series of related function templates for representing and manipulating arrays of values.

The valarray array classes are defined to be free of certain forms of aliasing, thus allowing operations on these classes to be optimized.

Any function returning a valarray<T> is permitted to return an object of another type, provided all the const member functions of valarray<T> are also applicable to this type.

This return type shall not add more than two levels of template nesting over the most deeply nested argument type.248

Implementations introducing such replacement types shall provide additional functions and operators as follows:

(4.1)
for every function taking a const valarray<T>& other than begin and end, identical functions taking the replacement types shall be added;
(4.2)
for every function taking two const valarray<T>& arguments, identical functions taking every combination of const valarray<T>& and replacement types shall be added.

In particular, an implementation shall allow a valarray<T> to be constructed from such replacement types and shall allow assignments and compound assignments of such types to valarray<T>, slice_array<T>, gslice_array<T>, mask_array<T> and indirect_array<T> objects.

These library functions are permitted to throw a bad_alloc exception if there are not sufficient resources available to carry out the operation.

Note that the exception is not mandated.

248)

[implimits] recommends a minimum number of recursively nested template instantiations.

This requirement thus indirectly suggests a minimum allowable complexity for valarray expressions.

⮥

26.7.2 Class template valarray [template.valarray]

26.7.2.1 Overview [template.valarray.overview]

namespace std {
  template<class T> class valarray {
  public:
    using value_type = T;

    // [valarray.cons], construct/destroy
    valarray();
    explicit valarray(size_t);
    valarray(const T&, size_t);
    valarray(const T*, size_t);
    valarray(const valarray&);
    valarray(valarray&&) noexcept;
    valarray(const slice_array<T>&);
    valarray(const gslice_array<T>&);
    valarray(const mask_array<T>&);
    valarray(const indirect_array<T>&);
    valarray(initializer_list<T>);
    ~valarray();

    // [valarray.assign], assignment
    valarray& operator=(const valarray&);
    valarray& operator=(valarray&&) noexcept;
    valarray& operator=(initializer_list<T>);
    valarray& operator=(const T&);
    valarray& operator=(const slice_array<T>&);
    valarray& operator=(const gslice_array<T>&);
    valarray& operator=(const mask_array<T>&);
    valarray& operator=(const indirect_array<T>&);

    // [valarray.access], element access
    const T&          operator[](size_t) const;
    T&                operator[](size_t);

    // [valarray.sub], subset operations
    valarray          operator[](slice) const;
    slice_array<T>    operator[](slice);
    valarray          operator[](const gslice&) const;
    gslice_array<T>   operator[](const gslice&);
    valarray          operator[](const valarray<bool>&) const;
    mask_array<T>     operator[](const valarray<bool>&);
    valarray          operator[](const valarray<size_t>&) const;
    indirect_array<T> operator[](const valarray<size_t>&);

    // [valarray.unary], unary operators
    valarray operator+() const;
    valarray operator-() const;
    valarray operator~() const;
    valarray<bool> operator!() const;

    // [valarray.cassign], compound assignment
    valarray& operator*= (const T&);
    valarray& operator/= (const T&);
    valarray& operator%= (const T&);
    valarray& operator+= (const T&);
    valarray& operator-= (const T&);
    valarray& operator^= (const T&);
    valarray& operator&= (const T&);
    valarray& operator|= (const T&);
    valarray& operator<<=(const T&);
    valarray& operator>>=(const T&);

    valarray& operator*= (const valarray&);
    valarray& operator/= (const valarray&);
    valarray& operator%= (const valarray&);
    valarray& operator+= (const valarray&);
    valarray& operator-= (const valarray&);
    valarray& operator^= (const valarray&);
    valarray& operator|= (const valarray&);
    valarray& operator&= (const valarray&);
    valarray& operator<<=(const valarray&);
    valarray& operator>>=(const valarray&);

    // [valarray.members], member functions
    void swap(valarray&) noexcept;

    size_t size() const;

    T sum() const;
    T min() const;
    T max() const;

    valarray shift (int) const;
    valarray cshift(int) const;
    valarray apply(T func(T)) const;
    valarray apply(T func(const T&)) const;
    void resize(size_t sz, T c = T());
  };

  template<class T, size_t cnt> valarray(const T(&)[cnt], size_t) -> valarray<T>;
}

The class template valarray<T> is a one-dimensional smart array, with elements numbered sequentially from zero.

It is a representation of the mathematical concept of an ordered set of values.

For convenience, an object of type valarray<T> is referred to as an “array” throughout the remainder of [numarray].

The illusion of higher dimensionality may be produced by the familiar idiom of computed indices, together with the powerful subsetting capabilities provided by the generalized subscript operators.249

249)

The intent is to specify an array template that has the minimum functionality necessary to address aliasing ambiguities and the proliferation of temporary objects.

Thus, the valarray template is neither a matrix class nor a field class.

However, it is a very useful building block for designing such classes.

⮥

26.7.2.2 Constructors [valarray.cons]

🔗

valarray();

Effects: Constructs a valarray that has zero length.250

🔗

explicit valarray(size_t n);

Effects: Constructs a valarray that has length n.

Each element of the array is value-initialized .

🔗

valarray(const T& v, size_t n);

Effects: Constructs a valarray that has length n.

Each element of the array is initialized with v.

🔗

valarray(const T* p, size_t n);

Preconditions: [p, p + n) is a valid range.

Effects: Constructs a valarray that has length n.

The values of the elements of the array are initialized with the first n values pointed to by the first argument.251

🔗

valarray(const valarray& v);

Effects: Constructs a valarray that has the same length as v.

The elements are initialized with the values of the corresponding elements of v.252

🔗

valarray(valarray&& v) noexcept;

Effects: Constructs a valarray that has the same length as v.

The elements are initialized with the values of the corresponding elements of v.

Complexity: Constant.

🔗

valarray(initializer_list<T> il);

Effects: Equivalent to valarray(il.begin(), il.size()).

🔗

valarray(const slice_array<T>&);
valarray(const gslice_array<T>&);
valarray(const mask_array<T>&);
valarray(const indirect_array<T>&);

These conversion constructors convert one of the four reference templates to a valarray.

🔗

~valarray();

Effects: The destructor is applied to every element of *this; an implementation may return all allocated memory.

250)

This default constructor is essential, since arrays of valarray may be useful.

After initialization, the length of an empty array can be increased with the resize member function.

⮥

251)

This constructor is the preferred method for converting a C array to a valarray object.

⮥

252)

This copy constructor creates a distinct array rather than an alias.

Implementations in which arrays share storage are permitted, but they would need to implement a copy-on-reference mechanism to ensure that arrays are conceptually distinct.

⮥

26.7.2.3 Assignment [valarray.assign]

🔗

valarray& operator=(const valarray& v);

Effects: Each element of the *this array is assigned the value of the corresponding element of v.

If the length of v is not equal to the length of *this, resizes *this to make the two arrays the same length, as if by calling resize(v.size()), before performing the assignment.

Postconditions: size() == v.size().

Returns: *this.

🔗

valarray& operator=(valarray&& v) noexcept;

Effects: *this obtains the value of v.

The value of v after the assignment is not specified.

Returns: *this.

Complexity: Linear.

🔗

valarray& operator=(initializer_list<T> il);

Effects: Equivalent to: return *this = valarray(il);

🔗

valarray& operator=(const T& v);

Effects: Assigns v to each element of *this.

Returns: *this.

🔗

valarray& operator=(const slice_array<T>&);
valarray& operator=(const gslice_array<T>&);
valarray& operator=(const mask_array<T>&);
valarray& operator=(const indirect_array<T>&);

Preconditions: The length of the array to which the argument refers equals size().

The value of an element in the left-hand side of a valarray assignment operator does not depend on the value of another element in that left-hand side.

These operators allow the results of a generalized subscripting operation to be assigned directly to a valarray.

26.7.2.4 Element access [valarray.access]

🔗

const T&  operator[](size_t n) const;
T& operator[](size_t n);

Preconditions: n < size() is true.

Returns: A reference to the corresponding element of the array.

[Note

The expression (a[i] = q, a[i]) == q evaluates to true for any non-constant valarray<T> a, any T q, and for any size_t i such that the value of i is less than the length of a.

— end note

]

Remarks: The expression addressof(a[i+j]) == addressof(a[i]) + j evaluates to true for all size_t i and size_t j such that i+j < a.size().

The expression addressof(a[i]) != addressof(b[j]) evaluates to true for any two arrays a and b and for any size_t i and size_t j such that i < a.size() and j < b.size().

[Note

This property indicates an absence of aliasing and may be used to advantage by optimizing compilers.

Compilers may take advantage of inlining, constant propagation, loop fusion, tracking of pointers obtained from operator new, and other techniques to generate efficient valarrays.

— end note

]

The reference returned by the subscript operator for an array shall be valid until the member function resize(size_t, T) is called for that array or until the lifetime of that array ends, whichever happens first.

26.7.2.5 Subset operations [valarray.sub]

The member operator[] is overloaded to provide several ways to select sequences of elements from among those controlled by *this.

Each of these operations returns a subset of the array.

The const-qualified versions return this subset as a new valarray object.

The non-const versions return a class template object which has reference semantics to the original array, working in conjunction with various overloads of operator= and other assigning operators to allow selective replacement (slicing) of the controlled sequence.

In each case the selected element(s) shall exist.

🔗

valarray operator[](slice slicearr) const;

Returns: A valarray containing those elements of the controlled sequence designated by slicearr.

[Example

const valarray<char> v0("abcdefghijklmnop", 16);
// v0[slice(2, 5, 3)] returns valarray<char>("cfilo", 5)

— end example

]

🔗

slice_array<T> operator[](slice slicearr);

Returns: An object that holds references to elements of the controlled sequence selected by slicearr.

[Example

valarray<char> v0("abcdefghijklmnop", 16);
valarray<char> v1("ABCDE", 5);
v0[slice(2, 5, 3)] = v1;
// v0 == valarray<char>("abAdeBghCjkDmnEp", 16);

— end example

]

🔗

valarray operator[](const gslice& gslicearr) const;

Returns: A valarray containing those elements of the controlled sequence designated by gslicearr.

[Example

const valarray<char> v0("abcdefghijklmnop", 16);
const size_t lv[] = { 2, 3 };
const size_t dv[] = { 7, 2 };
const valarray<size_t> len(lv, 2), str(dv, 2);
// v0[gslice(3, len, str)] returns
// valarray<char>("dfhkmo", 6)

— end example

]

🔗

gslice_array<T> operator[](const gslice& gslicearr);

Returns: An object that holds references to elements of the controlled sequence selected by gslicearr.

