<filename>rithm/__init__.py
"""Arbitrary precision arithmetic."""

__version__ = '5.2.0'

try:
    from ._rithm import (Fraction,
                         Int)
except ImportError:
    from math import gcd as _gcd
    from operator import mul as _mul
    from typing import (Any as _Any,
                        Optional as _Optional,
                        Tuple as _Tuple,
                        Union as _Union,
                        overload as _overload)


    class Int:
        def bit_length(self) -> 'Int':
            return Int(self._value.bit_length())

        def gcd(self, other: 'Int') -> 'Int':
            return Int(_gcd(self._value, other._value))

        __slots__ = '_value',

        def __new__(cls,
                    _value: _Union[str, int] = 0,
                    _base: _Optional[int] = None) -> 'Int':
            self = super().__new__(cls)
            self._value = (int(_value, _base)
                           if _base is not None
                           else int(_value))
            return self

        def __abs__(self) -> 'Int':
            return Int(abs(self._value))

        def __add__(self, other: 'Int') -> 'Int':
            return (Int(self._value + other._value)
                    if isinstance(other, Int)
                    else NotImplemented)

        def __and__(self, other: 'Int') -> 'Int':
            return (Int(self._value & other._value)
                    if isinstance(other, Int)
                    else NotImplemented)

        def __bool__(self) -> bool:
            return bool(self._value)

        def __ceil__(self) -> 'Int':
            return self

        def __divmod__(self, other: 'Int') -> _Tuple['Int', 'Int']:
            return (tuple(Int(value)
                          for value in divmod(self._value, other._value))
                    if isinstance(other, Int)
                    else NotImplemented)

        def __eq__(self, other: 'Int') -> bool:
            return (self._value == other._value
                    if isinstance(other, Int)
                    else NotImplemented)

        def __float__(self) -> float:
            return float(self._value)

        def __floor__(self) -> 'Int':
            return self

        def __floordiv__(self, other: 'Int') -> 'Int':
            return (Int(self._value // other._value)
                    if isinstance(other, Int)
                    else NotImplemented)

        def __ge__(self, other: 'Int') -> bool:
            return (self._value >= other._value
                    if isinstance(other, Int)
                    else NotImplemented)

        def __getstate__(self) -> int:
            return self._value

        def __gt__(self, other: 'Int') -> bool:
            return (self._value > other._value
                    if isinstance(other, Int)
                    else NotImplemented)

        def __hash__(self) -> int:
            return hash(self._value)

        def __invert__(self) -> 'Int':
            return Int(~self._value)

        def __int__(self) -> int:
            return self._value

        def __le__(self, other: 'Int') -> bool:
            return (self._value <= other._value
                    if isinstance(other, Int)
                    else NotImplemented)

        def __lshift__(self, other: 'Int') -> 'Int':
            return (Int(self._value << other._value)
                    if isinstance(other, Int)
                    else NotImplemented)

        def __lt__(self, other: 'Int') -> bool:
            return (self._value < other._value
                    if isinstance(other, Int)
                    else NotImplemented)

        def __mod__(self, other: 'Int') -> 'Int':
            return (Int(self._value % other._value)
                    if isinstance(other, Int)
                    else NotImplemented)

        def __mul__(self, other: 'Int') -> 'Int':
            return (Int(self._value * other._value)
                    if isinstance(other, Int)
                    else NotImplemented)

        def __neg__(self) -> 'Int':
            return Int(-self._value)

        def __or__(self, other: 'Int') -> 'Int':
            return (Int(self._value | other._value)
                    if isinstance(other, Int)
                    else NotImplemented)

        def __pow__(self,
                    exponent: 'Int',
                    divisor: _Optional['Int'] = None
                    ) -> _Union['Fraction', 'Int']:
            return (((Int(self._value ** exponent._value)
                      if exponent >= _ZERO
                      else Fraction(_ONE, self) ** -exponent)
                     if divisor is None
                     else (Int(pow(self._value, exponent._value,
                                   divisor._value))
                           if isinstance(divisor, Int)
                           else NotImplemented))
                    if isinstance(exponent, Int)
                    else NotImplemented)

