// -*- C++ -*-
//===----------------------------------------------------------------------===//
//
// Part of the LLVM Project, under the Apache License v2.0 with LLVM Exceptions.
// See https://llvm.org/LICENSE.txt for license information.
// SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception
//
//===----------------------------------------------------------------------===//

#ifndef _LIBCPP_COMPLEX
#define _LIBCPP_COMPLEX

/*
    complex synopsis

namespace std
{

template<class T>
class complex
{
public:
    typedef T value_type;

    complex(const T& re = T(), const T& im = T()); // constexpr in C++14
    complex(const complex&);  // constexpr in C++14
    template<class X> complex(const complex<X>&);  // constexpr in C++14

    T real() const; // constexpr in C++14
    T imag() const; // constexpr in C++14

    void real(T); // constexpr in C++20
    void imag(T); // constexpr in C++20

    complex<T>& operator= (const T&); // constexpr in C++20
    complex<T>& operator+=(const T&); // constexpr in C++20
    complex<T>& operator-=(const T&); // constexpr in C++20
    complex<T>& operator*=(const T&); // constexpr in C++20
    complex<T>& operator/=(const T&); // constexpr in C++20

    complex& operator=(const complex&); // constexpr in C++20
    template<class X> complex<T>& operator= (const complex<X>&); // constexpr in C++20
    template<class X> complex<T>& operator+=(const complex<X>&); // constexpr in C++20
    template<class X> complex<T>& operator-=(const complex<X>&); // constexpr in C++20
    template<class X> complex<T>& operator*=(const complex<X>&); // constexpr in C++20
    template<class X> complex<T>& operator/=(const complex<X>&); // constexpr in C++20
};

template<>
class complex<float>
{
public:
    typedef float value_type;

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

    constexpr float real() const;
    void real(float); // constexpr in C++20
    constexpr float imag() const;
    void imag(float); // constexpr in C++20

    complex<float>& operator= (float); // constexpr in C++20
    complex<float>& operator+=(float); // constexpr in C++20
    complex<float>& operator-=(float); // constexpr in C++20
    complex<float>& operator*=(float); // constexpr in C++20
    complex<float>& operator/=(float); // constexpr in C++20

    complex<float>& operator=(const complex<float>&); // constexpr in C++20
    template<class X> complex<float>& operator= (const complex<X>&); // constexpr in C++20
    template<class X> complex<float>& operator+=(const complex<X>&); // constexpr in C++20
    template<class X> complex<float>& operator-=(const complex<X>&); // constexpr in C++20
    template<class X> complex<float>& operator*=(const complex<X>&); // constexpr in C++20
    template<class X> complex<float>& operator/=(const complex<X>&); // constexpr in C++20
};

template<>
class complex<double>
{
public:
    typedef double value_type;

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

    constexpr double real() const;
    void real(double); // constexpr in C++20
    constexpr double imag() const;
    void imag(double); // constexpr in C++20

    complex<double>& operator= (double); // constexpr in C++20
    complex<double>& operator+=(double); // constexpr in C++20
    complex<double>& operator-=(double); // constexpr in C++20
    complex<double>& operator*=(double); // constexpr in C++20
    complex<double>& operator/=(double); // constexpr in C++20
    complex<double>& operator=(const complex<double>&); // constexpr in C++20

    template<class X> complex<double>& operator= (const complex<X>&); // constexpr in C++20
    template<class X> complex<double>& operator+=(const complex<X>&); // constexpr in C++20
    template<class X> complex<double>& operator-=(const complex<X>&); // constexpr in C++20
    template<class X> complex<double>& operator*=(const complex<X>&); // constexpr in C++20
    template<class X> complex<double>& operator/=(const complex<X>&); // constexpr in C++20
};

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

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

    constexpr long double real() const;
    void real(long double); // constexpr in C++20
    constexpr long double imag() const;
    void imag(long double); // constexpr in C++20

    complex<long double>& operator=(const complex<long double>&); // constexpr in C++20
    complex<long double>& operator= (long double); // constexpr in C++20
    complex<long double>& operator+=(long double); // constexpr in C++20
    complex<long double>& operator-=(long double); // constexpr in C++20
    complex<long double>& operator*=(long double); // constexpr in C++20
    complex<long double>& operator/=(long double); // constexpr in C++20

    template<class X> complex<long double>& operator= (const complex<X>&); // constexpr in C++20
    template<class X> complex<long double>& operator+=(const complex<X>&); // constexpr in C++20
    template<class X> complex<long double>& operator-=(const complex<X>&); // constexpr in C++20
    template<class X> complex<long double>& operator*=(const complex<X>&); // constexpr in C++20
    template<class X> complex<long double>& operator/=(const complex<X>&); // constexpr in C++20
};

// 26.3.6 operators:
template<class T> complex<T> operator+(const complex<T>&, const complex<T>&); // constexpr in C++20
template<class T> complex<T> operator+(const complex<T>&, const T&);          // constexpr in C++20
template<class T> complex<T> operator+(const T&, const complex<T>&);          // constexpr in C++20
template<class T> complex<T> operator-(const complex<T>&, const complex<T>&); // constexpr in C++20
template<class T> complex<T> operator-(const complex<T>&, const T&);          // constexpr in C++20
template<class T> complex<T> operator-(const T&, const complex<T>&);          // constexpr in C++20
template<class T> complex<T> operator*(const complex<T>&, const complex<T>&); // constexpr in C++20
template<class T> complex<T> operator*(const complex<T>&, const T&);          // constexpr in C++20
template<class T> complex<T> operator*(const T&, const complex<T>&);          // constexpr in C++20
template<class T> complex<T> operator/(const complex<T>&, const complex<T>&); // constexpr in C++20
template<class T> complex<T> operator/(const complex<T>&, const T&);          // constexpr in C++20
template<class T> complex<T> operator/(const T&, const complex<T>&);          // constexpr in C++20
template<class T> complex<T> operator+(const complex<T>&);                    // constexpr in C++20
template<class T> complex<T> operator-(const complex<T>&);                    // constexpr in C++20
template<class T> bool operator==(const complex<T>&, const complex<T>&);      // constexpr in C++14
template<class T> bool operator==(const complex<T>&, const T&);               // constexpr in C++14
template<class T> bool operator==(const T&, const complex<T>&);               // constexpr in C++14, removed in C++20
template<class T> bool operator!=(const complex<T>&, const complex<T>&);      // constexpr in C++14, removed in C++20
template<class T> bool operator!=(const complex<T>&, const T&);               // constexpr in C++14, removed in C++20
template<class T> bool operator!=(const T&, const complex<T>&);               // constexpr in C++14, removed in C++20

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>&);

// 26.3.7 values:

template<class T>              T real(const complex<T>&); // constexpr in C++14
                     long double real(long double);       // constexpr in C++14
                          double real(double);            // constexpr in C++14
template<Integral T>      double real(T);                 // constexpr in C++14
                          float  real(float);             // constexpr in C++14

template<class T>              T imag(const complex<T>&); // constexpr in C++14
                     long double imag(long double);       // constexpr in C++14
                          double imag(double);            // constexpr in C++14
template<Integral T>      double imag(T);                 // constexpr in C++14
                          float  imag(float);             // constexpr in C++14

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

template<class T>              T arg(const complex<T>&);
                     long double arg(long double);
                          double arg(double);
template<Integral T>      double arg(T);
                          float  arg(float);

template<class T>              T norm(const complex<T>&); // constexpr in C++20
                     long double norm(long double);       // constexpr in C++20
                          double norm(double);            // constexpr in C++20
template<Integral T>      double norm(T);                 // constexpr in C++20
                          float  norm(float);             // constexpr in C++20

template<class T>      complex<T>           conj(const complex<T>&); // constexpr in C++20
                       complex<long double> conj(long double);       // constexpr in C++20
                       complex<double>      conj(double);            // constexpr in C++20
template<Integral T>   complex<double>      conj(T);                 // constexpr in C++20
                       complex<float>       conj(float);             // constexpr in C++20

template<class T>    complex<T>           proj(const complex<T>&);
                     complex<long double> proj(long double);
                     complex<double>      proj(double);
template<Integral T> complex<double>      proj(T);
                     complex<float>       proj(float);

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

// 26.3.8 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.tuple], tuple interface
  template<class T> struct tuple_size;                               // Since C++26
  template<size_t I, class T> struct tuple_element;                  // Since C++26
  template<class T> struct tuple_size<complex<T>>;                   // Since C++26
  template<size_t I, class T> struct tuple_element<I, complex<T>>;   // Since C++26
  template<size_t I, class T>
    constexpr T& get(complex<T>&) noexcept;                          // Since C++26
  template<size_t I, class T>
    constexpr T&& get(complex<T>&&) noexcept;                        // Since C++26
  template<size_t I, class T>
    constexpr const T& get(const complex<T>&) noexcept;              // Since C++26
  template<size_t I, class T>
    constexpr const T&& get(const complex<T>&&) noexcept;            // Since C++26