[Example

valarray<char> v0("abcdefghijklmnop", 16);
valarray<char> v1("ABCDEF", 6);
const size_t lv[] = { 2, 3 };
const size_t dv[] = { 7, 2 };
const valarray<size_t> len(lv, 2), str(dv, 2);
v0[gslice(3, len, str)] = v1;
// v0 == valarray<char>("abcAeBgCijDlEnFp", 16)

— end example

]

🔗

valarray operator[](const valarray<bool>& boolarr) const;

Returns: A valarray containing those elements of the controlled sequence designated by boolarr.

[Example

const valarray<char> v0("abcdefghijklmnop", 16);
const bool vb[] = { false, false, true, true, false, true };
// v0[valarray<bool>(vb, 6)] returns
// valarray<char>("cdf", 3)

— end example

]

🔗

mask_array<T> operator[](const valarray<bool>& boolarr);

Returns: An object that holds references to elements of the controlled sequence selected by boolarr.

[Example

valarray<char> v0("abcdefghijklmnop", 16);
valarray<char> v1("ABC", 3);
const bool vb[] = { false, false, true, true, false, true };
v0[valarray<bool>(vb, 6)] = v1;
// v0 == valarray<char>("abABeCghijklmnop", 16)

— end example

]

🔗

valarray operator[](const valarray<size_t>& indarr) const;

Returns: A valarray containing those elements of the controlled sequence designated by indarr.

[Example

const valarray<char> v0("abcdefghijklmnop", 16);
const size_t vi[] = { 7, 5, 2, 3, 8 };
// v0[valarray<size_t>(vi, 5)] returns
// valarray<char>("hfcdi", 5)

— end example

]

🔗

indirect_array<T> operator[](const valarray<size_t>& indarr);

Returns: An object that holds references to elements of the controlled sequence selected by indarr.

[Example

valarray<char> v0("abcdefghijklmnop", 16);
valarray<char> v1("ABCDE", 5);
const size_t vi[] = { 7, 5, 2, 3, 8 };
v0[valarray<size_t>(vi, 5)] = v1;
// v0 == valarray<char>("abCDeBgAEjklmnop", 16)

— end example

]

26.7.2.6 Unary operators [valarray.unary]

🔗

valarray operator+() const;
valarray operator-() const;
valarray operator~() const;
valarray<bool> operator!() const;

Mandates: The indicated operator can be applied to operands of type T and returns a value of type T (bool for operator!) or which may be unambiguously implicitly converted to type T (bool for operator!).

Returns: A valarray whose length is size().

Each element of the returned array is initialized with the result of applying the indicated operator to the corresponding element of the array.

26.7.2.7 Compound assignment [valarray.cassign]

🔗

valarray& operator*= (const valarray& v);
valarray& operator/= (const valarray& v);
valarray& operator%= (const valarray& v);
valarray& operator+= (const valarray& v);
valarray& operator-= (const valarray& v);
valarray& operator^= (const valarray& v);
valarray& operator&= (const valarray& v);
valarray& operator|= (const valarray& v);
valarray& operator<<=(const valarray& v);
valarray& operator>>=(const valarray& v);

Mandates: The indicated operator can be applied to two operands of type T.

Preconditions: size() == v.size() is true.

The value of an element in the left-hand side of a valarray compound assignment operator does not depend on the value of another element in that left hand side.

Effects: Each of these operators performs the indicated operation on each of the elements of *this and the corresponding element of v.

Returns: *this.

Remarks: The appearance of an array on the left-hand side of a compound assignment does not invalidate references or pointers.

🔗

valarray& operator*= (const T& v);
valarray& operator/= (const T& v);
valarray& operator%= (const T& v);
valarray& operator+= (const T& v);
valarray& operator-= (const T& v);
valarray& operator^= (const T& v);
valarray& operator&= (const T& v);
valarray& operator|= (const T& v);
valarray& operator<<=(const T& v);
valarray& operator>>=(const T& v);

Mandates: The indicated operator can be applied to two operands of type T.

Effects: Each of these operators applies the indicated operation to each element of *this and v.

Returns: *this

Remarks: The appearance of an array on the left-hand side of a compound assignment does not invalidate references or pointers to the elements of the array.

26.7.2.8 Member functions [valarray.members]

🔗

void swap(valarray& v) noexcept;

Effects: *this obtains the value of v.

v obtains the value of *this.

Complexity: Constant.

🔗

size_t size() const;

Returns: The number of elements in the array.

Complexity: Constant time.

🔗

T sum() const;

Mandates: operator+= can be applied to operands of type T.

Preconditions: size() > 0 is true.

Returns: The sum of all the elements of the array.

If the array has length 1, returns the value of element 0.

Otherwise, the returned value is calculated by applying operator+= to a copy of an element of the array and all other elements of the array in an unspecified order.

🔗

T min() const;

Preconditions: size() > 0 is true.

Returns: The minimum value contained in *this.

For an array of length 1, the value of element 0 is returned.

For all other array lengths, the determination is made using operator<.

🔗

T max() const;

Preconditions: size() > 0 is true.

Returns: The maximum value contained in *this.

For an array of length 1, the value of element 0 is returned.

For all other array lengths, the determination is made using operator<.

🔗

valarray shift(int n) const;

Returns: A valarray of length size(), each of whose elements I is (*this)[I + n] if I + n is non-negative and less than size(), otherwise T().

[Note

If element zero is taken as the leftmost element, a positive value of n shifts the elements left n places, with zero fill.

— end note

]

[Example

If the argument has the value -2, the first two elements of the result will be value-initialized; the third element of the result will be assigned the value of the first element of the argument; etc.

— end example

]

🔗

valarray cshift(int n) const;

Returns: A valarray of length size() that is a circular shift of *this.

If element zero is taken as the leftmost element, a non-negative value of n shifts the elements circularly left n places and a negative value of n shifts the elements circularly right

- n

places.

🔗

valarray apply(T func(T)) const;
valarray apply(T func(const T&)) const;

Returns: A valarray whose length is size().

Each element of the returned array is assigned the value returned by applying the argument function to the corresponding element of *this.

🔗

void resize(size_t sz, T c = T());

Effects: Changes the length of the *this array to sz and then assigns to each element the value of the second argument.

Resizing invalidates all pointers and references to elements in the array.

26.7.3 valarray non-member operations [valarray.nonmembers]

26.7.3.1 Binary operators [valarray.binary]

🔗

template<class T> valarray<T> operator* (const valarray<T>&, const valarray<T>&);
template<class T> valarray<T> operator/ (const valarray<T>&, const valarray<T>&);
template<class T> valarray<T> operator% (const valarray<T>&, const valarray<T>&);
template<class T> valarray<T> operator+ (const valarray<T>&, const valarray<T>&);
template<class T> valarray<T> operator- (const valarray<T>&, const valarray<T>&);
template<class T> valarray<T> operator^ (const valarray<T>&, const valarray<T>&);
template<class T> valarray<T> operator& (const valarray<T>&, const valarray<T>&);
template<class T> valarray<T> operator| (const valarray<T>&, const valarray<T>&);
template<class T> valarray<T> operator<<(const valarray<T>&, const valarray<T>&);
template<class T> valarray<T> operator>>(const valarray<T>&, const valarray<T>&);

Mandates: The indicated operator can be applied to operands of type T and returns a value of type T or which can be unambiguously implicitly converted to T.

Preconditions: The argument arrays have the same length.

Returns: A valarray whose length is equal to the lengths of the argument arrays.

Each element of the returned array is initialized with the result of applying the indicated operator to the corresponding elements of the argument arrays.

🔗

template<class T> valarray<T> operator* (const valarray<T>&,
                                         const typename valarray<T>::value_type&);
template<class T> valarray<T> operator* (const typename valarray<T>::value_type&,
                                         const valarray<T>&);
template<class T> valarray<T> operator/ (const valarray<T>&,
                                         const typename valarray<T>::value_type&);
template<class T> valarray<T> operator/ (const typename valarray<T>::value_type&,
                                         const valarray<T>&);
template<class T> valarray<T> operator% (const valarray<T>&,
                                         const typename valarray<T>::value_type&);
template<class T> valarray<T> operator% (const typename valarray<T>::value_type&,
                                         const valarray<T>&);
template<class T> valarray<T> operator+ (const valarray<T>&,
                                         const typename valarray<T>::value_type&);
template<class T> valarray<T> operator+ (const typename valarray<T>::value_type&,
                                         const valarray<T>&);
template<class T> valarray<T> operator- (const valarray<T>&,
                                         const typename valarray<T>::value_type&);
template<class T> valarray<T> operator- (const typename valarray<T>::value_type&,
                                         const valarray<T>&);
template<class T> valarray<T> operator^ (const valarray<T>&,
                                         const typename valarray<T>::value_type&);
template<class T> valarray<T> operator^ (const typename valarray<T>::value_type&,
                                         const valarray<T>&);
template<class T> valarray<T> operator& (const valarray<T>&,
                                         const typename valarray<T>::value_type&);
template<class T> valarray<T> operator& (const typename valarray<T>::value_type&,
                                         const valarray<T>&);
template<class T> valarray<T> operator| (const valarray<T>&,
                                         const typename valarray<T>::value_type&);
template<class T> valarray<T> operator| (const typename valarray<T>::value_type&,
                                         const valarray<T>&);
template<class T> valarray<T> operator<<(const valarray<T>&,
                                         const typename valarray<T>::value_type&);
template<class T> valarray<T> operator<<(const typename valarray<T>::value_type&,
                                         const valarray<T>&);
template<class T> valarray<T> operator>>(const valarray<T>&,
                                         const typename valarray<T>::value_type&);
template<class T> valarray<T> operator>>(const typename valarray<T>::value_type&,
                                         const valarray<T>&);

Mandates: The indicated operator can be applied to operands of type T and returns a value of type T or which can be unambiguously implicitly converted to T.

Returns: A valarray whose length is equal to the length of the array argument.

Each element of the returned array is initialized with the result of applying the indicated operator to the corresponding element of the array argument and the non-array argument.