        def __repr__(self) -> str:
            return f'rithm.Int(\'{self._value}\')'

        def __rshift__(self, other: 'Int') -> 'Int':
            return (Int(self._value >> other._value)
                    if isinstance(other, Int)
                    else NotImplemented)

        def __setstate__(self, state: int) -> None:
            self._value = state

        def __str__(self) -> str:
            return str(self._value)

        def __sub__(self, other: 'Int') -> 'Int':
            return (Int(self._value - other._value)
                    if isinstance(other, Int)
                    else NotImplemented)

        def __truediv__(self, other: 'Int') -> 'Fraction':
            return (Fraction(self, other)
                    if isinstance(other, Int)
                    else NotImplemented)

        def __trunc__(self) -> 'Int':
            return self

        def __xor__(self, other: 'Int') -> 'Int':
            return (Int(self._value ^ other._value)
                    if isinstance(other, Int)
                    else NotImplemented)


    _ONE = Int('1')
    _ZERO = Int()


    class Fraction:
        @property
        def denominator(self) -> Int:
            return self._denominator

        @property
        def numerator(self) -> Int:
            return self._numerator

        __slots__ = '_denominator', '_numerator'

        def __new__(cls,
                    _numerator: _Union[Int, float] = _ZERO,
                    _denominator: _Optional[Int] = None,
                    *,
                    _normalize: bool = True) -> 'Fraction':
            self = super().__new__(cls)
            if _denominator is None:
                if isinstance(_numerator, float):
                    raw_numerator, raw_denominator = (
                        _numerator.as_integer_ratio())
                    numerator, denominator = (Int(raw_numerator),
                                              Int(raw_denominator))
                elif isinstance(_numerator, Int):
                    numerator, denominator = _numerator, _ONE
                else:
                    raise TypeError('First argument should be of '
                                    f'type {Int} or {float}, '
                                    f'but found: {type(_numerator)}.')
            elif isinstance(_denominator, Int):
                numerator, denominator = _numerator, _denominator
            else:
                raise TypeError(f'Denominator should be of type {Int}, '
                                f'but found: {type(_denominator)}.')
            if not denominator:
                raise ZeroDivisionError('Denominator should not be zero.')
            if _normalize:
                numerator, denominator = _normalize_components_sign(
                    *_normalize_components_moduli(numerator, denominator))
            self._numerator, self._denominator = numerator, denominator
            return self

        def __abs__(self) -> 'Fraction':
            return Fraction(abs(self.numerator), self.denominator,
                            _normalize=False)

        @_overload
        def __add__(self, other: _Any) -> _Any:
            ...

        @_overload
        def __add__(self, other: _Union['Fraction', Int]) -> 'Fraction':
            ...

        def __add__(self, other: _Union['Fraction', Int, _Any]
                    ) -> _Union['Fraction', _Any]:
            return (self._add_fraction(other)
                    if isinstance(other, Fraction)
                    else (self._add_int(other)
                          if isinstance(other, Int)
                          else NotImplemented))

        def __bool__(self) -> bool:
            return bool(self.numerator)

        def __eq__(self, other: 'Fraction') -> bool:
            return (self.numerator == other.numerator
                    and self.denominator == other.denominator
                    if isinstance(other, Fraction)
                    else (self.denominator == _ONE
                          and self.numerator == other
                          if isinstance(other, Int)
                          else NotImplemented))

        def __float__(self) -> float:
            return self._numerator._value / self._denominator._value

        @_overload
        def __floordiv__(self, other: _Any) -> _Any:
            ...

        @_overload
        def __floordiv__(self, other: _Union['Fraction', Int]) -> Int:
            ...