  // [complex.literals], complex literals
  inline namespace literals {
  inline namespace complex_literals {
    constexpr complex<long double> operator""il(long double);        // Since C++14
    constexpr complex<long double> operator""il(unsigned long long); // Since C++14
    constexpr complex<double> operator""i(long double);              // Since C++14
    constexpr complex<double> operator""i(unsigned long long);       // Since C++14
    constexpr complex<float> operator""if(long double);              // Since C++14
    constexpr complex<float> operator""if(unsigned long long);       // Since C++14
  }
  }
}  // std

*/

#include <__config>
#include <__fwd/complex.h>
#include <__fwd/tuple.h>
#include <__tuple/tuple_element.h>
#include <__tuple/tuple_size.h>
#include <__utility/move.h>
#include <cmath>
#include <version>

#if !defined(_LIBCPP_HAS_NO_LOCALIZATION)
#  include <sstream> // for std::basic_ostringstream
#endif

#if !defined(_LIBCPP_HAS_NO_PRAGMA_SYSTEM_HEADER)
#  pragma GCC system_header
#endif

_LIBCPP_PUSH_MACROS
#include <__undef_macros>

_LIBCPP_BEGIN_NAMESPACE_STD

template <class _Tp>
class _LIBCPP_TEMPLATE_VIS complex;

template <class _Tp>
_LIBCPP_HIDE_FROM_ABI _LIBCPP_CONSTEXPR_SINCE_CXX20 complex<_Tp>
operator*(const complex<_Tp>& __z, const complex<_Tp>& __w);
template <class _Tp>
_LIBCPP_HIDE_FROM_ABI _LIBCPP_CONSTEXPR_SINCE_CXX20 complex<_Tp>
operator/(const complex<_Tp>& __x, const complex<_Tp>& __y);

template <class _Tp>
class _LIBCPP_TEMPLATE_VIS complex {
public:
  typedef _Tp value_type;

private:
  value_type __re_;
  value_type __im_;

public:
  _LIBCPP_HIDE_FROM_ABI _LIBCPP_CONSTEXPR_SINCE_CXX14
  complex(const value_type& __re = value_type(), const value_type& __im = value_type())
      : __re_(__re), __im_(__im) {}
  template <class _Xp>
  _LIBCPP_HIDE_FROM_ABI _LIBCPP_CONSTEXPR_SINCE_CXX14 complex(const complex<_Xp>& __c)
      : __re_(__c.real()), __im_(__c.imag()) {}

  _LIBCPP_HIDE_FROM_ABI _LIBCPP_CONSTEXPR_SINCE_CXX14 value_type real() const { return __re_; }
  _LIBCPP_HIDE_FROM_ABI _LIBCPP_CONSTEXPR_SINCE_CXX14 value_type imag() const { return __im_; }

  _LIBCPP_HIDE_FROM_ABI _LIBCPP_CONSTEXPR_SINCE_CXX20 void real(value_type __re) { __re_ = __re; }
  _LIBCPP_HIDE_FROM_ABI _LIBCPP_CONSTEXPR_SINCE_CXX20 void imag(value_type __im) { __im_ = __im; }

  _LIBCPP_HIDE_FROM_ABI _LIBCPP_CONSTEXPR_SINCE_CXX20 complex& operator=(const value_type& __re) {
    __re_ = __re;
    __im_ = value_type();
    return *this;
  }
  _LIBCPP_HIDE_FROM_ABI _LIBCPP_CONSTEXPR_SINCE_CXX20 complex& operator+=(const value_type& __re) {
    __re_ += __re;
    return *this;
  }
  _LIBCPP_HIDE_FROM_ABI _LIBCPP_CONSTEXPR_SINCE_CXX20 complex& operator-=(const value_type& __re) {
    __re_ -= __re;
    return *this;
  }
  _LIBCPP_HIDE_FROM_ABI _LIBCPP_CONSTEXPR_SINCE_CXX20 complex& operator*=(const value_type& __re) {
    __re_ *= __re;
    __im_ *= __re;
    return *this;
  }
  _LIBCPP_HIDE_FROM_ABI _LIBCPP_CONSTEXPR_SINCE_CXX20 complex& operator/=(const value_type& __re) {
    __re_ /= __re;
    __im_ /= __re;
    return *this;
  }

  template <class _Xp>
  _LIBCPP_HIDE_FROM_ABI _LIBCPP_CONSTEXPR_SINCE_CXX20 complex& operator=(const complex<_Xp>& __c) {
    __re_ = __c.real();
    __im_ = __c.imag();
    return *this;
  }
  template <class _Xp>
  _LIBCPP_HIDE_FROM_ABI _LIBCPP_CONSTEXPR_SINCE_CXX20 complex& operator+=(const complex<_Xp>& __c) {
    __re_ += __c.real();
    __im_ += __c.imag();
    return *this;
  }
  template <class _Xp>
  _LIBCPP_HIDE_FROM_ABI _LIBCPP_CONSTEXPR_SINCE_CXX20 complex& operator-=(const complex<_Xp>& __c) {
    __re_ -= __c.real();
    __im_ -= __c.imag();
    return *this;
  }
  template <class _Xp>
  _LIBCPP_HIDE_FROM_ABI _LIBCPP_CONSTEXPR_SINCE_CXX20 complex& operator*=(const complex<_Xp>& __c) {
    *this = *this * complex(__c.real(), __c.imag());
    return *this;
  }
  template <class _Xp>
  _LIBCPP_HIDE_FROM_ABI _LIBCPP_CONSTEXPR_SINCE_CXX20 complex& operator/=(const complex<_Xp>& __c) {
    *this = *this / complex(__c.real(), __c.imag());
    return *this;
  }

#if _LIBCPP_STD_VER >= 26
  template <size_t _Ip, class _Xp>
  friend _LIBCPP_HIDE_FROM_ABI constexpr _Xp& get(complex<_Xp>&) noexcept;

  template <size_t _Ip, class _Xp>
  friend _LIBCPP_HIDE_FROM_ABI constexpr _Xp&& get(complex<_Xp>&&) noexcept;

  template <size_t _Ip, class _Xp>
  friend _LIBCPP_HIDE_FROM_ABI constexpr const _Xp& get(const complex<_Xp>&) noexcept;

  template <size_t _Ip, class _Xp>
  friend _LIBCPP_HIDE_FROM_ABI constexpr const _Xp&& get(const complex<_Xp>&&) noexcept;
#endif
};

template <>
class _LIBCPP_TEMPLATE_VIS complex<double>;
template <>
class _LIBCPP_TEMPLATE_VIS complex<long double>;

template <>
class _LIBCPP_TEMPLATE_VIS complex<float> {
  float __re_;
  float __im_;

public:
  typedef float value_type;

  _LIBCPP_HIDE_FROM_ABI _LIBCPP_CONSTEXPR complex(float __re = 0.0f, float __im = 0.0f) : __re_(__re), __im_(__im) {}
  _LIBCPP_HIDE_FROM_ABI explicit _LIBCPP_CONSTEXPR complex(const complex<double>& __c);
  _LIBCPP_HIDE_FROM_ABI explicit _LIBCPP_CONSTEXPR complex(const complex<long double>& __c);

  _LIBCPP_HIDE_FROM_ABI _LIBCPP_CONSTEXPR float real() const { return __re_; }
  _LIBCPP_HIDE_FROM_ABI _LIBCPP_CONSTEXPR float imag() const { return __im_; }

  _LIBCPP_HIDE_FROM_ABI _LIBCPP_CONSTEXPR_SINCE_CXX20 void real(value_type __re) { __re_ = __re; }
  _LIBCPP_HIDE_FROM_ABI _LIBCPP_CONSTEXPR_SINCE_CXX20 void imag(value_type __im) { __im_ = __im; }

  _LIBCPP_HIDE_FROM_ABI _LIBCPP_CONSTEXPR_SINCE_CXX20 complex& operator=(float __re) {
    __re_ = __re;
    __im_ = value_type();
    return *this;
  }
  _LIBCPP_HIDE_FROM_ABI _LIBCPP_CONSTEXPR_SINCE_CXX20 complex& operator+=(float __re) {
    __re_ += __re;
    return *this;
  }
  _LIBCPP_HIDE_FROM_ABI _LIBCPP_CONSTEXPR_SINCE_CXX20 complex& operator-=(float __re) {
    __re_ -= __re;
    return *this;
  }
  _LIBCPP_HIDE_FROM_ABI _LIBCPP_CONSTEXPR_SINCE_CXX20 complex& operator*=(float __re) {
    __re_ *= __re;
    __im_ *= __re;
    return *this;
  }
  _LIBCPP_HIDE_FROM_ABI _LIBCPP_CONSTEXPR_SINCE_CXX20 complex& operator/=(float __re) {
    __re_ /= __re;
    __im_ /= __re;
    return *this;
  }

  template <class _Xp>
  _LIBCPP_HIDE_FROM_ABI _LIBCPP_CONSTEXPR_SINCE_CXX20 complex& operator=(const complex<_Xp>& __c) {
    __re_ = __c.real();
    __im_ = __c.imag();
    return *this;
  }
  template <class _Xp>
  _LIBCPP_HIDE_FROM_ABI _LIBCPP_CONSTEXPR_SINCE_CXX20 complex& operator+=(const complex<_Xp>& __c) {
    __re_ += __c.real();
    __im_ += __c.imag();
    return *this;
  }
  template <class _Xp>
  _LIBCPP_HIDE_FROM_ABI _LIBCPP_CONSTEXPR_SINCE_CXX20 complex& operator-=(const complex<_Xp>& __c) {
    __re_ -= __c.real();
    __im_ -= __c.imag();
    return *this;
  }
  template <class _Xp>
  _LIBCPP_HIDE_FROM_ABI _LIBCPP_CONSTEXPR_SINCE_CXX20 complex& operator*=(const complex<_Xp>& __c) {
    *this = *this * complex(__c.real(), __c.imag());
    return *this;
  }
  template <class _Xp>
  _LIBCPP_HIDE_FROM_ABI _LIBCPP_CONSTEXPR_SINCE_CXX20 complex& operator/=(const complex<_Xp>& __c) {
    *this = *this / complex(__c.real(), __c.imag());
    return *this;
  }