26.7.3.2 Logical operators [valarray.comparison]

🔗

template<class T> valarray<bool> operator==(const valarray<T>&, const valarray<T>&);
template<class T> valarray<bool> operator!=(const valarray<T>&, const valarray<T>&);
template<class T> valarray<bool> operator< (const valarray<T>&, const valarray<T>&);
template<class T> valarray<bool> operator> (const valarray<T>&, const valarray<T>&);
template<class T> valarray<bool> operator<=(const valarray<T>&, const valarray<T>&);
template<class T> valarray<bool> operator>=(const valarray<T>&, const valarray<T>&);
template<class T> valarray<bool> operator&&(const valarray<T>&, const valarray<T>&);
template<class T> valarray<bool> operator||(const valarray<T>&, const valarray<T>&);

Mandates: The indicated operator can be applied to operands of type T and returns a value of type bool or which can be unambiguously implicitly converted to bool.

Preconditions: The two array arguments have the same length.

Returns: A valarray<bool> whose length is equal to the length of the array arguments.

Each element of the returned array is initialized with the result of applying the indicated operator to the corresponding elements of the argument arrays.

🔗

template<class T> valarray<bool> operator==(const valarray<T>&,
                                            const typename valarray<T>::value_type&);
template<class T> valarray<bool> operator==(const typename valarray<T>::value_type&,
                                            const valarray<T>&);
template<class T> valarray<bool> operator!=(const valarray<T>&,
                                            const typename valarray<T>::value_type&);
template<class T> valarray<bool> operator!=(const typename valarray<T>::value_type&,
                                            const valarray<T>&);
template<class T> valarray<bool> operator< (const valarray<T>&,
                                            const typename valarray<T>::value_type&);
template<class T> valarray<bool> operator< (const typename valarray<T>::value_type&,
                                            const valarray<T>&);
template<class T> valarray<bool> operator> (const valarray<T>&,
                                            const typename valarray<T>::value_type&);
template<class T> valarray<bool> operator> (const typename valarray<T>::value_type&,
                                            const valarray<T>&);
template<class T> valarray<bool> operator<=(const valarray<T>&,
                                            const typename valarray<T>::value_type&);
template<class T> valarray<bool> operator<=(const typename valarray<T>::value_type&,
                                            const valarray<T>&);
template<class T> valarray<bool> operator>=(const valarray<T>&,
                                            const typename valarray<T>::value_type&);
template<class T> valarray<bool> operator>=(const typename valarray<T>::value_type&,
                                            const valarray<T>&);
template<class T> valarray<bool> operator&&(const valarray<T>&,
                                            const typename valarray<T>::value_type&);
template<class T> valarray<bool> operator&&(const typename valarray<T>::value_type&,
                                            const valarray<T>&);
template<class T> valarray<bool> operator||(const valarray<T>&,
                                            const typename valarray<T>::value_type&);
template<class T> valarray<bool> operator||(const typename valarray<T>::value_type&,
                                            const valarray<T>&);

Mandates: The indicated operator can be applied to operands of type T and returns a value of type bool or which can be unambiguously implicitly converted to bool.

Returns: A valarray<bool> whose length is equal to the length of the array argument.

Each element of the returned array is initialized with the result of applying the indicated operator to the corresponding element of the array and the non-array argument.

26.7.3.3 Transcendentals [valarray.transcend]

🔗

template<class T> valarray<T> abs  (const valarray<T>&);
template<class T> valarray<T> acos (const valarray<T>&);
template<class T> valarray<T> asin (const valarray<T>&);
template<class T> valarray<T> atan (const valarray<T>&);
template<class T> valarray<T> atan2(const valarray<T>&, const valarray<T>&);
template<class T> valarray<T> atan2(const valarray<T>&, const typename valarray<T>::value_type&);
template<class T> valarray<T> atan2(const typename valarray<T>::value_type&, const valarray<T>&);
template<class T> valarray<T> cos  (const valarray<T>&);
template<class T> valarray<T> cosh (const valarray<T>&);
template<class T> valarray<T> exp  (const valarray<T>&);
template<class T> valarray<T> log  (const valarray<T>&);
template<class T> valarray<T> log10(const valarray<T>&);
template<class T> valarray<T> pow  (const valarray<T>&, const valarray<T>&);
template<class T> valarray<T> pow  (const valarray<T>&, const typename valarray<T>::value_type&);
template<class T> valarray<T> pow  (const typename valarray<T>::value_type&, const valarray<T>&);
template<class T> valarray<T> sin  (const valarray<T>&);
template<class T> valarray<T> sinh (const valarray<T>&);
template<class T> valarray<T> sqrt (const valarray<T>&);
template<class T> valarray<T> tan  (const valarray<T>&);
template<class T> valarray<T> tanh (const valarray<T>&);

Mandates: A unique function with the indicated name can be applied (unqualified) to an operand of type T.

This function returns a value of type T or which can be unambiguously implicitly converted to type T.

26.7.3.4 Specialized algorithms [valarray.special]

🔗

template<class T> void swap(valarray<T>& x, valarray<T>& y) noexcept;

Effects: Equivalent to x.swap(y).

26.7.4 Class slice [class.slice]

26.7.4.1 Overview [class.slice.overview]

namespace std {
  class slice {
  public:
    slice();
    slice(size_t, size_t, size_t);

    size_t start() const;
    size_t size() const;
    size_t stride() const;

    friend bool operator==(const slice& x, const slice& y);
  };
}

The slice class represents a BLAS-like slice from an array.

Such a slice is specified by a starting index, a length, and a stride.253

253)

BLAS stands for Basic Linear Algebra Subprograms. C++ programs may instantiate this class.

See, for example, Dongarra, Du Croz, Duff, and Hammerling: A set of Level 3 Basic Linear Algebra Subprograms; Technical Report MCS-P1-0888, Argonne National Laboratory (USA), Mathematics and Computer Science Division, August, 1988.

⮥

26.7.4.2 Constructors [cons.slice]

🔗

slice();
slice(size_t start, size_t length, size_t stride);
slice(const slice&);

The default constructor is equivalent to slice(0, 0, 0).

A default constructor is provided only to permit the declaration of arrays of slices.

The constructor with arguments for a slice takes a start, length, and stride parameter.

[Example

slice(3, 8, 2) constructs a slice which selects elements

3, 5, 7, \dots, 17

from an array.

— end example

]

26.7.4.3 Access functions [slice.access]

🔗

size_t start() const;
size_t size() const;
size_t stride() const;

Returns: The start, length, or stride specified by a slice object.

Complexity: Constant time.

26.7.4.4 Operators [slice.ops]

🔗

friend bool operator==(const slice& x, const slice& y);

Effects: Equivalent to:

return x.start() == y.start() && x.size() == y.size() && x.stride() == y.stride();

26.7.5 Class template slice_array [template.slice.array]

26.7.5.1 Overview [template.slice.array.overview]

namespace std {
  template<class T> class slice_array {
  public:
    using value_type = T;

    void operator=  (const valarray<T>&) const;
    void operator*= (const valarray<T>&) const;
    void operator/= (const valarray<T>&) const;
    void operator%= (const valarray<T>&) const;
    void operator+= (const valarray<T>&) const;
    void operator-= (const valarray<T>&) const;
    void operator^= (const valarray<T>&) const;
    void operator&= (const valarray<T>&) const;
    void operator|= (const valarray<T>&) const;
    void operator<<=(const valarray<T>&) const;
    void operator>>=(const valarray<T>&) const;

    slice_array(const slice_array&);
    ~slice_array();
    const slice_array& operator=(const slice_array&) const;
    void operator=(const T&) const;

    slice_array() = delete;     // as implied by declaring copy constructor above
  };
}

This template is a helper template used by the slice subscript operator

slice_array<T> valarray<T>::operator[](slice);

It has reference semantics to a subset of an array specified by a slice object.

[Example

The expression a[slice(1, 5, 3)] = b; has the effect of assigning the elements of b to a slice of the elements in a.

For the slice shown, the elements selected from a are

1, 4, \dots, 13

— end example

]

26.7.5.2 Assignment [slice.arr.assign]

🔗

void operator=(const valarray<T>&) const;
const slice_array& operator=(const slice_array&) const;

These assignment operators have reference semantics, assigning the values of the argument array elements to selected elements of the valarray<T> object to which the slice_array object refers.

26.7.5.3 Compound assignment [slice.arr.comp.assign]

🔗

void operator*= (const valarray<T>&) const;
void operator/= (const valarray<T>&) const;
void operator%= (const valarray<T>&) const;
void operator+= (const valarray<T>&) const;
void operator-= (const valarray<T>&) const;
void operator^= (const valarray<T>&) const;
void operator&= (const valarray<T>&) const;
void operator|= (const valarray<T>&) const;
void operator<<=(const valarray<T>&) const;
void operator>>=(const valarray<T>&) const;

These compound assignments have reference semantics, applying the indicated operation to the elements of the argument array and selected elements of the valarray<T> object to which the slice_array object refers.

26.7.5.4 Fill function [slice.arr.fill]

🔗

void operator=(const T&) const;

This function has reference semantics, assigning the value of its argument to the elements of the valarray<T> object to which the slice_array object refers.

26.7.6 The gslice class [class.gslice]

26.7.6.1 Overview [class.gslice.overview]

namespace std {
  class gslice {
  public:
    gslice();
    gslice(size_t s, const valarray<size_t>& l, const valarray<size_t>& d);

    size_t           start() const;
    valarray<size_t> size() const;
    valarray<size_t> stride() const;
  };
}

This class represents a generalized slice out of an array.

A gslice is defined by a starting offset (s), a set of lengths (

l_{j}

), and a set of strides (

d_{j}

The number of lengths shall equal the number of strides.

A gslice represents a mapping from a set of indices (

i_{j}

), equal in number to the number of strides, to a single index k.

It is useful for building multidimensional array classes using the valarray template, which is one-dimensional.

The set of one-dimensional index values specified by a gslice are

k = s + \sum j i_{j} d_{j}

where the multidimensional indices

i_{j}

range in value from 0 to

l_{i j} - 1

[Example

The gslice specification

start  = 3
length = {2, 4, 3}
stride = {19, 4, 1}

yields the sequence of one-dimensional indices

k = 3 + (0, 1) \times 19 + (0, 1, 2, 3) \times 4 + (0, 1, 2) \times 1

which are ordered as shown in the following table:

	 $(i_{0}, i_{1}, i_{2}, k) =$ 
		 $(0, 0, 0, 3)$ ,    
		 $(0, 0, 1, 4)$ ,    
		 $(0, 0, 2, 5)$ ,    
		 $(0, 1, 0, 7)$ ,    
		 $(0, 1, 1, 8)$ ,    
		 $(0, 1, 2, 9)$ ,    
		 $(0, 2, 0, 11)$ , 
		 $(0, 2, 1, 12)$ , 
		 $(0, 2, 2, 13)$ , 
		 $(0, 3, 0, 15)$ , 
		 $(0, 3, 1, 16)$ , 
		 $(0, 3, 2, 17)$ , 
		 $(1, 0, 0, 22)$ , 
		 $(1, 0, 1, 23)$ , 
		…      
		 $(1, 3, 2, 36)$

That is, the highest-ordered index turns fastest.