        def __floordiv__(self, other: _Union['Fraction', Int, _Any]
                         ) -> _Union[Int, _Any]:
            return ((self.numerator * other.denominator)
                    // (self.denominator * other.numerator)
                    if isinstance(other, Fraction)
                    else (self.numerator // (self.denominator * other)
                          if isinstance(other, Int)
                          else NotImplemented))

        def __ge__(self, other: 'Fraction') -> bool:
            return (self.numerator * other.denominator
                    >= other.numerator * self.denominator
                    if isinstance(other, Fraction)
                    else (self.numerator >= other * self.denominator
                          if isinstance(other, Int)
                          else NotImplemented))

        def __getstate__(self) -> _Tuple[Int, Int]:
            return self._numerator, self._denominator

        def __gt__(self, other: 'Fraction') -> bool:
            return (self.numerator * other.denominator
                    > other.numerator * self.denominator
                    if isinstance(other, Fraction)
                    else (self.numerator > other * self.denominator
                          if isinstance(other, Int)
                          else NotImplemented))

        def __le__(self, other: 'Fraction') -> bool:
            return (self.numerator * other.denominator
                    <= other.numerator * self.denominator
                    if isinstance(other, Fraction)
                    else (self.numerator <= other * self.denominator
                          if isinstance(other, Int)
                          else NotImplemented))

        def __lt__(self, other: 'Fraction') -> bool:
            return (self.numerator * other.denominator
                    < other.numerator * self.denominator
                    if isinstance(other, Fraction)
                    else (self.numerator < other * self.denominator
                          if isinstance(other, Int)
                          else NotImplemented))

        @_overload
        def __mod__(self, other: _Any) -> _Any:
            ...

        @_overload
        def __mod__(self, other: _Union['Fraction', Int]) -> 'Fraction':
            ...

        def __mod__(self, other: _Union['Fraction', Int, _Any]
                    ) -> _Union['Fraction', _Any]:
            return (Fraction((self.numerator * other.denominator)
                             % (self.denominator * other.numerator),
                             self.denominator * other.denominator)
                    if isinstance(other, Fraction)
                    else (Fraction(self.numerator % (self.denominator * other),
                                   self.denominator)
                          if isinstance(other, Int)
                          else NotImplemented))

        @_overload
        def __mul__(self, other: _Any) -> _Any:
            ...

        @_overload
        def __mul__(self, other: _Union['Fraction', Int]) -> 'Fraction':
            ...

        def __mul__(self, other: _Union['Fraction', Int, _Any]
                    ) -> _Union['Fraction', _Any]:
            return (self._mul_by_fraction(other)
                    if isinstance(other, Fraction)
                    else (self._mul_by_int(other)
                          if isinstance(other, Int)
                          else NotImplemented))

        def __neg__(self) -> 'Fraction':
            return Fraction(-self.numerator, self.denominator,
                            _normalize=False)

        def __pow__(self, exponent: 'Int', divisor: None = None) -> 'Fraction':
            return (
                (Fraction(self.numerator ** exponent,
                          self.denominator ** exponent,
                          _normalize=False)
                 if exponent >= _ZERO
                 else Fraction(
                    *_normalize_components_sign(self.denominator ** -exponent,
                                                self.numerator ** -exponent),
                    _normalize=False))
                if isinstance(exponent, Int) and divisor is None
                else NotImplemented)

        def __repr__(self) -> str:
            return f'rithm.Fraction({self.numerator!r}, {self.denominator!r})'

        @_overload
        def __radd__(self, other: _Any) -> _Any:
            ...

        @_overload
        def __radd__(self, other: Int) -> 'Fraction':
            ...

        def __radd__(self, other: _Union[Int, _Any]
                     ) -> _Union['Fraction', _Any]:
            return (self._add_int(other)
                    if isinstance(other, Int)
                    else NotImplemented)

        @_overload
        def __rfloordiv__(self, other: _Any) -> _Any:
            ...

        @_overload
        def __rfloordiv__(self, other: Int) -> Int:
            ...

        def __rfloordiv__(self, other: _Union[Int, _Any]) -> _Union[Int, _Any]:
            return ((other * self.denominator) // self.numerator
                    if isinstance(other, Int)
                    else NotImplemented)

        @_overload
        def __rmod__(self, other: _Any) -> _Any:
            ...

        @_overload
        def __rmod__(self, other: Int) -> 'Fraction':
            ...

        def __rmod__(self, other: _Union[Int, _Any]
                     ) -> _Union['Fraction', _Any]:
            return (Fraction((other * self.denominator) % self.numerator,
                             self.denominator)
                    if isinstance(other, Int)
                    else NotImplemented)

        @_overload
        def __rmul__(self, other: _Any) -> _Any:
            ...