#if _LIBCPP_STD_VER >= 26
  template <size_t _Ip, class _Xp>
  friend _LIBCPP_HIDE_FROM_ABI constexpr _Xp& get(complex<_Xp>&) noexcept;

  template <size_t _Ip, class _Xp>
  friend _LIBCPP_HIDE_FROM_ABI constexpr _Xp&& get(complex<_Xp>&&) noexcept;

  template <size_t _Ip, class _Xp>
  friend _LIBCPP_HIDE_FROM_ABI constexpr const _Xp& get(const complex<_Xp>&) noexcept;

  template <size_t _Ip, class _Xp>
  friend _LIBCPP_HIDE_FROM_ABI constexpr const _Xp&& get(const complex<_Xp>&&) noexcept;
#endif
};

template <>
class _LIBCPP_TEMPLATE_VIS complex<double> {
  double __re_;
  double __im_;

public:
  typedef double value_type;

  _LIBCPP_HIDE_FROM_ABI _LIBCPP_CONSTEXPR complex(double __re = 0.0, double __im = 0.0) : __re_(__re), __im_(__im) {}
  _LIBCPP_HIDE_FROM_ABI _LIBCPP_CONSTEXPR complex(const complex<float>& __c);
  _LIBCPP_HIDE_FROM_ABI explicit _LIBCPP_CONSTEXPR complex(const complex<long double>& __c);

  _LIBCPP_HIDE_FROM_ABI _LIBCPP_CONSTEXPR double real() const { return __re_; }
  _LIBCPP_HIDE_FROM_ABI _LIBCPP_CONSTEXPR double imag() const { return __im_; }

  _LIBCPP_HIDE_FROM_ABI _LIBCPP_CONSTEXPR_SINCE_CXX20 void real(value_type __re) { __re_ = __re; }
  _LIBCPP_HIDE_FROM_ABI _LIBCPP_CONSTEXPR_SINCE_CXX20 void imag(value_type __im) { __im_ = __im; }

  _LIBCPP_HIDE_FROM_ABI _LIBCPP_CONSTEXPR_SINCE_CXX20 complex& operator=(double __re) {
    __re_ = __re;
    __im_ = value_type();
    return *this;
  }
  _LIBCPP_HIDE_FROM_ABI _LIBCPP_CONSTEXPR_SINCE_CXX20 complex& operator+=(double __re) {
    __re_ += __re;
    return *this;
  }
  _LIBCPP_HIDE_FROM_ABI _LIBCPP_CONSTEXPR_SINCE_CXX20 complex& operator-=(double __re) {
    __re_ -= __re;
    return *this;
  }
  _LIBCPP_HIDE_FROM_ABI _LIBCPP_CONSTEXPR_SINCE_CXX20 complex& operator*=(double __re) {
    __re_ *= __re;
    __im_ *= __re;
    return *this;
  }
  _LIBCPP_HIDE_FROM_ABI _LIBCPP_CONSTEXPR_SINCE_CXX20 complex& operator/=(double __re) {
    __re_ /= __re;
    __im_ /= __re;
    return *this;
  }

  template <class _Xp>
  _LIBCPP_HIDE_FROM_ABI _LIBCPP_CONSTEXPR_SINCE_CXX20 complex& operator=(const complex<_Xp>& __c) {
    __re_ = __c.real();
    __im_ = __c.imag();
    return *this;
  }
  template <class _Xp>
  _LIBCPP_HIDE_FROM_ABI _LIBCPP_CONSTEXPR_SINCE_CXX20 complex& operator+=(const complex<_Xp>& __c) {
    __re_ += __c.real();
    __im_ += __c.imag();
    return *this;
  }
  template <class _Xp>
  _LIBCPP_HIDE_FROM_ABI _LIBCPP_CONSTEXPR_SINCE_CXX20 complex& operator-=(const complex<_Xp>& __c) {
    __re_ -= __c.real();
    __im_ -= __c.imag();
    return *this;
  }
  template <class _Xp>
  _LIBCPP_HIDE_FROM_ABI _LIBCPP_CONSTEXPR_SINCE_CXX20 complex& operator*=(const complex<_Xp>& __c) {
    *this = *this * complex(__c.real(), __c.imag());
    return *this;
  }
  template <class _Xp>
  _LIBCPP_HIDE_FROM_ABI _LIBCPP_CONSTEXPR_SINCE_CXX20 complex& operator/=(const complex<_Xp>& __c) {
    *this = *this / complex(__c.real(), __c.imag());
    return *this;
  }

#if _LIBCPP_STD_VER >= 26
  template <size_t _Ip, class _Xp>
  friend _LIBCPP_HIDE_FROM_ABI constexpr _Xp& get(complex<_Xp>&) noexcept;

  template <size_t _Ip, class _Xp>
  friend _LIBCPP_HIDE_FROM_ABI constexpr _Xp&& get(complex<_Xp>&&) noexcept;

  template <size_t _Ip, class _Xp>
  friend _LIBCPP_HIDE_FROM_ABI constexpr const _Xp& get(const complex<_Xp>&) noexcept;

  template <size_t _Ip, class _Xp>
  friend _LIBCPP_HIDE_FROM_ABI constexpr const _Xp&& get(const complex<_Xp>&&) noexcept;
#endif
};

template <>
class _LIBCPP_TEMPLATE_VIS complex<long double> {
  long double __re_;
  long double __im_;

public:
  typedef long double value_type;

  _LIBCPP_HIDE_FROM_ABI _LIBCPP_CONSTEXPR complex(long double __re = 0.0L, long double __im = 0.0L)
      : __re_(__re), __im_(__im) {}
  _LIBCPP_HIDE_FROM_ABI _LIBCPP_CONSTEXPR complex(const complex<float>& __c);
  _LIBCPP_HIDE_FROM_ABI _LIBCPP_CONSTEXPR complex(const complex<double>& __c);

  _LIBCPP_HIDE_FROM_ABI _LIBCPP_CONSTEXPR long double real() const { return __re_; }
  _LIBCPP_HIDE_FROM_ABI _LIBCPP_CONSTEXPR long double imag() const { return __im_; }

  _LIBCPP_HIDE_FROM_ABI _LIBCPP_CONSTEXPR_SINCE_CXX20 void real(value_type __re) { __re_ = __re; }
  _LIBCPP_HIDE_FROM_ABI _LIBCPP_CONSTEXPR_SINCE_CXX20 void imag(value_type __im) { __im_ = __im; }

  _LIBCPP_HIDE_FROM_ABI _LIBCPP_CONSTEXPR_SINCE_CXX20 complex& operator=(long double __re) {
    __re_ = __re;
    __im_ = value_type();
    return *this;
  }
  _LIBCPP_HIDE_FROM_ABI _LIBCPP_CONSTEXPR_SINCE_CXX20 complex& operator+=(long double __re) {
    __re_ += __re;
    return *this;
  }
  _LIBCPP_HIDE_FROM_ABI _LIBCPP_CONSTEXPR_SINCE_CXX20 complex& operator-=(long double __re) {
    __re_ -= __re;
    return *this;
  }
  _LIBCPP_HIDE_FROM_ABI _LIBCPP_CONSTEXPR_SINCE_CXX20 complex& operator*=(long double __re) {
    __re_ *= __re;
    __im_ *= __re;
    return *this;
  }
  _LIBCPP_HIDE_FROM_ABI _LIBCPP_CONSTEXPR_SINCE_CXX20 complex& operator/=(long double __re) {
    __re_ /= __re;
    __im_ /= __re;
    return *this;
  }

  template <class _Xp>
  _LIBCPP_HIDE_FROM_ABI _LIBCPP_CONSTEXPR_SINCE_CXX20 complex& operator=(const complex<_Xp>& __c) {
    __re_ = __c.real();
    __im_ = __c.imag();
    return *this;
  }
  template <class _Xp>
  _LIBCPP_HIDE_FROM_ABI _LIBCPP_CONSTEXPR_SINCE_CXX20 complex& operator+=(const complex<_Xp>& __c) {
    __re_ += __c.real();
    __im_ += __c.imag();
    return *this;
  }
  template <class _Xp>
  _LIBCPP_HIDE_FROM_ABI _LIBCPP_CONSTEXPR_SINCE_CXX20 complex& operator-=(const complex<_Xp>& __c) {
    __re_ -= __c.real();
    __im_ -= __c.imag();
    return *this;
  }
  template <class _Xp>
  _LIBCPP_HIDE_FROM_ABI _LIBCPP_CONSTEXPR_SINCE_CXX20 complex& operator*=(const complex<_Xp>& __c) {
    *this = *this * complex(__c.real(), __c.imag());
    return *this;
  }
  template <class _Xp>
  _LIBCPP_HIDE_FROM_ABI _LIBCPP_CONSTEXPR_SINCE_CXX20 complex& operator/=(const complex<_Xp>& __c) {
    *this = *this / complex(__c.real(), __c.imag());
    return *this;
  }