— end example

]

It is possible to have degenerate generalized slices in which an address is repeated.

[Example

If the stride parameters in the previous example are changed to {1, 1, 1}, the first few elements of the resulting sequence of indices will be

	 $(0, 0, 0, 3)$ ,  
	 $(0, 0, 1, 4)$ ,  
	 $(0, 0, 2, 5)$ ,  
	 $(0, 1, 0, 4)$ ,  
	 $(0, 1, 1, 5)$ ,  
	 $(0, 1, 2, 6)$ ,  
	…

— end example

]

If a degenerate slice is used as the argument to the non-const version of operator[](const gslice&), the behavior is undefined.

26.7.6.2 Constructors [gslice.cons]

🔗

gslice();
gslice(size_t start, const valarray<size_t>& lengths,
         const valarray<size_t>& strides);
gslice(const gslice&);

The default constructor is equivalent to gslice(0, valarray<size_t>(), valarray<size_t>()).

The constructor with arguments builds a gslice based on a specification of start, lengths, and strides, as explained in the previous subclause.

26.7.6.3 Access functions [gslice.access]

🔗

size_t           start()  const;
valarray<size_t> size() const;
valarray<size_t> stride() const;

Returns: The representation of the start, lengths, or strides specified for the gslice.

Complexity: start() is constant time.

size() and stride() are linear in the number of strides.

26.7.7 Class template gslice_array [template.gslice.array]

26.7.7.1 Overview [template.gslice.array.overview]

namespace std {
  template<class T> class gslice_array {
  public:
    using value_type = T;

    void operator=  (const valarray<T>&) const;
    void operator*= (const valarray<T>&) const;
    void operator/= (const valarray<T>&) const;
    void operator%= (const valarray<T>&) const;
    void operator+= (const valarray<T>&) const;
    void operator-= (const valarray<T>&) const;
    void operator^= (const valarray<T>&) const;
    void operator&= (const valarray<T>&) const;
    void operator|= (const valarray<T>&) const;
    void operator<<=(const valarray<T>&) const;
    void operator>>=(const valarray<T>&) const;

    gslice_array(const gslice_array&);
    ~gslice_array();
    const gslice_array& operator=(const gslice_array&) const;
    void operator=(const T&) const;

    gslice_array() = delete;    // as implied by declaring copy constructor above
  };
}

This template is a helper template used by the gslice subscript operator

🔗

gslice_array<T> valarray<T>::operator[](const gslice&);

It has reference semantics to a subset of an array specified by a gslice object.

Thus, the expression a[gslice(1, length, stride)] = b has the effect of assigning the elements of b to a generalized slice of the elements in a.

26.7.7.2 Assignment [gslice.array.assign]

🔗

void operator=(const valarray<T>&) const;
const gslice_array& operator=(const gslice_array&) const;

These assignment operators have reference semantics, assigning the values of the argument array elements to selected elements of the valarray<T> object to which the gslice_array refers.

26.7.7.3 Compound assignment [gslice.array.comp.assign]

🔗

void operator*= (const valarray<T>&) const;
void operator/= (const valarray<T>&) const;
void operator%= (const valarray<T>&) const;
void operator+= (const valarray<T>&) const;
void operator-= (const valarray<T>&) const;
void operator^= (const valarray<T>&) const;
void operator&= (const valarray<T>&) const;
void operator|= (const valarray<T>&) const;
void operator<<=(const valarray<T>&) const;
void operator>>=(const valarray<T>&) const;

26.7.7.4 Fill function [gslice.array.fill]

🔗

void operator=(const T&) const;

This function has reference semantics, assigning the value of its argument to the elements of the valarray<T> object to which the gslice_array object refers.

26.7.8 Class template mask_array [template.mask.array]

26.7.8.1 Overview [template.mask.array.overview]

namespace std {
  template<class T> class mask_array {
  public:
    using value_type = T;

    void operator=  (const valarray<T>&) const;
    void operator*= (const valarray<T>&) const;
    void operator/= (const valarray<T>&) const;
    void operator%= (const valarray<T>&) const;
    void operator+= (const valarray<T>&) const;
    void operator-= (const valarray<T>&) const;
    void operator^= (const valarray<T>&) const;
    void operator&= (const valarray<T>&) const;
    void operator|= (const valarray<T>&) const;
    void operator<<=(const valarray<T>&) const;
    void operator>>=(const valarray<T>&) const;

    mask_array(const mask_array&);
    ~mask_array();
    const mask_array& operator=(const mask_array&) const;
    void operator=(const T&) const;

    mask_array() = delete;      // as implied by declaring copy constructor above
  };
}

This template is a helper template used by the mask subscript operator:

🔗

mask_array<T> valarray<T>::operator[](const valarray<bool>&).

It has reference semantics to a subset of an array specified by a boolean mask.

Thus, the expression a[mask] = b; has the effect of assigning the elements of b to the masked elements in a (those for which the corresponding element in mask is true).

26.7.8.2 Assignment [mask.array.assign]

🔗

void operator=(const valarray<T>&) const;
const mask_array& operator=(const mask_array&) const;

These assignment operators have reference semantics, assigning the values of the argument array elements to selected elements of the valarray<T> object to which it refers.

26.7.8.3 Compound assignment [mask.array.comp.assign]

🔗

void operator*= (const valarray<T>&) const;
void operator/= (const valarray<T>&) const;
void operator%= (const valarray<T>&) const;
void operator+= (const valarray<T>&) const;
void operator-= (const valarray<T>&) const;
void operator^= (const valarray<T>&) const;
void operator&= (const valarray<T>&) const;
void operator|= (const valarray<T>&) const;
void operator<<=(const valarray<T>&) const;
void operator>>=(const valarray<T>&) const;

26.7.8.4 Fill function [mask.array.fill]

🔗

void operator=(const T&) const;

This function has reference semantics, assigning the value of its argument to the elements of the valarray<T> object to which the mask_array object refers.

26.7.9 Class template indirect_array [template.indirect.array]

26.7.9.1 Overview [template.indirect.array.overview]

namespace std {
  template<class T> class indirect_array {
  public:
    using value_type = T;

    void operator=  (const valarray<T>&) const;
    void operator*= (const valarray<T>&) const;
    void operator/= (const valarray<T>&) const;
    void operator%= (const valarray<T>&) const;
    void operator+= (const valarray<T>&) const;
    void operator-= (const valarray<T>&) const;
    void operator^= (const valarray<T>&) const;
    void operator&= (const valarray<T>&) const;
    void operator|= (const valarray<T>&) const;
    void operator<<=(const valarray<T>&) const;
    void operator>>=(const valarray<T>&) const;

    indirect_array(const indirect_array&);
    ~indirect_array();
    const indirect_array& operator=(const indirect_array&) const;
    void operator=(const T&) const;

    indirect_array() = delete;  // as implied by declaring copy constructor above
  };
}

This template is a helper template used by the indirect subscript operator

🔗

indirect_array<T> valarray<T>::operator[](const valarray<size_t>&).

It has reference semantics to a subset of an array specified by an indirect_array.

Thus, the expression a[indirect] = b; has the effect of assigning the elements of b to the elements in a whose indices appear in indirect.

26.7.9.2 Assignment [indirect.array.assign]

🔗

void operator=(const valarray<T>&) const;
const indirect_array& operator=(const indirect_array&) const;

These assignment operators have reference semantics, assigning the values of the argument array elements to selected elements of the valarray<T> object to which it refers.

If the indirect_array specifies an element in the valarray<T> object to which it refers more than once, the behavior is undefined.

[Example

int addr[] = {2, 3, 1, 4, 4};
valarray<size_t> indirect(addr, 5);
valarray<double> a(0., 10), b(1., 5);
a[indirect] = b;

results in undefined behavior since element 4 is specified twice in the indirection.

— end example

]

26.7.9.3 Compound assignment [indirect.array.comp.assign]

🔗

void operator*= (const valarray<T>&) const;
void operator/= (const valarray<T>&) const;
void operator%= (const valarray<T>&) const;
void operator+= (const valarray<T>&) const;
void operator-= (const valarray<T>&) const;
void operator^= (const valarray<T>&) const;
void operator&= (const valarray<T>&) const;
void operator|= (const valarray<T>&) const;
void operator<<=(const valarray<T>&) const;
void operator>>=(const valarray<T>&) const;

If the indirect_array specifies an element in the valarray<T> object to which it refers more than once, the behavior is undefined.

26.7.9.4 Fill function [indirect.array.fill]

🔗

void operator=(const T&) const;

This function has reference semantics, assigning the value of its argument to the elements of the valarray<T> object to which the indirect_array object refers.

26.7.10 valarray range access [valarray.range]

In the begin and end function templates that follow, unspecified1 is a type that meets the requirements of a mutable Cpp17RandomAccessIterator ([random.access.iterators]) and models contiguous_iterator ([iterator.concept.contiguous]), whose value_type is the template parameter T and whose reference type is T&.

unspecified2 is a type that meets the requirements of a constant Cpp17RandomAccessIterator and models contiguous_iterator, whose value_type is the template parameter T and whose reference type is const T&.

The iterators returned by begin and end for an array are guaranteed to be valid until the member function resize(size_t, T) is called for that array or until the lifetime of that array ends, whichever happens first.

🔗

template<class T> unspecified1 begin(valarray<T>& v);
template<class T> unspecified2 begin(const valarray<T>& v);

Returns: An iterator referencing the first value in the array.

🔗

template<class T> unspecified1 end(valarray<T>& v);
template<class T> unspecified2 end(const valarray<T>& v);

Returns: An iterator referencing one past the last value in the array.