        @_overload
        def __rmul__(self, other: Int) -> 'Fraction':
            ...

        def __rmul__(self, other: _Union[Int, _Any]
                     ) -> _Union['Fraction', _Any]:
            return (self._mul_by_int(other)
                    if isinstance(other, Int)
                    else NotImplemented)

        @_overload
        def __rsub__(self, other: _Any) -> _Any:
            ...

        @_overload
        def __rsub__(self, other: Int) -> 'Fraction':
            ...

        def __rsub__(self, other: _Union[Int, _Any]
                     ) -> _Union['Fraction', _Any]:
            return (
                Fraction(
                    *_normalize_components_moduli(other * self.denominator
                                                  - self.numerator,
                                                  self.denominator),
                    _normalize=False)
                if isinstance(other, Int)
                else NotImplemented)

        def __setstate__(self, state: _Tuple[Int, Int]) -> None:
            self._numerator, self._denominator = state

        def __str__(self) -> str:
            return (str(self.numerator)
                    if self.denominator == _ONE
                    else f'{self.numerator}/{self.denominator}')

        @_overload
        def __sub__(self, other: _Any) -> _Any:
            ...

        @_overload
        def __sub__(self, other: _Union['Fraction', Int]) -> 'Fraction':
            ...

        def __sub__(self, other: _Union['Fraction', Int, _Any]
                    ) -> _Union['Fraction', _Any]:
            return (
                Fraction(
                    *_normalize_components_moduli(
                        self.numerator * other.denominator
                        - other.numerator * self.denominator,
                        self.denominator * other.denominator),
                    _normalize=False)
                if isinstance(other, Fraction)
                else
                (Fraction(
                    *_normalize_components_moduli(self.numerator
                                                  - other * self.denominator,
                                                  self.denominator),
                    _normalize=False)
                 if isinstance(other, Int)
                 else NotImplemented))

        @_overload
        def __rtruediv__(self, other: _Any) -> _Any:
            ...

        @_overload
        def __rtruediv__(self, other: Int) -> 'Fraction':
            ...

        def __rtruediv__(self, other: _Union[Int, _Any]
                         ) -> _Union['Fraction', _Any]:
            return (self._rtruediv_by_int(other)
                    if isinstance(other, Int)
                    else NotImplemented)

        @_overload
        def __truediv__(self, other: _Any) -> _Any:
            ...

        @_overload
        def __truediv__(self, other: _Union['Fraction', Int]) -> 'Fraction':
            ...

        def __truediv__(self, other: _Union['Fraction', Int, _Any]
                        ) -> _Union['Fraction', _Any]:
            return (
                Fraction(*_normalize_components_sign(
                    *map(_mul,
                         _normalize_components_moduli(self.numerator,
                                                      other.numerator),
                         _normalize_components_moduli(other.denominator,
                                                      self.denominator))),
                         _normalize=False)
                if isinstance(other, Fraction)
                else (self._truediv_by_int(other)
                      if isinstance(other, Int)
                      else NotImplemented))

        def _add_fraction(self, other: 'Fraction') -> 'Fraction':
            return Fraction(
                *_normalize_components_moduli(
                    self.numerator * other.denominator
                    + other.numerator * self.denominator,
                    self.denominator * other.denominator),
                _normalize=False)

        def _add_int(self, other: Int) -> 'Fraction':
            return Fraction(
                *_normalize_components_moduli(self.numerator
                                              + other * self.denominator,
                                              self.denominator),
                _normalize=False)

        def _mul_by_fraction(self, other: 'Fraction') -> 'Fraction':
            numerator, other_denominator = _normalize_components_moduli(
                self.numerator, other.denominator)
            other_numerator, denominator = _normalize_components_moduli(
                other.numerator, self.denominator)
            return Fraction(numerator * other_numerator,
                            denominator * other_denominator,
                            _normalize=False)

        def _mul_by_int(self, other: Int) -> 'Fraction':
            other, denominator = _normalize_components_moduli(other,
                                                              self.denominator)
            return Fraction(self.numerator * other, denominator,
                            _normalize=False)