#if _LIBCPP_STD_VER >= 26
  template <size_t _Ip, class _Xp>
  friend _LIBCPP_HIDE_FROM_ABI constexpr _Xp& get(complex<_Xp>&) noexcept;

  template <size_t _Ip, class _Xp>
  friend _LIBCPP_HIDE_FROM_ABI constexpr _Xp&& get(complex<_Xp>&&) noexcept;

  template <size_t _Ip, class _Xp>
  friend _LIBCPP_HIDE_FROM_ABI constexpr const _Xp& get(const complex<_Xp>&) noexcept;

  template <size_t _Ip, class _Xp>
  friend _LIBCPP_HIDE_FROM_ABI constexpr const _Xp&& get(const complex<_Xp>&&) noexcept;
#endif
};

inline _LIBCPP_CONSTEXPR complex<float>::complex(const complex<double>& __c) : __re_(__c.real()), __im_(__c.imag()) {}

inline _LIBCPP_CONSTEXPR complex<float>::complex(const complex<long double>& __c)
    : __re_(__c.real()), __im_(__c.imag()) {}

inline _LIBCPP_CONSTEXPR complex<double>::complex(const complex<float>& __c) : __re_(__c.real()), __im_(__c.imag()) {}

inline _LIBCPP_CONSTEXPR complex<double>::complex(const complex<long double>& __c)
    : __re_(__c.real()), __im_(__c.imag()) {}

inline _LIBCPP_CONSTEXPR complex<long double>::complex(const complex<float>& __c)
    : __re_(__c.real()), __im_(__c.imag()) {}

inline _LIBCPP_CONSTEXPR complex<long double>::complex(const complex<double>& __c)
    : __re_(__c.real()), __im_(__c.imag()) {}

// 26.3.6 operators:

template <class _Tp>
inline _LIBCPP_HIDE_FROM_ABI _LIBCPP_CONSTEXPR_SINCE_CXX20 complex<_Tp>
operator+(const complex<_Tp>& __x, const complex<_Tp>& __y) {
  complex<_Tp> __t(__x);
  __t += __y;
  return __t;
}

template <class _Tp>
inline _LIBCPP_HIDE_FROM_ABI _LIBCPP_CONSTEXPR_SINCE_CXX20 complex<_Tp>
operator+(const complex<_Tp>& __x, const _Tp& __y) {
  complex<_Tp> __t(__x);
  __t += __y;
  return __t;
}

template <class _Tp>
inline _LIBCPP_HIDE_FROM_ABI _LIBCPP_CONSTEXPR_SINCE_CXX20 complex<_Tp>
operator+(const _Tp& __x, const complex<_Tp>& __y) {
  complex<_Tp> __t(__y);
  __t += __x;
  return __t;
}

template <class _Tp>
inline _LIBCPP_HIDE_FROM_ABI _LIBCPP_CONSTEXPR_SINCE_CXX20 complex<_Tp>
operator-(const complex<_Tp>& __x, const complex<_Tp>& __y) {
  complex<_Tp> __t(__x);
  __t -= __y;
  return __t;
}

template <class _Tp>
inline _LIBCPP_HIDE_FROM_ABI _LIBCPP_CONSTEXPR_SINCE_CXX20 complex<_Tp>
operator-(const complex<_Tp>& __x, const _Tp& __y) {
  complex<_Tp> __t(__x);
  __t -= __y;
  return __t;
}

template <class _Tp>
inline _LIBCPP_HIDE_FROM_ABI _LIBCPP_CONSTEXPR_SINCE_CXX20 complex<_Tp>
operator-(const _Tp& __x, const complex<_Tp>& __y) {
  complex<_Tp> __t(-__y);
  __t += __x;
  return __t;
}

template <class _Tp>
_LIBCPP_HIDE_FROM_ABI _LIBCPP_CONSTEXPR_SINCE_CXX20 complex<_Tp>
operator*(const complex<_Tp>& __z, const complex<_Tp>& __w) {
  _Tp __a = __z.real();
  _Tp __b = __z.imag();
  _Tp __c = __w.real();
  _Tp __d = __w.imag();

  // Avoid floating point operations that are invalid during constant evaluation
  if (__libcpp_is_constant_evaluated()) {
    bool __z_zero = __a == _Tp(0) && __b == _Tp(0);
    bool __w_zero = __c == _Tp(0) && __d == _Tp(0);
    bool __z_inf  = std::__constexpr_isinf(__a) || std::__constexpr_isinf(__b);
    bool __w_inf  = std::__constexpr_isinf(__c) || std::__constexpr_isinf(__d);
    bool __z_nan =
        !__z_inf && ((std::__constexpr_isnan(__a) && std::__constexpr_isnan(__b)) ||
                     (std::__constexpr_isnan(__a) && __b == _Tp(0)) || (__a == _Tp(0) && std::__constexpr_isnan(__b)));
    bool __w_nan =
        !__w_inf && ((std::__constexpr_isnan(__c) && std::__constexpr_isnan(__d)) ||
                     (std::__constexpr_isnan(__c) && __d == _Tp(0)) || (__c == _Tp(0) && std::__constexpr_isnan(__d)));
    if (__z_nan || __w_nan) {
      return complex<_Tp>(_Tp(numeric_limits<_Tp>::quiet_NaN()), _Tp(0));
    }
    if (__z_inf || __w_inf) {
      if (__z_zero || __w_zero) {
        return complex<_Tp>(_Tp(numeric_limits<_Tp>::quiet_NaN()), _Tp(0));
      }
      return complex<_Tp>(_Tp(numeric_limits<_Tp>::infinity()), _Tp(numeric_limits<_Tp>::infinity()));
    }
    bool __z_nonzero_nan = !__z_inf && !__z_nan && (std::__constexpr_isnan(__a) || std::__constexpr_isnan(__b));
    bool __w_nonzero_nan = !__w_inf && !__w_nan && (std::__constexpr_isnan(__c) || std::__constexpr_isnan(__d));
    if (__z_nonzero_nan || __w_nonzero_nan) {
      return complex<_Tp>(_Tp(numeric_limits<_Tp>::quiet_NaN()), _Tp(0));
    }
  }

  _Tp __ac = __a * __c;
  _Tp __bd = __b * __d;
  _Tp __ad = __a * __d;
  _Tp __bc = __b * __c;
  _Tp __x  = __ac - __bd;
  _Tp __y  = __ad + __bc;
  if (std::__constexpr_isnan(__x) && std::__constexpr_isnan(__y)) {
    bool __recalc = false;
    if (std::__constexpr_isinf(__a) || std::__constexpr_isinf(__b)) {
      __a = std::__constexpr_copysign(std::__constexpr_isinf(__a) ? _Tp(1) : _Tp(0), __a);
      __b = std::__constexpr_copysign(std::__constexpr_isinf(__b) ? _Tp(1) : _Tp(0), __b);
      if (std::__constexpr_isnan(__c))
        __c = std::__constexpr_copysign(_Tp(0), __c);
      if (std::__constexpr_isnan(__d))
        __d = std::__constexpr_copysign(_Tp(0), __d);
      __recalc = true;
    }
    if (std::__constexpr_isinf(__c) || std::__constexpr_isinf(__d)) {
      __c = std::__constexpr_copysign(std::__constexpr_isinf(__c) ? _Tp(1) : _Tp(0), __c);
      __d = std::__constexpr_copysign(std::__constexpr_isinf(__d) ? _Tp(1) : _Tp(0), __d);
      if (std::__constexpr_isnan(__a))
        __a = std::__constexpr_copysign(_Tp(0), __a);
      if (std::__constexpr_isnan(__b))
        __b = std::__constexpr_copysign(_Tp(0), __b);
      __recalc = true;
    }
    if (!__recalc && (std::__constexpr_isinf(__ac) || std::__constexpr_isinf(__bd) || std::__constexpr_isinf(__ad) ||
                      std::__constexpr_isinf(__bc))) {
      if (std::__constexpr_isnan(__a))
        __a = std::__constexpr_copysign(_Tp(0), __a);
      if (std::__constexpr_isnan(__b))
        __b = std::__constexpr_copysign(_Tp(0), __b);
      if (std::__constexpr_isnan(__c))
        __c = std::__constexpr_copysign(_Tp(0), __c);
      if (std::__constexpr_isnan(__d))
        __d = std::__constexpr_copysign(_Tp(0), __d);
      __recalc = true;
    }
    if (__recalc) {
      __x = _Tp(INFINITY) * (__a * __c - __b * __d);
      __y = _Tp(INFINITY) * (__a * __d + __b * __c);
    }
  }
  return complex<_Tp>(__x, __y);
}