26.8 Mathematical functions for floating-point types [c.math]

26.8.1 Header <cmath> synopsis [cmath.syn]

namespace std {
  using float_t = see below;
  using double_t = see below;
}

#define HUGE_VAL see below
#define HUGE_VALF see below
#define HUGE_VALL see below
#define INFINITY see below
#define NAN see below
#define FP_INFINITE see below
#define FP_NAN see below
#define FP_NORMAL see below
#define FP_SUBNORMAL see below
#define FP_ZERO see below
#define FP_FAST_FMA see below
#define FP_FAST_FMAF see below
#define FP_FAST_FMAL see below
#define FP_ILOGB0 see below
#define FP_ILOGBNAN see below
#define MATH_ERRNO see below
#define MATH_ERREXCEPT see below

#define math_errhandling see below

namespace std {
  float acos(float x);  // see [library.c]
  double acos(double x);
  long double acos(long double x);  // see [library.c]
  float acosf(float x);
  long double acosl(long double x);

  float asin(float x);  // see [library.c]
  double asin(double x);
  long double asin(long double x);  // see [library.c]
  float asinf(float x);
  long double asinl(long double x);

  float atan(float x);                  // see [library.c]
  double atan(double x);
  long double atan(long double x);      // see [library.c]
  float atanf(float x);
  long double atanl(long double x);

  float atan2(float y, float x);        // see [library.c]
  double atan2(double y, double x);
  long double atan2(long double y, long double x);  // see [library.c]
  float atan2f(float y, float x);
  long double atan2l(long double y, long double x);

  float cos(float x);                   // see [library.c]
  double cos(double x);
  long double cos(long double x);       // see [library.c]
  float cosf(float x);
  long double cosl(long double x);

  float sin(float x);                   // see [library.c]
  double sin(double x);
  long double sin(long double x);       // see [library.c]
  float sinf(float x);
  long double sinl(long double x);

  float tan(float x);                   // see [library.c]
  double tan(double x);
  long double tan(long double x);       // see [library.c]
  float tanf(float x);
  long double tanl(long double x);

  float acosh(float x);                 // see [library.c]
  double acosh(double x);
  long double acosh(long double x);     // see [library.c]
  float acoshf(float x);
  long double acoshl(long double x);

  float asinh(float x);                 // see [library.c]
  double asinh(double x);
  long double asinh(long double x);     // see [library.c]
  float asinhf(float x);
  long double asinhl(long double x);

  float atanh(float x);                 // see [library.c]
  double atanh(double x);
  long double atanh(long double x);     // see [library.c]
  float atanhf(float x);
  long double atanhl(long double x);

  float cosh(float x);                  // see [library.c]
  double cosh(double x);
  long double cosh(long double x);      // see [library.c]
  float coshf(float x);
  long double coshl(long double x);

  float sinh(float x);                  // see [library.c]
  double sinh(double x);
  long double sinh(long double x);      // see [library.c]
  float sinhf(float x);
  long double sinhl(long double x);

  float tanh(float x);                  // see [library.c]
  double tanh(double x);
  long double tanh(long double x);      // see [library.c]
  float tanhf(float x);
  long double tanhl(long double x);

  float exp(float x);                   // see [library.c]
  double exp(double x);
  long double exp(long double x);       // see [library.c]
  float expf(float x);
  long double expl(long double x);

  float exp2(float x);                  // see [library.c]
  double exp2(double x);
  long double exp2(long double x);      // see [library.c]
  float exp2f(float x);
  long double exp2l(long double x);

  float expm1(float x);                 // see [library.c]
  double expm1(double x);
  long double expm1(long double x);     // see [library.c]
  float expm1f(float x);
  long double expm1l(long double x);

  float frexp(float value, int* exp);   // see [library.c]
  double frexp(double value, int* exp);
  long double frexp(long double value, int* exp);   // see [library.c]
  float frexpf(float value, int* exp);
  long double frexpl(long double value, int* exp);

  int ilogb(float x);                   // see [library.c]
  int ilogb(double x);
  int ilogb(long double x);             // see [library.c]
  int ilogbf(float x);
  int ilogbl(long double x);

  float ldexp(float x, int exp);        // see [library.c]
  double ldexp(double x, int exp);
  long double ldexp(long double x, int exp);    // see [library.c]
  float ldexpf(float x, int exp);
  long double ldexpl(long double x, int exp);

  float log(float x);                   // see [library.c]
  double log(double x);
  long double log(long double x);       // see [library.c]
  float logf(float x);
  long double logl(long double x);

  float log10(float x);                 // see [library.c]
  double log10(double x);
  long double log10(long double x);     // see [library.c]
  float log10f(float x);
  long double log10l(long double x);

  float log1p(float x);                 // see [library.c]
  double log1p(double x);
  long double log1p(long double x);     // see [library.c]
  float log1pf(float x);
  long double log1pl(long double x);

  float log2(float x);                  // see [library.c]
  double log2(double x);
  long double log2(long double x);      // see [library.c]
  float log2f(float x);
  long double log2l(long double x);

  float logb(float x);                  // see [library.c]
  double logb(double x);
  long double logb(long double x);      // see [library.c]
  float logbf(float x);
  long double logbl(long double x);

  float modf(float value, float* iptr); // see [library.c]
  double modf(double value, double* iptr);
  long double modf(long double value, long double* iptr);   // see [library.c]
  float modff(float value, float* iptr);
  long double modfl(long double value, long double* iptr);

  float scalbn(float x, int n);         // see [library.c]
  double scalbn(double x, int n);
  long double scalbn(long double x, int n);     // see [library.c]
  float scalbnf(float x, int n);
  long double scalbnl(long double x, int n);

  float scalbln(float x, long int n);   // see [library.c]
  double scalbln(double x, long int n);
  long double scalbln(long double x, long int n);   // see [library.c]
  float scalblnf(float x, long int n);
  long double scalblnl(long double x, long int n);

  float cbrt(float x);                  // see [library.c]
  double cbrt(double x);
  long double cbrt(long double x);      // see [library.c]
  float cbrtf(float x);
  long double cbrtl(long double x);

  // [c.math.abs], absolute values
  int abs(int j);
  long int abs(long int j);
  long long int abs(long long int j);
  float abs(float j);
  double abs(double j);
  long double abs(long double j);

  float fabs(float x);                  // see [library.c]
  double fabs(double x);
  long double fabs(long double x);      // see [library.c]
  float fabsf(float x);
  long double fabsl(long double x);

  float hypot(float x, float y);        // see [library.c]
  double hypot(double x, double y);
  long double hypot(long double x, long double y);  // see [library.c]
  float hypotf(float x, float y);
  long double hypotl(long double x, long double y);

  // [c.math.hypot3], three-dimensional hypotenuse
  float hypot(float x, float y, float z);
  double hypot(double x, double y, double z);
  long double hypot(long double x, long double y, long double z);

  float pow(float x, float y);          // see [library.c]
  double pow(double x, double y);
  long double pow(long double x, long double y);    // see [library.c]
  float powf(float x, float y);
  long double powl(long double x, long double y);

  float sqrt(float x);                  // see [library.c]
  double sqrt(double x);
  long double sqrt(long double x);      // see [library.c]
  float sqrtf(float x);
  long double sqrtl(long double x);

  float erf(float x);                   // see [library.c]
  double erf(double x);
  long double erf(long double x);       // see [library.c]
  float erff(float x);
  long double erfl(long double x);

  float erfc(float x);                  // see [library.c]
  double erfc(double x);
  long double erfc(long double x);      // see [library.c]
  float erfcf(float x);
  long double erfcl(long double x);

  float lgamma(float x);                // see [library.c]
  double lgamma(double x);
  long double lgamma(long double x);    // see [library.c]
  float lgammaf(float x);
  long double lgammal(long double x);

  float tgamma(float x);                // see [library.c]
  double tgamma(double x);
  long double tgamma(long double x);    // see [library.c]
  float tgammaf(float x);
  long double tgammal(long double x);

  float ceil(float x);                  // see [library.c]
  double ceil(double x);
  long double ceil(long double x);      // see [library.c]
  float ceilf(float x);
  long double ceill(long double x);

  float floor(float x);                 // see [library.c]
  double floor(double x);
  long double floor(long double x);     // see [library.c]
  float floorf(float x);
  long double floorl(long double x);

  float nearbyint(float x);             // see [library.c]
  double nearbyint(double x);
  long double nearbyint(long double x); // see [library.c]
  float nearbyintf(float x);
  long double nearbyintl(long double x);

  float rint(float x);                  // see [library.c]
  double rint(double x);
  long double rint(long double x);      // see [library.c]
  float rintf(float x);
  long double rintl(long double x);

  long int lrint(float x);              // see [library.c]
  long int lrint(double x);
  long int lrint(long double x);        // see [library.c]
  long int lrintf(float x);
  long int lrintl(long double x);

  long long int llrint(float x);        // see [library.c]
  long long int llrint(double x);
  long long int llrint(long double x);  // see [library.c]
  long long int llrintf(float x);
  long long int llrintl(long double x);

  float round(float x);                 // see [library.c]
  double round(double x);
  long double round(long double x);     // see [library.c]
  float roundf(float x);
  long double roundl(long double x);

  long int lround(float x);             // see [library.c]
  long int lround(double x);
  long int lround(long double x);       // see [library.c]
  long int lroundf(float x);
  long int lroundl(long double x);

  long long int llround(float x);       // see [library.c]
  long long int llround(double x);
  long long int llround(long double x); // see [library.c]
  long long int llroundf(float x);
  long long int llroundl(long double x);

  float trunc(float x);                 // see [library.c]
  double trunc(double x);
  long double trunc(long double x);     // see [library.c]
  float truncf(float x);
  long double truncl(long double x);

  float fmod(float x, float y);         // see [library.c]
  double fmod(double x, double y);
  long double fmod(long double x, long double y);   // see [library.c]
  float fmodf(float x, float y);
  long double fmodl(long double x, long double y);

  float remainder(float x, float y);    // see [library.c]
  double remainder(double x, double y);
  long double remainder(long double x, long double y);  // see [library.c]
  float remainderf(float x, float y);
  long double remainderl(long double x, long double y);

  float remquo(float x, float y, int* quo);     // see [library.c]
  double remquo(double x, double y, int* quo);
  long double remquo(long double x, long double y, int* quo);   // see [library.c]
  float remquof(float x, float y, int* quo);
  long double remquol(long double x, long double y, int* quo);

  float copysign(float x, float y);     // see [library.c]
  double copysign(double x, double y);
  long double copysign(long double x, long double y);   // see [library.c]
  float copysignf(float x, float y);
  long double copysignl(long double x, long double y);

  double nan(const char* tagp);
  float nanf(const char* tagp);
  long double nanl(const char* tagp);

  float nextafter(float x, float y);    // see [library.c]
  double nextafter(double x, double y);
  long double nextafter(long double x, long double y);  // see [library.c]
  float nextafterf(float x, float y);
  long double nextafterl(long double x, long double y);

  float nexttoward(float x, long double y);     // see [library.c]
  double nexttoward(double x, long double y);
  long double nexttoward(long double x, long double y);     // see [library.c]
  float nexttowardf(float x, long double y);
  long double nexttowardl(long double x, long double y);

  float fdim(float x, float y);         // see [library.c]
  double fdim(double x, double y);
  long double fdim(long double x, long double y);   // see [library.c]
  float fdimf(float x, float y);
  long double fdiml(long double x, long double y);

  float fmax(float x, float y);         // see [library.c]
  double fmax(double x, double y);
  long double fmax(long double x, long double y);   // see [library.c]
  float fmaxf(float x, float y);
  long double fmaxl(long double x, long double y);

  float fmin(float x, float y);         // see [library.c]
  double fmin(double x, double y);
  long double fmin(long double x, long double y);   // see [library.c]
  float fminf(float x, float y);
  long double fminl(long double x, long double y);

  float fma(float x, float y, float z); // see [library.c]
  double fma(double x, double y, double z);
  long double fma(long double x, long double y, long double z); // see [library.c]
  float fmaf(float x, float y, float z);
  long double fmal(long double x, long double y, long double z);