        def _rtruediv_by_int(self, other: Int) -> 'Fraction':
            other, numerator = _normalize_components_moduli(other,
                                                            self.numerator)
            return Fraction(
                *_normalize_components_sign(other * self.denominator,
                                            numerator),
                _normalize=False)

        def _truediv_by_int(self, other: Int) -> 'Fraction':
            numerator, other = _normalize_components_moduli(self.numerator,
                                                            other)
            return Fraction(
                *_normalize_components_sign(numerator,
                                            other * self.denominator),
                _normalize=False)


    def _normalize_components_moduli(numerator: Int,
                                     denominator: Int) -> _Tuple[Int, Int]:
        gcd = numerator.gcd(denominator)
        return numerator // gcd, denominator // gcd


    def _normalize_components_sign(numerator: Int,
                                   denominator: Int) -> _Tuple[Int, Int]:
        return ((-numerator, -denominator)
                if denominator < _ZERO
                else (numerator, denominator))

<filename>rithm/__init__.py
"""Arbitrary precision arithmetic."""

__version__ = '5.2.0'

try:
    from ._rithm import (Fraction,
                         Int)
except ImportError:
    from math import gcd as _gcd
    from operator import mul as _mul
    from typing import (Any as _Any,
                        Optional as _Optional,
                        Tuple as _Tuple,
                        Union as _Union,
                        overload as _overload)


    class Int:
        def bit_length(self) -> 'Int':
            return Int(self._value.bit_length())

        def gcd(self, other: 'Int') -> 'Int':
            return Int(_gcd(self._value, other._value))

        __slots__ = '_value',

        def __new__(cls,
                    _value: _Union[str, int] = 0,
                    _base: _Optional[int] = None) -> 'Int':
            self = super().__new__(cls)
            self._value = (int(_value, _base)
                           if _base is not None
                           else int(_value))
            return self

        def __abs__(self) -> 'Int':
            return Int(abs(self._value))

        def __add__(self, other: 'Int') -> 'Int':
            return (Int(self._value + other._value)
                    if isinstance(other, Int)
                    else NotImplemented)

        def __and__(self, other: 'Int') -> 'Int':
            return (Int(self._value & other._value)
                    if isinstance(other, Int)
                    else NotImplemented)

        def __bool__(self) -> bool:
            return bool(self._value)

        def __ceil__(self) -> 'Int':
            return self

        def __divmod__(self, other: 'Int') -> _Tuple['Int', 'Int']:
            return (tuple(Int(value)
                          for value in divmod(self._value, other._value))
                    if isinstance(other, Int)
                    else NotImplemented)

        def __eq__(self, other: 'Int') -> bool:
            return (self._value == other._value
                    if isinstance(other, Int)
                    else NotImplemented)

        def __float__(self) -> float:
            return float(self._value)

        def __floor__(self) -> 'Int':
            return self

        def __floordiv__(self, other: 'Int') -> 'Int':
            return (Int(self._value // other._value)
                    if isinstance(other, Int)
                    else NotImplemented)

        def __ge__(self, other: 'Int') -> bool:
            return (self._value >= other._value
                    if isinstance(other, Int)
                    else NotImplemented)

        def __getstate__(self) -> int:
            return self._value

        def __gt__(self, other: 'Int') -> bool:
            return (self._value > other._value
                    if isinstance(other, Int)
                    else NotImplemented)

        def __hash__(self) -> int:
            return hash(self._value)

        def __invert__(self) -> 'Int':
            return Int(~self._value)

        def __int__(self) -> int:
            return self._value

        def __le__(self, other: 'Int') -> bool:
            return (self._value <= other._value
                    if isinstance(other, Int)
                    else NotImplemented)

        def __lshift__(self, other: 'Int') -> 'Int':
            return (Int(self._value << other._value)
                    if isinstance(other, Int)
                    else NotImplemented)

        def __lt__(self, other: 'Int') -> bool:
            return (self._value < other._value
                    if isinstance(other, Int)
                    else NotImplemented)

        def __mod__(self, other: 'Int') -> 'Int':
            return (Int(self._value % other._value)
                    if isinstance(other, Int)
                    else NotImplemented)