template <class _Tp>
inline _LIBCPP_HIDE_FROM_ABI _LIBCPP_CONSTEXPR_SINCE_CXX20 complex<_Tp>
operator*(const complex<_Tp>& __x, const _Tp& __y) {
  complex<_Tp> __t(__x);
  __t *= __y;
  return __t;
}

template <class _Tp>
inline _LIBCPP_HIDE_FROM_ABI _LIBCPP_CONSTEXPR_SINCE_CXX20 complex<_Tp>
operator*(const _Tp& __x, const complex<_Tp>& __y) {
  complex<_Tp> __t(__y);
  __t *= __x;
  return __t;
}

template <class _Tp>
_LIBCPP_HIDE_FROM_ABI _LIBCPP_CONSTEXPR_SINCE_CXX20 complex<_Tp>
operator/(const complex<_Tp>& __z, const complex<_Tp>& __w) {
  int __ilogbw = 0;
  _Tp __a      = __z.real();
  _Tp __b      = __z.imag();
  _Tp __c      = __w.real();
  _Tp __d      = __w.imag();
  _Tp __logbw  = std::__constexpr_logb(std::__constexpr_fmax(std::__constexpr_fabs(__c), std::__constexpr_fabs(__d)));
  if (std::__constexpr_isfinite(__logbw)) {
    __ilogbw = static_cast<int>(__logbw);
    __c      = std::__constexpr_scalbn(__c, -__ilogbw);
    __d      = std::__constexpr_scalbn(__d, -__ilogbw);
  }

  // Avoid floating point operations that are invalid during constant evaluation
  if (__libcpp_is_constant_evaluated()) {
    bool __z_zero = __a == _Tp(0) && __b == _Tp(0);
    bool __w_zero = __c == _Tp(0) && __d == _Tp(0);
    bool __z_inf  = std::__constexpr_isinf(__a) || std::__constexpr_isinf(__b);
    bool __w_inf  = std::__constexpr_isinf(__c) || std::__constexpr_isinf(__d);
    bool __z_nan =
        !__z_inf && ((std::__constexpr_isnan(__a) && std::__constexpr_isnan(__b)) ||
                     (std::__constexpr_isnan(__a) && __b == _Tp(0)) || (__a == _Tp(0) && std::__constexpr_isnan(__b)));
    bool __w_nan =
        !__w_inf && ((std::__constexpr_isnan(__c) && std::__constexpr_isnan(__d)) ||
                     (std::__constexpr_isnan(__c) && __d == _Tp(0)) || (__c == _Tp(0) && std::__constexpr_isnan(__d)));
    if ((__z_nan || __w_nan) || (__z_inf && __w_inf)) {
      return complex<_Tp>(_Tp(numeric_limits<_Tp>::quiet_NaN()), _Tp(0));
    }
    bool __z_nonzero_nan = !__z_inf && !__z_nan && (std::__constexpr_isnan(__a) || std::__constexpr_isnan(__b));
    bool __w_nonzero_nan = !__w_inf && !__w_nan && (std::__constexpr_isnan(__c) || std::__constexpr_isnan(__d));
    if (__z_nonzero_nan || __w_nonzero_nan) {
      if (__w_zero) {
        return complex<_Tp>(_Tp(numeric_limits<_Tp>::infinity()), _Tp(numeric_limits<_Tp>::infinity()));
      }
      return complex<_Tp>(_Tp(numeric_limits<_Tp>::quiet_NaN()), _Tp(0));
    }
    if (__w_inf) {
      return complex<_Tp>(_Tp(0), _Tp(0));
    }
    if (__z_inf) {
      return complex<_Tp>(_Tp(numeric_limits<_Tp>::infinity()), _Tp(numeric_limits<_Tp>::infinity()));
    }
    if (__w_zero) {
      if (__z_zero) {
        return complex<_Tp>(_Tp(numeric_limits<_Tp>::quiet_NaN()), _Tp(0));
      }
      return complex<_Tp>(_Tp(numeric_limits<_Tp>::infinity()), _Tp(numeric_limits<_Tp>::infinity()));
    }
  }

  _Tp __denom = __c * __c + __d * __d;
  _Tp __x     = std::__constexpr_scalbn((__a * __c + __b * __d) / __denom, -__ilogbw);
  _Tp __y     = std::__constexpr_scalbn((__b * __c - __a * __d) / __denom, -__ilogbw);
  if (std::__constexpr_isnan(__x) && std::__constexpr_isnan(__y)) {
    if ((__denom == _Tp(0)) && (!std::__constexpr_isnan(__a) || !std::__constexpr_isnan(__b))) {
      __x = std::__constexpr_copysign(_Tp(INFINITY), __c) * __a;
      __y = std::__constexpr_copysign(_Tp(INFINITY), __c) * __b;
    } else if ((std::__constexpr_isinf(__a) || std::__constexpr_isinf(__b)) && std::__constexpr_isfinite(__c) &&
               std::__constexpr_isfinite(__d)) {
      __a = std::__constexpr_copysign(std::__constexpr_isinf(__a) ? _Tp(1) : _Tp(0), __a);
      __b = std::__constexpr_copysign(std::__constexpr_isinf(__b) ? _Tp(1) : _Tp(0), __b);
      __x = _Tp(INFINITY) * (__a * __c + __b * __d);
      __y = _Tp(INFINITY) * (__b * __c - __a * __d);
    } else if (std::__constexpr_isinf(__logbw) && __logbw > _Tp(0) && std::__constexpr_isfinite(__a) &&
               std::__constexpr_isfinite(__b)) {
      __c = std::__constexpr_copysign(std::__constexpr_isinf(__c) ? _Tp(1) : _Tp(0), __c);
      __d = std::__constexpr_copysign(std::__constexpr_isinf(__d) ? _Tp(1) : _Tp(0), __d);
      __x = _Tp(0) * (__a * __c + __b * __d);
      __y = _Tp(0) * (__b * __c - __a * __d);
    }
  }
  return complex<_Tp>(__x, __y);
}

template <class _Tp>
inline _LIBCPP_HIDE_FROM_ABI _LIBCPP_CONSTEXPR_SINCE_CXX20 complex<_Tp>
operator/(const complex<_Tp>& __x, const _Tp& __y) {
  return complex<_Tp>(__x.real() / __y, __x.imag() / __y);
}

template <class _Tp>
inline _LIBCPP_HIDE_FROM_ABI _LIBCPP_CONSTEXPR_SINCE_CXX20 complex<_Tp>
operator/(const _Tp& __x, const complex<_Tp>& __y) {
  complex<_Tp> __t(__x);
  __t /= __y;
  return __t;
}

template <class _Tp>
inline _LIBCPP_HIDE_FROM_ABI _LIBCPP_CONSTEXPR_SINCE_CXX20 complex<_Tp> operator+(const complex<_Tp>& __x) {
  return __x;
}

template <class _Tp>
inline _LIBCPP_HIDE_FROM_ABI _LIBCPP_CONSTEXPR_SINCE_CXX20 complex<_Tp> operator-(const complex<_Tp>& __x) {
  return complex<_Tp>(-__x.real(), -__x.imag());
}

template <class _Tp>
inline _LIBCPP_HIDE_FROM_ABI _LIBCPP_CONSTEXPR_SINCE_CXX14 bool
operator==(const complex<_Tp>& __x, const complex<_Tp>& __y) {
  return __x.real() == __y.real() && __x.imag() == __y.imag();
}

template <class _Tp>
inline _LIBCPP_HIDE_FROM_ABI _LIBCPP_CONSTEXPR_SINCE_CXX14 bool operator==(const complex<_Tp>& __x, const _Tp& __y) {
  return __x.real() == __y && __x.imag() == 0;
}