  // [c.math.lerp], linear interpolation
  constexpr float lerp(float a, float b, float t) noexcept;
  constexpr double lerp(double a, double b, double t) noexcept;
  constexpr long double lerp(long double a, long double b, long double t) noexcept;

  // [c.math.fpclass], classification / comparison functions
  int fpclassify(float x);
  int fpclassify(double x);
  int fpclassify(long double x);

  bool isfinite(float x);
  bool isfinite(double x);
  bool isfinite(long double x);

  bool isinf(float x);
  bool isinf(double x);
  bool isinf(long double x);

  bool isnan(float x);
  bool isnan(double x);
  bool isnan(long double x);

  bool isnormal(float x);
  bool isnormal(double x);
  bool isnormal(long double x);

  bool signbit(float x);
  bool signbit(double x);
  bool signbit(long double x);

  bool isgreater(float x, float y);
  bool isgreater(double x, double y);
  bool isgreater(long double x, long double y);

  bool isgreaterequal(float x, float y);
  bool isgreaterequal(double x, double y);
  bool isgreaterequal(long double x, long double y);

  bool isless(float x, float y);
  bool isless(double x, double y);
  bool isless(long double x, long double y);

  bool islessequal(float x, float y);
  bool islessequal(double x, double y);
  bool islessequal(long double x, long double y);

  bool islessgreater(float x, float y);
  bool islessgreater(double x, double y);
  bool islessgreater(long double x, long double y);

  bool isunordered(float x, float y);
  bool isunordered(double x, double y);
  bool isunordered(long double x, long double y);

  // [sf.cmath], mathematical special functions

  // [sf.cmath.assoc.laguerre], associated Laguerre polynomials
  double       assoc_laguerre(unsigned n, unsigned m, double x);
  float        assoc_laguerref(unsigned n, unsigned m, float x);
  long double  assoc_laguerrel(unsigned n, unsigned m, long double x);

  // [sf.cmath.assoc.legendre], associated Legendre functions
  double       assoc_legendre(unsigned l, unsigned m, double x);
  float        assoc_legendref(unsigned l, unsigned m, float x);
  long double  assoc_legendrel(unsigned l, unsigned m, long double x);

  // [sf.cmath.beta], beta function
  double       beta(double x, double y);
  float        betaf(float x, float y);
  long double  betal(long double x, long double y);

  // [sf.cmath.comp.ellint.1], complete elliptic integral of the first kind
  double       comp_ellint_1(double k);
  float        comp_ellint_1f(float k);
  long double  comp_ellint_1l(long double k);

  // [sf.cmath.comp.ellint.2], complete elliptic integral of the second kind
  double       comp_ellint_2(double k);
  float        comp_ellint_2f(float k);
  long double  comp_ellint_2l(long double k);

  // [sf.cmath.comp.ellint.3], complete elliptic integral of the third kind
  double       comp_ellint_3(double k, double nu);
  float        comp_ellint_3f(float k, float nu);
  long double  comp_ellint_3l(long double k, long double nu);

  // [sf.cmath.cyl.bessel.i], regular modified cylindrical Bessel functions
  double       cyl_bessel_i(double nu, double x);
  float        cyl_bessel_if(float nu, float x);
  long double  cyl_bessel_il(long double nu, long double x);

  // [sf.cmath.cyl.bessel.j], cylindrical Bessel functions of the first kind
  double       cyl_bessel_j(double nu, double x);
  float        cyl_bessel_jf(float nu, float x);
  long double  cyl_bessel_jl(long double nu, long double x);

  // [sf.cmath.cyl.bessel.k], irregular modified cylindrical Bessel functions
  double       cyl_bessel_k(double nu, double x);
  float        cyl_bessel_kf(float nu, float x);
  long double  cyl_bessel_kl(long double nu, long double x);

  // [sf.cmath.cyl.neumann], cylindrical Neumann functions;
  // cylindrical Bessel functions of the second kind
  double       cyl_neumann(double nu, double x);
  float        cyl_neumannf(float nu, float x);
  long double  cyl_neumannl(long double nu, long double x);

  // [sf.cmath.ellint.1], incomplete elliptic integral of the first kind
  double       ellint_1(double k, double phi);
  float        ellint_1f(float k, float phi);
  long double  ellint_1l(long double k, long double phi);

  // [sf.cmath.ellint.2], incomplete elliptic integral of the second kind
  double       ellint_2(double k, double phi);
  float        ellint_2f(float k, float phi);
  long double  ellint_2l(long double k, long double phi);

  // [sf.cmath.ellint.3], incomplete elliptic integral of the third kind
  double       ellint_3(double k, double nu, double phi);
  float        ellint_3f(float k, float nu, float phi);
  long double  ellint_3l(long double k, long double nu, long double phi);

  // [sf.cmath.expint], exponential integral
  double       expint(double x);
  float        expintf(float x);
  long double  expintl(long double x);

  // [sf.cmath.hermite], Hermite polynomials
  double       hermite(unsigned n, double x);
  float        hermitef(unsigned n, float x);
  long double  hermitel(unsigned n, long double x);

  // [sf.cmath.laguerre], Laguerre polynomials
  double       laguerre(unsigned n, double x);
  float        laguerref(unsigned n, float x);
  long double  laguerrel(unsigned n, long double x);

  // [sf.cmath.legendre], Legendre polynomials
  double       legendre(unsigned l, double x);
  float        legendref(unsigned l, float x);
  long double  legendrel(unsigned l, long double x);

  // [sf.cmath.riemann.zeta], Riemann zeta function
  double       riemann_zeta(double x);
  float        riemann_zetaf(float x);
  long double  riemann_zetal(long double x);

  // [sf.cmath.sph.bessel], spherical Bessel functions of the first kind
  double       sph_bessel(unsigned n, double x);
  float        sph_besself(unsigned n, float x);
  long double  sph_bessell(unsigned n, long double x);

  // [sf.cmath.sph.legendre], spherical associated Legendre functions
  double       sph_legendre(unsigned l, unsigned m, double theta);
  float        sph_legendref(unsigned l, unsigned m, float theta);
  long double  sph_legendrel(unsigned l, unsigned m, long double theta);

  // [sf.cmath.sph.neumann], spherical Neumann functions;
  // spherical Bessel functions of the second kind
  double       sph_neumann(unsigned n, double x);
  float        sph_neumannf(unsigned n, float x);
  long double  sph_neumannl(unsigned n, long double x);
}

The contents and meaning of the header <cmath> are the same as the C standard library header <math.h>, with the addition of a three-dimensional hypotenuse function and the mathematical special functions described in [sf.cmath].

[Note

Several functions have additional overloads in this document, but they have the same behavior as in the C standard library .

— end note

]

For each set of overloaded functions within <cmath>, with the exception of abs, there shall be additional overloads sufficient to ensure:

(2.1)
If any argument of arithmetic type corresponding to a double parameter has type long double, then all arguments of arithmetic type corresponding to double parameters are effectively cast to long double.
(2.2)
Otherwise, if any argument of arithmetic type corresponding to a double parameter has type double or an integer type, then all arguments of arithmetic type corresponding to double parameters are effectively cast to double.
(2.3)
[Note
: Otherwise, all arguments of arithmetic type corresponding to double parameters have type float. — end note
]

[Note

: abs is exempted from these rules in order to stay compatible with C. — end note

]

26.8.2 Absolute values [c.math.abs]

[Note

The headers <cstdlib> ([cstdlib.syn]) and <cmath> ([cmath.syn]) declare the functions described in this subclause.

— end note

]

🔗

int abs(int j);
long int abs(long int j);
long long int abs(long long int j);
float abs(float j);
double abs(double j);
long double abs(long double j);

Effects: The abs functions have the semantics specified in the C standard library for the functions abs, labs, llabs, fabsf, fabs, and fabsl.

Remarks: If abs() is called with an argument of type X for which is_unsigned_v<X> is true and if X cannot be converted to int by integral promotion, the program is ill-formed.

[Note

Arguments that can be promoted to int are permitted for compatibility with C.

— end note

]

26.8.3 Three-dimensional hypotenuse [c.math.hypot3]

🔗

float hypot(float x, float y, float z);
double hypot(double x, double y, double z);
long double hypot(long double x, long double y, long double z);

Returns:

\sqrt{x^{2} + y^{2} + z^{2}}

26.8.4 Linear interpolation [c.math.lerp]

🔗

constexpr float lerp(float a, float b, float t) noexcept;
constexpr double lerp(double a, double b, double t) noexcept;
constexpr long double lerp(long double a, long double b, long double t) noexcept;

Returns:

a + t (b - a)

Remarks: Let r be the value returned.

If isfinite(a) && isfinite(b), then:

(2.1)
If t == 0, then r == a.
(2.2)
If t == 1, then r == b.
(2.3)
If t >= 0 && t <= 1, then isfinite(r).
(2.4)
If isfinite(t) && a == b, then r == a.
(2.5)
If isfinite(t) || !isnan(t) && b-a != 0, then !isnan(r).

Let CMP(x,y) be 1 if x > y, -1 if x < y, and 0 otherwise.

For any t1 and t2, the product of CMP(lerp(a, b, t2), lerp(a, b, t1)), CMP(t2, t1), and CMP(b, a) is non-negative.

26.8.5 Classification / comparison functions [c.math.fpclass]

The classification / comparison functions behave the same as the C macros with the corresponding names defined in the C standard library.

Each function is overloaded for the three floating-point types.

26.8.6 Mathematical special functions [sf.cmath]

If any argument value to any of the functions specified in this subclause is a NaN (Not a Number), the function shall return a NaN but it shall not report a domain error.

