libcxx/include/complex

// -*- 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

*/

#if __cplusplus < 201103L && defined(_LIBCPP_USE_FROZEN_CXX03_HEADERS)
#  include <__cxx03/complex>
#else
#  include <__config>
#  include <__cstddef/size_t.h>
#  include <__fwd/complex.h>
#  include <__fwd/tuple.h>
#  include <__tuple/tuple_element.h>
#  include <__tuple/tuple_size.h>
#  include <__type_traits/conditional.h>
#  include <__utility/move.h>
#  include <cmath>
#  include <version>

#  if _LIBCPP_HAS_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 complex;

template <class _Tp, __enable_if_t<is_floating_point<_Tp>::value, int> = 0>
_LIBCPP_HIDE_FROM_ABI _LIBCPP_CONSTEXPR_SINCE_CXX20 complex<_Tp>
operator*(const complex<_Tp>& __z, const complex<_Tp>& __w);

template <class _Tp, __enable_if_t<!is_floating_point<_Tp>::value, int> = 0>
_LIBCPP_HIDE_FROM_ABI _LIBCPP_CONSTEXPR_SINCE_CXX20 complex<_Tp>
operator*(const complex<_Tp>& __z, const complex<_Tp>& __w);

template <class _Tp, __enable_if_t<is_floating_point<_Tp>::value, int> = 0>
_LIBCPP_HIDE_FROM_ABI _LIBCPP_CONSTEXPR_SINCE_CXX20 complex<_Tp>
operator/(const complex<_Tp>& __x, const complex<_Tp>& __y);

template <class _Tp, __enable_if_t<!is_floating_point<_Tp>::value, int> = 0>
_LIBCPP_HIDE_FROM_ABI _LIBCPP_CONSTEXPR_SINCE_CXX20 complex<_Tp>
operator/(const complex<_Tp>& __x, const complex<_Tp>& __y);

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

struct __from_builtin_tag {};

template <class _Tp>
using __complex_t _LIBCPP_NODEBUG =
    __conditional_t<is_same<_Tp, float>::value,
                    _Complex float,
                    __conditional_t<is_same<_Tp, double>::value, _Complex double, _Complex long double> >;

template <class _Tp>
_LIBCPP_HIDE_FROM_ABI _LIBCPP_CONSTEXPR __complex_t<_Tp> __make_complex(_Tp __re, _Tp __im) {
#  if __has_builtin(__builtin_complex)
  return __builtin_complex(__re, __im);
#  else
  return __complex_t<_Tp>{__re, __im};
#  endif
}

template <>
class 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) {}

  template <class _Tag, __enable_if_t<_IsSame<_Tag, __from_builtin_tag>::value, int> = 0>
  _LIBCPP_HIDE_FROM_ABI _LIBCPP_CONSTEXPR explicit complex(_Tag, _Complex float __v)
      : __re_(__real__ __v), __im_(__imag__ __v) {}

  _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 _Complex float __builtin() const { return std::__make_complex(__re_, __im_); }
  _LIBCPP_HIDE_FROM_ABI _LIBCPP_CONSTEXPR_SINCE_CXX14 void __builtin(_Complex float __f) {
    __re_ = __real__ __f;
    __im_ = __imag__ __f;
  }

  _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 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) {}

  template <class _Tag, __enable_if_t<_IsSame<_Tag, __from_builtin_tag>::value, int> = 0>
  _LIBCPP_HIDE_FROM_ABI _LIBCPP_CONSTEXPR explicit complex(_Tag, _Complex double __v)
      : __re_(__real__ __v), __im_(__imag__ __v) {}

  _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 _Complex double __builtin() const {
    return std::__make_complex(__re_, __im_);
  }

  _LIBCPP_HIDE_FROM_ABI _LIBCPP_CONSTEXPR_SINCE_CXX14 void __builtin(_Complex double __f) {
    __re_ = __real__ __f;
    __im_ = __imag__ __f;
  }

  _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 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) {}

  template <class _Tag, __enable_if_t<_IsSame<_Tag, __from_builtin_tag>::value, int> = 0>
  _LIBCPP_HIDE_FROM_ABI _LIBCPP_CONSTEXPR explicit complex(_Tag, _Complex long double __v)
      : __re_(__real__ __v), __im_(__imag__ __v) {}

  _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 _Complex long double __builtin() const {
    return std::__make_complex(__re_, __im_);
  }

  _LIBCPP_HIDE_FROM_ABI _LIBCPP_CONSTEXPR_SINCE_CXX14 void __builtin(_Complex long double __f) {
    __re_ = __real__ __f;
    __im_ = __imag__ __f;
  }