#if _LIBCPP_STD_VER <= 17

template <class _Tp>
inline _LIBCPP_HIDE_FROM_ABI _LIBCPP_CONSTEXPR_SINCE_CXX14 bool operator==(const _Tp& __x, const complex<_Tp>& __y) {
  return __x == __y.real() && 0 == __y.imag();
}

template <class _Tp>
inline _LIBCPP_HIDE_FROM_ABI _LIBCPP_CONSTEXPR_SINCE_CXX14 bool
operator!=(const complex<_Tp>& __x, const complex<_Tp>& __y) {
  return !(__x == __y);
}

template <class _Tp>
inline _LIBCPP_HIDE_FROM_ABI _LIBCPP_CONSTEXPR_SINCE_CXX14 bool operator!=(const complex<_Tp>& __x, const _Tp& __y) {
  return !(__x == __y);
}

template <class _Tp>
inline _LIBCPP_HIDE_FROM_ABI _LIBCPP_CONSTEXPR_SINCE_CXX14 bool operator!=(const _Tp& __x, const complex<_Tp>& __y) {
  return !(__x == __y);
}

#endif

// 26.3.7 values:

template <class _Tp, bool = is_integral<_Tp>::value, bool = is_floating_point<_Tp>::value >
struct __libcpp_complex_overload_traits {};

// Integral Types
template <class _Tp>
struct __libcpp_complex_overload_traits<_Tp, true, false> {
  typedef double _ValueType;
  typedef complex<double> _ComplexType;
};

// Floating point types
template <class _Tp>
struct __libcpp_complex_overload_traits<_Tp, false, true> {
  typedef _Tp _ValueType;
  typedef complex<_Tp> _ComplexType;
};

// real

template <class _Tp>
inline _LIBCPP_HIDE_FROM_ABI _LIBCPP_CONSTEXPR_SINCE_CXX14 _Tp real(const complex<_Tp>& __c) {
  return __c.real();
}

template <class _Tp>
inline _LIBCPP_HIDE_FROM_ABI _LIBCPP_CONSTEXPR_SINCE_CXX14 typename __libcpp_complex_overload_traits<_Tp>::_ValueType
real(_Tp __re) {
  return __re;
}

// imag

template <class _Tp>
inline _LIBCPP_HIDE_FROM_ABI _LIBCPP_CONSTEXPR_SINCE_CXX14 _Tp imag(const complex<_Tp>& __c) {
  return __c.imag();
}

template <class _Tp>
inline _LIBCPP_HIDE_FROM_ABI _LIBCPP_CONSTEXPR_SINCE_CXX14 typename __libcpp_complex_overload_traits<_Tp>::_ValueType
imag(_Tp) {
  return 0;
}

// abs

template <class _Tp>
inline _LIBCPP_HIDE_FROM_ABI _Tp abs(const complex<_Tp>& __c) {
  return std::hypot(__c.real(), __c.imag());
}

// arg

template <class _Tp>
inline _LIBCPP_HIDE_FROM_ABI _Tp arg(const complex<_Tp>& __c) {
  return std::atan2(__c.imag(), __c.real());
}

template <class _Tp, __enable_if_t<is_same<_Tp, long double>::value, int> = 0>
inline _LIBCPP_HIDE_FROM_ABI long double arg(_Tp __re) {
  return std::atan2l(0.L, __re);
}

template <class _Tp, __enable_if_t<is_integral<_Tp>::value || is_same<_Tp, double>::value, int> = 0>
inline _LIBCPP_HIDE_FROM_ABI double arg(_Tp __re) {
  return std::atan2(0., __re);
}

template <class _Tp, __enable_if_t<is_same<_Tp, float>::value, int> = 0>
inline _LIBCPP_HIDE_FROM_ABI float arg(_Tp __re) {
  return std::atan2f(0.F, __re);
}

// norm

template <class _Tp>
inline _LIBCPP_HIDE_FROM_ABI _LIBCPP_CONSTEXPR_SINCE_CXX20 _Tp norm(const complex<_Tp>& __c) {
  if (std::__constexpr_isinf(__c.real()))
    return std::abs(__c.real());
  if (std::__constexpr_isinf(__c.imag()))
    return std::abs(__c.imag());
  return __c.real() * __c.real() + __c.imag() * __c.imag();
}

template <class _Tp>
inline _LIBCPP_HIDE_FROM_ABI _LIBCPP_CONSTEXPR_SINCE_CXX20 typename __libcpp_complex_overload_traits<_Tp>::_ValueType
norm(_Tp __re) {
  typedef typename __libcpp_complex_overload_traits<_Tp>::_ValueType _ValueType;
  return static_cast<_ValueType>(__re) * __re;
}

// conj

template <class _Tp>
inline _LIBCPP_HIDE_FROM_ABI _LIBCPP_CONSTEXPR_SINCE_CXX20 complex<_Tp> conj(const complex<_Tp>& __c) {
  return complex<_Tp>(__c.real(), -__c.imag());
}

template <class _Tp>
inline _LIBCPP_HIDE_FROM_ABI _LIBCPP_CONSTEXPR_SINCE_CXX20 typename __libcpp_complex_overload_traits<_Tp>::_ComplexType
conj(_Tp __re) {
  typedef typename __libcpp_complex_overload_traits<_Tp>::_ComplexType _ComplexType;
  return _ComplexType(__re);
}

// proj

template <class _Tp>
inline _LIBCPP_HIDE_FROM_ABI complex<_Tp> proj(const complex<_Tp>& __c) {
  complex<_Tp> __r = __c;
  if (std::__constexpr_isinf(__c.real()) || std::__constexpr_isinf(__c.imag()))
    __r = complex<_Tp>(INFINITY, std::copysign(_Tp(0), __c.imag()));
  return __r;
}

template <class _Tp, __enable_if_t<is_floating_point<_Tp>::value, int> = 0>
inline _LIBCPP_HIDE_FROM_ABI typename __libcpp_complex_overload_traits<_Tp>::_ComplexType proj(_Tp __re) {
  if (std::__constexpr_isinf(__re))
    __re = std::abs(__re);
  return complex<_Tp>(__re);
}

template <class _Tp, __enable_if_t<is_integral<_Tp>::value, int> = 0>
inline _LIBCPP_HIDE_FROM_ABI typename __libcpp_complex_overload_traits<_Tp>::_ComplexType proj(_Tp __re) {
  typedef typename __libcpp_complex_overload_traits<_Tp>::_ComplexType _ComplexType;
  return _ComplexType(__re);
}

// polar

template <class _Tp>
_LIBCPP_HIDE_FROM_ABI complex<_Tp> polar(const _Tp& __rho, const _Tp& __theta = _Tp()) {
  if (std::__constexpr_isnan(__rho) || std::signbit(__rho))
    return complex<_Tp>(_Tp(NAN), _Tp(NAN));
  if (std::__constexpr_isnan(__theta)) {
    if (std::__constexpr_isinf(__rho))
      return complex<_Tp>(__rho, __theta);
    return complex<_Tp>(__theta, __theta);
  }
  if (std::__constexpr_isinf(__theta)) {
    if (std::__constexpr_isinf(__rho))
      return complex<_Tp>(__rho, _Tp(NAN));
    return complex<_Tp>(_Tp(NAN), _Tp(NAN));
  }
  _Tp __x = __rho * std::cos(__theta);
  if (std::__constexpr_isnan(__x))
    __x = 0;
  _Tp __y = __rho * std::sin(__theta);
  if (std::__constexpr_isnan(__y))
    __y = 0;
  return complex<_Tp>(__x, __y);
}

// log

template <class _Tp>
inline _LIBCPP_HIDE_FROM_ABI complex<_Tp> log(const complex<_Tp>& __x) {
  return complex<_Tp>(std::log(std::abs(__x)), std::arg(__x));
}

// log10

template <class _Tp>
inline _LIBCPP_HIDE_FROM_ABI complex<_Tp> log10(const complex<_Tp>& __x) {
  return std::log(__x) / std::log(_Tp(10));
}

// sqrt

template <class _Tp>
_LIBCPP_HIDE_FROM_ABI complex<_Tp> sqrt(const complex<_Tp>& __x) {
  if (std::__constexpr_isinf(__x.imag()))
    return complex<_Tp>(_Tp(INFINITY), __x.imag());
  if (std::__constexpr_isinf(__x.real())) {
    if (__x.real() > _Tp(0))
      return complex<_Tp>(
          __x.real(), std::__constexpr_isnan(__x.imag()) ? __x.imag() : std::copysign(_Tp(0), __x.imag()));
    return complex<_Tp>(
        std::__constexpr_isnan(__x.imag()) ? __x.imag() : _Tp(0), std::copysign(__x.real(), __x.imag()));
  }
  return std::polar(std::sqrt(std::abs(__x)), std::arg(__x) / _Tp(2));
}

// exp

template <class _Tp>
_LIBCPP_HIDE_FROM_ABI complex<_Tp> exp(const complex<_Tp>& __x) {
  _Tp __i = __x.imag();
  if (__i == 0) {
    return complex<_Tp>(std::exp(__x.real()), std::copysign(_Tp(0), __x.imag()));
  }
  if (std::__constexpr_isinf(__x.real())) {
    if (__x.real() < _Tp(0)) {
      if (!std::__constexpr_isfinite(__i))
        __i = _Tp(1);
    } else if (__i == 0 || !std::__constexpr_isfinite(__i)) {
      if (std::__constexpr_isinf(__i))
        __i = _Tp(NAN);
      return complex<_Tp>(__x.real(), __i);
    }
  }
  _Tp __e = std::exp(__x.real());
  return complex<_Tp>(__e * std::cos(__i), __e * std::sin(__i));
}