Otherwise, the function shall report a domain error for just those argument values for which:

(1.1)
the function description's Returns: clause explicitly specifies a domain and those argument values fall outside the specified domain, or
(1.2)
the corresponding mathematical function value has a nonzero imaginary component, or
(1.3)
the corresponding mathematical function is not mathematically defined.254

Unless otherwise specified, each function is defined for all finite values, for negative infinity, and for positive infinity.

254)

A mathematical function is mathematically defined for a given set of argument values (a) if it is explicitly defined for that set of argument values, or (b) if its limiting value exists and does not depend on the direction of approach.

⮥

26.8.6.1 Associated Laguerre polynomials [sf.cmath.assoc.laguerre]

🔗

double       assoc_laguerre(unsigned n, unsigned m, double x);
float        assoc_laguerref(unsigned n, unsigned m, float x);
long double  assoc_laguerrel(unsigned n, unsigned m, long double x);

Effects: These functions compute the associated Laguerre polynomials of their respective arguments n, m, and x.

Returns:

L_{n}^{m} (x) = (- 1)^{m} \frac{d^{m}}{d x^{m}} L_{n + m} (x), for x \geq 0,

where n is n, m is m, and x is x.

Remarks: The effect of calling each of these functions is implementation-defined if n >= 128 or if m >= 128.

26.8.6.2 Associated Legendre functions [sf.cmath.assoc.legendre]

🔗

double       assoc_legendre(unsigned l, unsigned m, double x);
float        assoc_legendref(unsigned l, unsigned m, float x);
long double  assoc_legendrel(unsigned l, unsigned m, long double x);

Effects: These functions compute the associated Legendre functions of their respective arguments l, m, and x.

Returns:

P_{ℓ}^{m} (x) = (1 - x^{2})^{m / 2} \frac{d^{m}}{d x^{m}} P_{ℓ} (x), for | x | \leq 1,

where l is l, m is m, and x is x.

Remarks: The effect of calling each of these functions is implementation-defined if l >= 128.

26.8.6.3 Beta function [sf.cmath.beta]

🔗

double       beta(double x, double y);
float        betaf(float x, float y);
long double  betal(long double x, long double y);

Effects: These functions compute the beta function of their respective arguments x and y.

Returns:

B (x, y) = \frac{Γ (x) Γ (y)}{Γ (x + y)}, for x > 0, y > 0,

where x is x and y is y.

26.8.6.4 Complete elliptic integral of the first kind [sf.cmath.comp.ellint.1]

🔗

double       comp_ellint_1(double k);
float        comp_ellint_1f(float k);
long double  comp_ellint_1l(long double k);

Effects: These functions compute the complete elliptic integral of the first kind of their respective arguments k.

Returns:

K (k) = F (k, π / 2), for | k | \leq 1,

where k is k.

26.8.6.5 Complete elliptic integral of the second kind [sf.cmath.comp.ellint.2]

🔗

double       comp_ellint_2(double k);
float        comp_ellint_2f(float k);
long double  comp_ellint_2l(long double k);

Effects: These functions compute the complete elliptic integral of the second kind of their respective arguments k.

Returns:

E (k) = E (k, π / 2), for | k | \leq 1,

where k is k.

26.8.6.6 Complete elliptic integral of the third kind [sf.cmath.comp.ellint.3]

🔗

double       comp_ellint_3(double k, double nu);
float        comp_ellint_3f(float k, float nu);
long double  comp_ellint_3l(long double k, long double nu);

Effects: These functions compute the complete elliptic integral of the third kind of their respective arguments k and nu.

Returns:

Π (ν, k) = Π (ν, k, π / 2), for | k | \leq 1,

where k is k and ν is nu.

26.8.6.7 Regular modified cylindrical Bessel functions [sf.cmath.cyl.bessel.i]

🔗

double       cyl_bessel_i(double nu, double x);
float        cyl_bessel_if(float nu, float x);
long double  cyl_bessel_il(long double nu, long double x);

Effects: These functions compute the regular modified cylindrical Bessel functions of their respective arguments nu and x.

Returns:

I_{ν} (x) = i^{- ν} J_{ν} (i x) = \infty \sum k = 0 \frac{(x / 2)^{ν + 2 k}}{k! Γ (ν + k + 1)}, for x \geq 0,

where ν is nu and x is x.

Remarks: The effect of calling each of these functions is implementation-defined if nu >= 128.

26.8.6.8 Cylindrical Bessel functions of the first kind [sf.cmath.cyl.bessel.j]

🔗

double       cyl_bessel_j(double nu, double x);
float        cyl_bessel_jf(float nu, float x);
long double  cyl_bessel_jl(long double nu, long double x);

Effects: These functions compute the cylindrical Bessel functions of the first kind of their respective arguments nu and x.

Returns:

J_{ν} (x) = \infty \sum k = 0 \frac{(- 1)^{k} (x / 2)^{ν + 2 k}}{k! Γ (ν + k + 1)}, for x \geq 0,

where ν is nu and x is x.

Remarks: The effect of calling each of these functions is implementation-defined if nu >= 128.

26.8.6.9 Irregular modified cylindrical Bessel functions [sf.cmath.cyl.bessel.k]

🔗

double       cyl_bessel_k(double nu, double x);
float        cyl_bessel_kf(float nu, float x);
long double  cyl_bessel_kl(long double nu, long double x);

Effects: These functions compute the irregular modified cylindrical Bessel functions of their respective arguments nu and x.

Returns:

K_{ν} (x) = (π / 2) i^{ν + 1} (J_{ν} (i x) + i N_{ν} (i x)) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ \begin{matrix} \frac{π}{2} \frac{I_{- ν} (x) - I_{ν} (x)}{sin ν π}, & for x \geq 0 and non-integral ν \frac{π}{2} lim_{μ \to ν} \frac{I_{- μ} (x) - I_{μ} (x)}{sin μ π}, & for x \geq 0 and integral ν \end{matrix}

where ν is nu and x is x.

Remarks: The effect of calling each of these functions is implementation-defined if nu >= 128.

26.8.6.10 Cylindrical Neumann functions [sf.cmath.cyl.neumann]

🔗

double       cyl_neumann(double nu, double x);
float        cyl_neumannf(float nu, float x);
long double  cyl_neumannl(long double nu, long double x);

Effects: These functions compute the cylindrical Neumann functions, also known as the cylindrical Bessel functions of the second kind, of their respective arguments nu and x.

Returns:

N_{ν} (x) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ \begin{matrix} \frac{J_{ν} (x) cos ν π - J_{- ν} (x)}{sin ν π}, & for x \geq 0 and non-integral ν lim_{μ \to ν} \frac{J_{μ} (x) cos μ π - J_{- μ} (x)}{sin μ π}, & for x \geq 0 and integral ν \end{matrix}

where ν is nu and x is x.

Remarks: The effect of calling each of these functions is implementation-defined if nu >= 128.

26.8.6.11 Incomplete elliptic integral of the first kind [sf.cmath.ellint.1]

🔗

double       ellint_1(double k, double phi);
float        ellint_1f(float k, float phi);
long double  ellint_1l(long double k, long double phi);

Effects: These functions compute the incomplete elliptic integral of the first kind of their respective arguments k and phi (phi measured in radians).

Returns:

F (k, ϕ) = \int_{0}^{ϕ} \frac{d θ}{\sqrt{1 - k^{2} {sin}^{2} θ}}, for | k | \leq 1,

where k is k and φ is phi.

26.8.6.12 Incomplete elliptic integral of the second kind [sf.cmath.ellint.2]

🔗

double       ellint_2(double k, double phi);
float        ellint_2f(float k, float phi);
long double  ellint_2l(long double k, long double phi);

Effects: These functions compute the incomplete elliptic integral of the second kind of their respective arguments k and phi (phi measured in radians).

Returns:

E (k, ϕ) = \int_{0}^{ϕ} \sqrt{1 - k^{2} {sin}^{2} θ} d θ, for | k | \leq 1,

where k is k and φ is phi.

26.8.6.13 Incomplete elliptic integral of the third kind [sf.cmath.ellint.3]

🔗

double       ellint_3(double k, double nu, double phi);
float        ellint_3f(float k, float nu, float phi);
long double  ellint_3l(long double k, long double nu, long double phi);

Effects: These functions compute the incomplete elliptic integral of the third kind of their respective arguments k, nu, and phi (phi measured in radians).

Returns:

Π (ν, k, ϕ) = \int_{0}^{ϕ} \frac{d θ}{(1 - ν {sin}^{2} θ) \sqrt{1 - k^{2} {sin}^{2} θ}}, for | k | \leq 1,

where ν is nu, k is k, and φ is phi.

26.8.6.14 Exponential integral [sf.cmath.expint]

🔗

double       expint(double x);
float        expintf(float x);
long double  expintl(long double x);

Effects: These functions compute the exponential integral of their respective arguments x.

Returns:

E i (x) = - \int_{- x}^{\infty} \frac{e^{- t}}{t} d t

where x is x.

26.8.6.15 Hermite polynomials [sf.cmath.hermite]

🔗

double       hermite(unsigned n, double x);
float        hermitef(unsigned n, float x);
long double  hermitel(unsigned n, long double x);

Effects: These functions compute the Hermite polynomials of their respective arguments n and x.

Returns:

H_{n} (x) = (- 1)^{n} e^{x^{2}} \frac{d^{n}}{d x^{n}} e^{- x^{2}}

where n is n and x is x.

Remarks: The effect of calling each of these functions is implementation-defined if n >= 128.

26.8.6.16 Laguerre polynomials [sf.cmath.laguerre]

🔗

double       laguerre(unsigned n, double x);
float        laguerref(unsigned n, float x);
long double  laguerrel(unsigned n, long double x);

Effects: These functions compute the Laguerre polynomials of their respective arguments n and x.

Returns:

L_{n} (x) = \frac{e^{x}}{n!} \frac{d^{n}}{d x^{n}} (x^{n} e^{- x}), for x \geq 0,

where n is n and x is x.

Remarks: The effect of calling each of these functions is implementation-defined if n >= 128.

26.8.6.17 Legendre polynomials [sf.cmath.legendre]

🔗

double       legendre(unsigned l, double x);
float        legendref(unsigned l, float x);
long double  legendrel(unsigned l, long double x);

Effects: These functions compute the Legendre polynomials of their respective arguments l and x.

Returns:

P_{ℓ} (x) = \frac{1}{2^{ℓ} ℓ!} \frac{d^{ℓ}}{d x^{ℓ}} (x^{2} - 1)^{ℓ}, for | x | \leq 1,

where l is l and x is x.