  _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, __enable_if_t<is_floating_point<_Tp>::value, int> >
_LIBCPP_HIDE_FROM_ABI _LIBCPP_CONSTEXPR_SINCE_CXX20 complex<_Tp>
operator*(const complex<_Tp>& __lhs, const complex<_Tp>& __rhs) {
  return complex<_Tp>(__from_builtin_tag(), __lhs.__builtin() * __rhs.__builtin());
}

template <class _Tp, __enable_if_t<!is_floating_point<_Tp>::value, int> >
_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();

  return complex<_Tp>((__a * __c) - (__b * __d), (__a * __d) + (__b * __c));
}

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, __enable_if_t<is_floating_point<_Tp>::value, int> >
_LIBCPP_HIDE_FROM_ABI _LIBCPP_CONSTEXPR_SINCE_CXX20 complex<_Tp>
operator/(const complex<_Tp>& __lhs, const complex<_Tp>& __rhs) {
  return complex<_Tp>(__from_builtin_tag(), __lhs.__builtin() / __rhs.__builtin());
}

template <class _Tp, __enable_if_t<!is_floating_point<_Tp>::value, int> >
_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();

  _Tp __denom = __c * __c + __d * __d;
  return complex<_Tp>((__a * __c + __b * __d) / __denom, (__b * __c - __a * __d) / __denom);
}

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::isinf(__c.real()) || std::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::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::isnan(__rho) || std::signbit(__rho))
    return complex<_Tp>(_Tp(NAN), _Tp(NAN));
  if (std::isnan(__theta)) {
    if (std::isinf(__rho))
      return complex<_Tp>(__rho, __theta);
    return complex<_Tp>(__theta, __theta);
  }
  if (std::isinf(__theta)) {
    if (std::isinf(__rho))
      return complex<_Tp>(__rho, _Tp(NAN));
    return complex<_Tp>(_Tp(NAN), _Tp(NAN));
  }
  _Tp __x = __rho * std::cos(__theta);
  if (std::isnan(__x))
    __x = 0;
  _Tp __y = __rho * std::sin(__theta);
  if (std::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::isinf(__x.imag()))
    return complex<_Tp>(_Tp(INFINITY), __x.imag());
  if (std::isinf(__x.real())) {
    if (__x.real() > _Tp(0))
      return complex<_Tp>(__x.real(), std::isnan(__x.imag()) ? __x.imag() : std::copysign(_Tp(0), __x.imag()));
    return complex<_Tp>(std::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::isinf(__x.real())) {
    if (__x.real() < _Tp(0)) {
      if (!std::isfinite(__i))
        __i = _Tp(1);
    } else if (__i == 0 || !std::isfinite(__i)) {
      if (std::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, __enable_if_t<is_floating_point<_Tp>::value && is_floating_point<_Up>::value, int> = 0>
inline _LIBCPP_HIDE_FROM_ABI complex<__promote_t<_Tp, _Up> > pow(const complex<_Tp>& __x, const complex<_Up>& __y) {
  typedef complex<__promote_t<_Tp, _Up> > result_type;
  return std::pow(result_type(__x), result_type(__y));
}

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

template <class _Tp, class _Up, __enable_if_t<is_arithmetic<_Tp>::value && is_floating_point<_Up>::value, int> = 0>
inline _LIBCPP_HIDE_FROM_ABI complex<__promote_t<_Tp, _Up> > pow(const _Tp& __x, const complex<_Up>& __y) {
  typedef complex<__promote_t<_Tp, _Up> > 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::isinf(__x.real())) {
    if (std::isnan(__x.imag()))
      return __x;
    if (std::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::isnan(__x.real())) {
    if (std::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::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::isinf(__x.real())) {
    if (std::isnan(__x.imag()))
      return complex<_Tp>(std::abs(__x.real()), __x.imag());
    if (std::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::isnan(__x.real())) {
    if (std::isinf(__x.imag()))
      return complex<_Tp>(std::abs(__x.imag()), __x.real());
    return complex<_Tp>(__x.real(), __x.real());
  }
  if (std::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::isinf(__x.imag())) {
    return complex<_Tp>(std::copysign(_Tp(0), __x.real()), std::copysign(__pi / _Tp(2), __x.imag()));
  }
  if (std::isnan(__x.imag())) {
    if (std::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::isnan(__x.real())) {
    return complex<_Tp>(__x.real(), __x.real());
  }
  if (std::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::isinf(__x.real()) && !std::isfinite(__x.imag()))
    return complex<_Tp>(__x.real(), _Tp(NAN));
  if (__x.real() == 0 && !std::isfinite(__x.imag()))
    return complex<_Tp>(__x.real(), _Tp(NAN));
  if (__x.imag() == 0 && !std::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::isinf(__x.real()) && !std::isfinite(__x.imag()))
    return complex<_Tp>(std::abs(__x.real()), _Tp(NAN));
  if (__x.real() == 0 && !std::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::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::isinf(__x.real())) {
    if (!std::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::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::isinf(__2rsh) && std::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::isinf(__x.real())) {
    if (std::isnan(__x.imag()))
      return complex<_Tp>(__x.imag(), __x.real());
    if (std::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::isnan(__x.real())) {
    if (std::isinf(__x.imag()))
      return complex<_Tp>(__x.real(), -__x.imag());
    return complex<_Tp>(__x.real(), __x.real());
  }
  if (std::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 _LIBCPP_HAS_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_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 _LIBCPP_NODEBUG = _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 // __cplusplus < 201103L && defined(_LIBCPP_USE_FROZEN_CXX03_HEADERS)

#endif // _LIBCPP_COMPLEX