// pow

template <class _Tp>
inline _LIBCPP_HIDE_FROM_ABI complex<_Tp> pow(const complex<_Tp>& __x, const complex<_Tp>& __y) {
  return std::exp(__y * std::log(__x));
}

template <class _Tp, class _Up>
inline _LIBCPP_HIDE_FROM_ABI complex<typename __promote<_Tp, _Up>::type>
pow(const complex<_Tp>& __x, const complex<_Up>& __y) {
  typedef complex<typename __promote<_Tp, _Up>::type> result_type;
  return std::pow(result_type(__x), result_type(__y));
}

template <class _Tp, class _Up, __enable_if_t<is_arithmetic<_Up>::value, int> = 0>
inline _LIBCPP_HIDE_FROM_ABI complex<typename __promote<_Tp, _Up>::type> pow(const complex<_Tp>& __x, const _Up& __y) {
  typedef complex<typename __promote<_Tp, _Up>::type> result_type;
  return std::pow(result_type(__x), result_type(__y));
}

template <class _Tp, class _Up, __enable_if_t<is_arithmetic<_Tp>::value, int> = 0>
inline _LIBCPP_HIDE_FROM_ABI complex<typename __promote<_Tp, _Up>::type> pow(const _Tp& __x, const complex<_Up>& __y) {
  typedef complex<typename __promote<_Tp, _Up>::type> result_type;
  return std::pow(result_type(__x), result_type(__y));
}

// __sqr, computes pow(x, 2)

template <class _Tp>
inline _LIBCPP_HIDE_FROM_ABI complex<_Tp> __sqr(const complex<_Tp>& __x) {
  return complex<_Tp>((__x.real() - __x.imag()) * (__x.real() + __x.imag()), _Tp(2) * __x.real() * __x.imag());
}

// asinh

template <class _Tp>
_LIBCPP_HIDE_FROM_ABI complex<_Tp> asinh(const complex<_Tp>& __x) {
  const _Tp __pi(atan2(+0., -0.));
  if (std::__constexpr_isinf(__x.real())) {
    if (std::__constexpr_isnan(__x.imag()))
      return __x;
    if (std::__constexpr_isinf(__x.imag()))
      return complex<_Tp>(__x.real(), std::copysign(__pi * _Tp(0.25), __x.imag()));
    return complex<_Tp>(__x.real(), std::copysign(_Tp(0), __x.imag()));
  }
  if (std::__constexpr_isnan(__x.real())) {
    if (std::__constexpr_isinf(__x.imag()))
      return complex<_Tp>(__x.imag(), __x.real());
    if (__x.imag() == 0)
      return __x;
    return complex<_Tp>(__x.real(), __x.real());
  }
  if (std::__constexpr_isinf(__x.imag()))
    return complex<_Tp>(std::copysign(__x.imag(), __x.real()), std::copysign(__pi / _Tp(2), __x.imag()));
  complex<_Tp> __z = std::log(__x + std::sqrt(std::__sqr(__x) + _Tp(1)));
  return complex<_Tp>(std::copysign(__z.real(), __x.real()), std::copysign(__z.imag(), __x.imag()));
}

// acosh

template <class _Tp>
_LIBCPP_HIDE_FROM_ABI complex<_Tp> acosh(const complex<_Tp>& __x) {
  const _Tp __pi(atan2(+0., -0.));
  if (std::__constexpr_isinf(__x.real())) {
    if (std::__constexpr_isnan(__x.imag()))
      return complex<_Tp>(std::abs(__x.real()), __x.imag());
    if (std::__constexpr_isinf(__x.imag())) {
      if (__x.real() > 0)
        return complex<_Tp>(__x.real(), std::copysign(__pi * _Tp(0.25), __x.imag()));
      else
        return complex<_Tp>(-__x.real(), std::copysign(__pi * _Tp(0.75), __x.imag()));
    }
    if (__x.real() < 0)
      return complex<_Tp>(-__x.real(), std::copysign(__pi, __x.imag()));
    return complex<_Tp>(__x.real(), std::copysign(_Tp(0), __x.imag()));
  }
  if (std::__constexpr_isnan(__x.real())) {
    if (std::__constexpr_isinf(__x.imag()))
      return complex<_Tp>(std::abs(__x.imag()), __x.real());
    return complex<_Tp>(__x.real(), __x.real());
  }
  if (std::__constexpr_isinf(__x.imag()))
    return complex<_Tp>(std::abs(__x.imag()), std::copysign(__pi / _Tp(2), __x.imag()));
  complex<_Tp> __z = std::log(__x + std::sqrt(std::__sqr(__x) - _Tp(1)));
  return complex<_Tp>(std::copysign(__z.real(), _Tp(0)), std::copysign(__z.imag(), __x.imag()));
}

// atanh

template <class _Tp>
_LIBCPP_HIDE_FROM_ABI complex<_Tp> atanh(const complex<_Tp>& __x) {
  const _Tp __pi(atan2(+0., -0.));
  if (std::__constexpr_isinf(__x.imag())) {
    return complex<_Tp>(std::copysign(_Tp(0), __x.real()), std::copysign(__pi / _Tp(2), __x.imag()));
  }
  if (std::__constexpr_isnan(__x.imag())) {
    if (std::__constexpr_isinf(__x.real()) || __x.real() == 0)
      return complex<_Tp>(std::copysign(_Tp(0), __x.real()), __x.imag());
    return complex<_Tp>(__x.imag(), __x.imag());
  }
  if (std::__constexpr_isnan(__x.real())) {
    return complex<_Tp>(__x.real(), __x.real());
  }
  if (std::__constexpr_isinf(__x.real())) {
    return complex<_Tp>(std::copysign(_Tp(0), __x.real()), std::copysign(__pi / _Tp(2), __x.imag()));
  }
  if (std::abs(__x.real()) == _Tp(1) && __x.imag() == _Tp(0)) {
    return complex<_Tp>(std::copysign(_Tp(INFINITY), __x.real()), std::copysign(_Tp(0), __x.imag()));
  }
  complex<_Tp> __z = std::log((_Tp(1) + __x) / (_Tp(1) - __x)) / _Tp(2);
  return complex<_Tp>(std::copysign(__z.real(), __x.real()), std::copysign(__z.imag(), __x.imag()));
}

// sinh

template <class _Tp>
_LIBCPP_HIDE_FROM_ABI complex<_Tp> sinh(const complex<_Tp>& __x) {
  if (std::__constexpr_isinf(__x.real()) && !std::__constexpr_isfinite(__x.imag()))
    return complex<_Tp>(__x.real(), _Tp(NAN));
  if (__x.real() == 0 && !std::__constexpr_isfinite(__x.imag()))
    return complex<_Tp>(__x.real(), _Tp(NAN));
  if (__x.imag() == 0 && !std::__constexpr_isfinite(__x.real()))
    return __x;
  return complex<_Tp>(std::sinh(__x.real()) * std::cos(__x.imag()), std::cosh(__x.real()) * std::sin(__x.imag()));
}

// cosh

template <class _Tp>
_LIBCPP_HIDE_FROM_ABI complex<_Tp> cosh(const complex<_Tp>& __x) {
  if (std::__constexpr_isinf(__x.real()) && !std::__constexpr_isfinite(__x.imag()))
    return complex<_Tp>(std::abs(__x.real()), _Tp(NAN));
  if (__x.real() == 0 && !std::__constexpr_isfinite(__x.imag()))
    return complex<_Tp>(_Tp(NAN), __x.real());
  if (__x.real() == 0 && __x.imag() == 0)
    return complex<_Tp>(_Tp(1), __x.imag());
  if (__x.imag() == 0 && !std::__constexpr_isfinite(__x.real()))
    return complex<_Tp>(std::abs(__x.real()), __x.imag());
  return complex<_Tp>(std::cosh(__x.real()) * std::cos(__x.imag()), std::sinh(__x.real()) * std::sin(__x.imag()));
}

// tanh

template <class _Tp>
_LIBCPP_HIDE_FROM_ABI complex<_Tp> tanh(const complex<_Tp>& __x) {
  if (std::__constexpr_isinf(__x.real())) {
    if (!std::__constexpr_isfinite(__x.imag()))
      return complex<_Tp>(std::copysign(_Tp(1), __x.real()), _Tp(0));
    return complex<_Tp>(std::copysign(_Tp(1), __x.real()), std::copysign(_Tp(0), std::sin(_Tp(2) * __x.imag())));
  }
  if (std::__constexpr_isnan(__x.real()) && __x.imag() == 0)
    return __x;
  _Tp __2r(_Tp(2) * __x.real());
  _Tp __2i(_Tp(2) * __x.imag());
  _Tp __d(std::cosh(__2r) + std::cos(__2i));
  _Tp __2rsh(std::sinh(__2r));
  if (std::__constexpr_isinf(__2rsh) && std::__constexpr_isinf(__d))
    return complex<_Tp>(__2rsh > _Tp(0) ? _Tp(1) : _Tp(-1), __2i > _Tp(0) ? _Tp(0) : _Tp(-0.));
  return complex<_Tp>(__2rsh / __d, std::sin(__2i) / __d);
}