Remarks: The effect of calling each of these functions is implementation-defined if l >= 128.

26.8.6.18 Riemann zeta function [sf.cmath.riemann.zeta]

🔗

double       riemann_zeta(double x);
float        riemann_zetaf(float x);
long double  riemann_zetal(long double x);

Effects: These functions compute the Riemann zeta function of their respective arguments x.

Returns:

ζ (x) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ \begin{matrix} \infty \sum k = 1 k^{- x}, & for x > 1 \frac{1}{1 - 2^{1 - x}} \infty \sum k = 1 (- 1)^{k - 1} k^{- x}, & for 0 \leq x \leq 1 2^{x} π^{x - 1} sin (\frac{π x}{2}) Γ (1 - x) ζ (1 - x), & for x < 0 \end{matrix}

where x is x.

26.8.6.19 Spherical Bessel functions of the first kind [sf.cmath.sph.bessel]

🔗

double       sph_bessel(unsigned n, double x);
float        sph_besself(unsigned n, float x);
long double  sph_bessell(unsigned n, long double x);

Effects: These functions compute the spherical Bessel functions of the first kind of their respective arguments n and x.

Returns:

j_{n} (x) = (π / 2 x)^{1 / 2} J_{n + 1 / 2} (x), for x \geq 0,

where n is n and x is x.

Remarks: The effect of calling each of these functions is implementation-defined if n >= 128.

26.8.6.20 Spherical associated Legendre functions [sf.cmath.sph.legendre]

🔗

double       sph_legendre(unsigned l, unsigned m, double theta);
float        sph_legendref(unsigned l, unsigned m, float theta);
long double  sph_legendrel(unsigned l, unsigned m, long double theta);

Effects: These functions compute the spherical associated Legendre functions of their respective arguments l, m, and theta (theta measured in radians).

Returns:

Y_{ℓ}^{m} (θ, 0)

where

Y_{ℓ}^{m} (θ, ϕ) = (- 1)^{m} {[\frac{(2 ℓ + 1)}{4 π} \frac{(ℓ - m)!}{(ℓ + m)!}]}^{1 / 2} P_{ℓ}^{m} (cos θ) e^{i m ϕ}, for | m | \leq ℓ,

and l is l, m is m, and θ is theta.

Remarks: The effect of calling each of these functions is implementation-defined if l >= 128.

26.8.6.21 Spherical Neumann functions [sf.cmath.sph.neumann]

🔗

double       sph_neumann(unsigned n, double x);
float        sph_neumannf(unsigned n, float x);
long double  sph_neumannl(unsigned n, long double x);

Effects: These functions compute the spherical Neumann functions, also known as the spherical Bessel functions of the second kind, of their respective arguments n and x.

Returns:

n_{n} (x) = (π / 2 x)^{1 / 2} N_{n + 1 / 2} (x), for x \geq 0,

where n is n and x is x.

Remarks: The effect of calling each of these functions is implementation-defined if n >= 128.

26.9 Numbers [numbers]

26.9.1 Header <numbers> synopsis [numbers.syn]

namespace std::numbers {
  template<class T> inline constexpr T e_v          = unspecified;
  template<class T> inline constexpr T log2e_v      = unspecified;
  template<class T> inline constexpr T log10e_v     = unspecified;
  template<class T> inline constexpr T pi_v         = unspecified;
  template<class T> inline constexpr T inv_pi_v     = unspecified;
  template<class T> inline constexpr T inv_sqrtpi_v = unspecified;
  template<class T> inline constexpr T ln2_v        = unspecified;
  template<class T> inline constexpr T ln10_v       = unspecified;
  template<class T> inline constexpr T sqrt2_v      = unspecified;
  template<class T> inline constexpr T sqrt3_v      = unspecified;
  template<class T> inline constexpr T inv_sqrt3_v  = unspecified;
  template<class T> inline constexpr T egamma_v     = unspecified;
  template<class T> inline constexpr T phi_v        = unspecified;

  template<floating_point T> inline constexpr T e_v<T>          = see below;
  template<floating_point T> inline constexpr T log2e_v<T>      = see below;
  template<floating_point T> inline constexpr T log10e_v<T>     = see below;
  template<floating_point T> inline constexpr T pi_v<T>         = see below;
  template<floating_point T> inline constexpr T inv_pi_v<T>     = see below;
  template<floating_point T> inline constexpr T inv_sqrtpi_v<T> = see below;
  template<floating_point T> inline constexpr T ln2_v<T>        = see below;
  template<floating_point T> inline constexpr T ln10_v<T>       = see below;
  template<floating_point T> inline constexpr T sqrt2_v<T>      = see below;
  template<floating_point T> inline constexpr T sqrt3_v<T>      = see below;
  template<floating_point T> inline constexpr T inv_sqrt3_v<T>  = see below;
  template<floating_point T> inline constexpr T egamma_v<T>     = see below;
  template<floating_point T> inline constexpr T phi_v<T>        = see below;

  inline constexpr double e          = e_v<double>;
  inline constexpr double log2e      = log2e_v<double>;
  inline constexpr double log10e     = log10e_v<double>;
  inline constexpr double pi         = pi_v<double>;
  inline constexpr double inv_pi     = inv_pi_v<double>;
  inline constexpr double inv_sqrtpi = inv_sqrtpi_v<double>;
  inline constexpr double ln2        = ln2_v<double>;
  inline constexpr double ln10       = ln10_v<double>;
  inline constexpr double sqrt2      = sqrt2_v<double>;
  inline constexpr double sqrt3      = sqrt3_v<double>;
  inline constexpr double inv_sqrt3  = inv_sqrt3_v<double>;
  inline constexpr double egamma     = egamma_v<double>;
  inline constexpr double phi        = phi_v<double>;
}

26.9.2 Mathematical constants [math.constants]

The library-defined partial specializations of mathematical constant variable templates are initialized with the nearest representable values of e,

log_{2} e

log_{10} e

, π,

\frac{1}{π}

\frac{1}{\sqrt{π}}

, ln2, ln10,

\sqrt{2}

\sqrt{3}

\frac{1}{\sqrt{3}}

, the Euler-Mascheroni γ constant, and the golden ratio φ constant

\frac{1 + \sqrt{5}}{2}

, respectively.

Pursuant to [namespace.std], a program may partially or explicitly specialize a mathematical constant variable template provided that the specialization depends on a program-defined type.

A program that instantiates a primary template of a mathematical constant variable template is ill-formed.

26 Numerics library [numerics]

26.1 General [numerics.general]

26.2 Numeric type requirements [numeric.requirements]

26.3 The floating-point environment [cfenv]

26.3.1 Header <cfenv> synopsis [cfenv.syn]

26.4 Complex numbers [complex.numbers]

26.4.1 Header <complex> synopsis [complex.syn]

26.4.2 Class template complex [complex]

26.4.3 Specializations [complex.special]

26.4.4 Member functions [complex.members]

26.4.5 Member operators [complex.member.ops]

26.4.6 Non-member operations [complex.ops]

26.4.7 Value operations [complex.value.ops]

26.4.8 Transcendentals [complex.transcendentals]

26.4.9 Additional overloads [cmplx.over]

26.4.10 Suffixes for complex number literals [complex.literals]

26.5 Bit manipulation [bit]

26.5.1 General [bit.general]

26.5.2 Header <bit> synopsis [bit.syn]

26.5.3 Function template bit_­cast [bit.cast]

26.5.4 Integral powers of 2 [bit.pow.two]

26.5.5 Rotating [bit.rotate]

26.5.6 Counting [bit.count]

26.5.7 Endian [bit.endian]

26.6 Random number generation [rand]

26.6.1 Header <random> synopsis [rand.synopsis]

26.6.2 Requirements [rand.req]

26.6.2.1 General requirements [rand.req.genl]

26.6.2.2 Seed sequence requirements [rand.req.seedseq]

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

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

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

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

26.6.3 Random number engine class templates [rand.eng]

26.6.3.1 Class template linear_­congruential_­engine [rand.eng.lcong]

26.6.3.2 Class template mersenne_­twister_­engine [rand.eng.mers]

26.6.3.3 Class template subtract_­with_­carry_­engine [rand.eng.sub]

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

26.6.4.1 In general [rand.adapt.general]

26.6.4.2 Class template discard_­block_­engine [rand.adapt.disc]

26.6.4.3 Class template independent_­bits_­engine [rand.adapt.ibits]

26.6.4.4 Class template shuffle_­order_­engine [rand.adapt.shuf]

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

26.6.6 Class random_­device [rand.device]

26.6.7 Utilities [rand.util]

26.6.7.1 Class seed_­seq [rand.util.seedseq]

26.6.7.2 Function template generate_­canonical [rand.util.canonical]

26.6.8 Random number distribution class templates [rand.dist]

26.6.8.1 In general [rand.dist.general]

26.6.8.2 Uniform distributions [rand.dist.uni]

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

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

26.6.8.3 Bernoulli distributions [rand.dist.bern]

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

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

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

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

26.6.8.4 Poisson distributions [rand.dist.pois]

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

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

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

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

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

26.6.8.5 Normal distributions [rand.dist.norm]

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

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

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

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

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

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

26.6.8.6 Sampling distributions [rand.dist.samp]

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

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

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

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

26.7 Numeric arrays [numarray]

26.7.1 Header <valarray> synopsis [valarray.syn]

26.7.2 Class template valarray [template.valarray]

26.7.2.1 Overview [template.valarray.overview]

26.7.2.2 Constructors [valarray.cons]

26.5.3 Function template bit_cast [bit.cast]

26.6.3.1 Class template linear_congruential_engine [rand.eng.lcong]

26.6.3.2 Class template mersenne_twister_engine [rand.eng.mers]

26.6.3.3 Class template subtract_with_carry_engine [rand.eng.sub]

26.6.4.2 Class template discard_block_engine [rand.adapt.disc]

26.6.4.3 Class template independent_bits_engine [rand.adapt.ibits]

26.6.4.4 Class template shuffle_order_engine [rand.adapt.shuf]

26.6.6 Class random_device [rand.device]

26.6.7.1 Class seed_seq [rand.util.seedseq]

26.6.7.2 Function template generate_canonical [rand.util.canonical]

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

26.7.5 Class template slice_array [template.slice.array]

26.7.7 Class template gslice_array [template.gslice.array]

26.7.8 Class template mask_array [template.mask.array]

26.7.9 Class template indirect_array [template.indirect.array]