// asin

template <class _Tp>
_LIBCPP_HIDE_FROM_ABI complex<_Tp> asin(const complex<_Tp>& __x) {
  complex<_Tp> __z = std::asinh(complex<_Tp>(-__x.imag(), __x.real()));
  return complex<_Tp>(__z.imag(), -__z.real());
}

// acos

template <class _Tp>
_LIBCPP_HIDE_FROM_ABI complex<_Tp> acos(const complex<_Tp>& __x) {
  const _Tp __pi(atan2(+0., -0.));
  if (std::__constexpr_isinf(__x.real())) {
    if (std::__constexpr_isnan(__x.imag()))
      return complex<_Tp>(__x.imag(), __x.real());
    if (std::__constexpr_isinf(__x.imag())) {
      if (__x.real() < _Tp(0))
        return complex<_Tp>(_Tp(0.75) * __pi, -__x.imag());
      return complex<_Tp>(_Tp(0.25) * __pi, -__x.imag());
    }
    if (__x.real() < _Tp(0))
      return complex<_Tp>(__pi, std::signbit(__x.imag()) ? -__x.real() : __x.real());
    return complex<_Tp>(_Tp(0), std::signbit(__x.imag()) ? __x.real() : -__x.real());
  }
  if (std::__constexpr_isnan(__x.real())) {
    if (std::__constexpr_isinf(__x.imag()))
      return complex<_Tp>(__x.real(), -__x.imag());
    return complex<_Tp>(__x.real(), __x.real());
  }
  if (std::__constexpr_isinf(__x.imag()))
    return complex<_Tp>(__pi / _Tp(2), -__x.imag());
  if (__x.real() == 0 && (__x.imag() == 0 || std::isnan(__x.imag())))
    return complex<_Tp>(__pi / _Tp(2), -__x.imag());
  complex<_Tp> __z = std::log(__x + std::sqrt(std::__sqr(__x) - _Tp(1)));
  if (std::signbit(__x.imag()))
    return complex<_Tp>(std::abs(__z.imag()), std::abs(__z.real()));
  return complex<_Tp>(std::abs(__z.imag()), -std::abs(__z.real()));
}

// atan

template <class _Tp>
_LIBCPP_HIDE_FROM_ABI complex<_Tp> atan(const complex<_Tp>& __x) {
  complex<_Tp> __z = std::atanh(complex<_Tp>(-__x.imag(), __x.real()));
  return complex<_Tp>(__z.imag(), -__z.real());
}

// sin

template <class _Tp>
_LIBCPP_HIDE_FROM_ABI complex<_Tp> sin(const complex<_Tp>& __x) {
  complex<_Tp> __z = std::sinh(complex<_Tp>(-__x.imag(), __x.real()));
  return complex<_Tp>(__z.imag(), -__z.real());
}

// cos

template <class _Tp>
inline _LIBCPP_HIDE_FROM_ABI complex<_Tp> cos(const complex<_Tp>& __x) {
  return std::cosh(complex<_Tp>(-__x.imag(), __x.real()));
}

// tan

template <class _Tp>
_LIBCPP_HIDE_FROM_ABI complex<_Tp> tan(const complex<_Tp>& __x) {
  complex<_Tp> __z = std::tanh(complex<_Tp>(-__x.imag(), __x.real()));
  return complex<_Tp>(__z.imag(), -__z.real());
}

#if !defined(_LIBCPP_HAS_NO_LOCALIZATION)
template <class _Tp, class _CharT, class _Traits>
_LIBCPP_HIDE_FROM_ABI basic_istream<_CharT, _Traits>&
operator>>(basic_istream<_CharT, _Traits>& __is, complex<_Tp>& __x) {
  if (__is.good()) {
    std::ws(__is);
    if (__is.peek() == _CharT('(')) {
      __is.get();
      _Tp __r;
      __is >> __r;
      if (!__is.fail()) {
        std::ws(__is);
        _CharT __c = __is.peek();
        if (__c == _CharT(',')) {
          __is.get();
          _Tp __i;
          __is >> __i;
          if (!__is.fail()) {
            std::ws(__is);
            __c = __is.peek();
            if (__c == _CharT(')')) {
              __is.get();
              __x = complex<_Tp>(__r, __i);
            } else
              __is.setstate(__is.failbit);
          } else
            __is.setstate(__is.failbit);
        } else if (__c == _CharT(')')) {
          __is.get();
          __x = complex<_Tp>(__r, _Tp(0));
        } else
          __is.setstate(__is.failbit);
      } else
        __is.setstate(__is.failbit);
    } else {
      _Tp __r;
      __is >> __r;
      if (!__is.fail())
        __x = complex<_Tp>(__r, _Tp(0));
      else
        __is.setstate(__is.failbit);
    }
  } else
    __is.setstate(__is.failbit);
  return __is;
}

template <class _Tp, class _CharT, class _Traits>
_LIBCPP_HIDE_FROM_ABI basic_ostream<_CharT, _Traits>&
operator<<(basic_ostream<_CharT, _Traits>& __os, const complex<_Tp>& __x) {
  basic_ostringstream<_CharT, _Traits> __s;
  __s.flags(__os.flags());
  __s.imbue(__os.getloc());
  __s.precision(__os.precision());
  __s << '(' << __x.real() << ',' << __x.imag() << ')';
  return __os << __s.str();
}
#endif // !_LIBCPP_HAS_NO_LOCALIZATION

#if _LIBCPP_STD_VER >= 26

// [complex.tuple], tuple interface

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

template <size_t _Ip, class _Tp>
struct tuple_element<_Ip, complex<_Tp>> {
  static_assert(_Ip < 2, "Index value is out of range.");
  using type = _Tp;
};

template <size_t _Ip, class _Xp>
_LIBCPP_HIDE_FROM_ABI constexpr _Xp& get(complex<_Xp>& __z) noexcept {
  static_assert(_Ip < 2, "Index value is out of range.");
  if constexpr (_Ip == 0) {
    return __z.__re_;
  } else {
    return __z.__im_;
  }
}

template <size_t _Ip, class _Xp>
_LIBCPP_HIDE_FROM_ABI constexpr _Xp&& get(complex<_Xp>&& __z) noexcept {
  static_assert(_Ip < 2, "Index value is out of range.");
  if constexpr (_Ip == 0) {
    return std::move(__z.__re_);
  } else {
    return std::move(__z.__im_);
  }
}

template <size_t _Ip, class _Xp>
_LIBCPP_HIDE_FROM_ABI constexpr const _Xp& get(const complex<_Xp>& __z) noexcept {
  static_assert(_Ip < 2, "Index value is out of range.");
  if constexpr (_Ip == 0) {
    return __z.__re_;
  } else {
    return __z.__im_;
  }
}

template <size_t _Ip, class _Xp>
_LIBCPP_HIDE_FROM_ABI constexpr const _Xp&& get(const complex<_Xp>&& __z) noexcept {
  static_assert(_Ip < 2, "Index value is out of range.");
  if constexpr (_Ip == 0) {
    return std::move(__z.__re_);
  } else {
    return std::move(__z.__im_);
  }
}

#endif // _LIBCPP_STD_VER >= 26

#if _LIBCPP_STD_VER >= 14
// Literal suffix for complex number literals [complex.literals]
inline namespace literals {
inline namespace complex_literals {
_LIBCPP_HIDE_FROM_ABI inline constexpr complex<long double> operator""il(long double __im) { return {0.0l, __im}; }

_LIBCPP_HIDE_FROM_ABI inline constexpr complex<long double> operator""il(unsigned long long __im) {
  return {0.0l, static_cast<long double>(__im)};
}

_LIBCPP_HIDE_FROM_ABI inline constexpr complex<double> operator""i(long double __im) {
  return {0.0, static_cast<double>(__im)};
}

_LIBCPP_HIDE_FROM_ABI inline constexpr complex<double> operator""i(unsigned long long __im) {
  return {0.0, static_cast<double>(__im)};
}

_LIBCPP_HIDE_FROM_ABI inline constexpr complex<float> operator""if(long double __im) {
  return {0.0f, static_cast<float>(__im)};
}

_LIBCPP_HIDE_FROM_ABI inline constexpr complex<float> operator""if(unsigned long long __im) {
  return {0.0f, static_cast<float>(__im)};
}
} // namespace complex_literals
} // namespace literals
#endif

_LIBCPP_END_NAMESPACE_STD

_LIBCPP_POP_MACROS

#if !defined(_LIBCPP_REMOVE_TRANSITIVE_INCLUDES) && _LIBCPP_STD_VER <= 20
#  include <iosfwd>
#  include <stdexcept>
#  include <type_traits>
#endif

#endif // _LIBCPP_COMPLEX