        def __mul__(self, other: 'Int') -> 'Int':
            return (Int(self._value * other._value)
                    if isinstance(other, Int)
                    else NotImplemented)

        def __neg__(self) -> 'Int':
            return Int(-self._value)

        def __or__(self, other: 'Int') -> 'Int':
            return (Int(self._value | other._value)
                    if isinstance(other, Int)
                    else NotImplemented)

        def __pow__(self,
                    exponent: 'Int',
                    divisor: _Optional['Int'] = None
                    ) -> _Union['Fraction', 'Int']:
            return (((Int(self._value ** exponent._value)
                      if exponent >= _ZERO
                      else Fraction(_ONE, self) ** -exponent)
                     if divisor is None
                     else (Int(pow(self._value, exponent._value,
                                   divisor._value))
                           if isinstance(divisor, Int)
                           else NotImplemented))
                    if isinstance(exponent, Int)
                    else NotImplemented)

        def __repr__(self) -> str:
            return f'rithm.Int(\'{self._value}\')'

        def __rshift__(self, other: 'Int') -> 'Int':
            return (Int(self._value >> other._value)
                    if isinstance(other, Int)
                    else NotImplemented)

        def __setstate__(self, state: int) -> None:
            self._value = state

        def __str__(self) -> str:
            return str(self._value)

        def __sub__(self, other: 'Int') -> 'Int':
            return (Int(self._value - other._value)
                    if isinstance(other, Int)
                    else NotImplemented)

        def __truediv__(self, other: 'Int') -> 'Fraction':
            return (Fraction(self, other)
                    if isinstance(other, Int)
                    else NotImplemented)

        def __trunc__(self) -> 'Int':
            return self

        def __xor__(self, other: 'Int') -> 'Int':
            return (Int(self._value ^ other._value)
                    if isinstance(other, Int)
                    else NotImplemented)


    _ONE = Int('1')
    _ZERO = Int()


    class Fraction:
        @property
        def denominator(self) -> Int:
            return self._denominator

        @property
        def numerator(self) -> Int:
            return self._numerator

        __slots__ = '_denominator', '_numerator'

        def __new__(cls,
                    _numerator: _Union[Int, float] = _ZERO,
                    _denominator: _Optional[Int] = None,
                    *,
                    _normalize: bool = True) -> 'Fraction':
            self = super().__new__(cls)
            if _denominator is None:
                if isinstance(_numerator, float):
                    raw_numerator, raw_denominator = (
                        _numerator.as_integer_ratio())
                    numerator, denominator = (Int(raw_numerator),
                                              Int(raw_denominator))
                elif isinstance(_numerator, Int):
                    numerator, denominator = _numerator, _ONE
                else:
                    raise TypeError('First argument should be of '
                                    f'type {Int} or {float}, '
                                    f'but found: {type(_numerator)}.')
            elif isinstance(_denominator, Int):
                numerator, denominator = _numerator, _denominator
            else:
                raise TypeError(f'Denominator should be of type {Int}, '
                                f'but found: {type(_denominator)}.')
            if not denominator:
                raise ZeroDivisionError('Denominator should not be zero.')
            if _normalize:
                numerator, denominator = _normalize_components_sign(
                    *_normalize_components_moduli(numerator, denominator))
            self._numerator, self._denominator = numerator, denominator
            return self

        def __abs__(self) -> 'Fraction':
            return Fraction(abs(self.numerator), self.denominator,
                            _normalize=False)

        @_overload
        def __add__(self, other: _Any) -> _Any:
            ...

        @_overload
        def __add__(self, other: _Union['Fraction', Int]) -> 'Fraction':
            ...

        def __add__(self, other: _Union['Fraction', Int, _Any]
                    ) -> _Union['Fraction', _Any]:
            return (self._add_fraction(other)
                    if isinstance(other, Fraction)
                    else (self._add_int(other)
                          if isinstance(other, Int)
                          else NotImplemented))

        def __bool__(self) -> bool:
            return bool(self.numerator)

        def __eq__(self, other: 'Fraction') -> bool:
            return (self.numerator == other.numerator
                    and self.denominator == other.denominator
                    if isinstance(other, Fraction)
                    else (self.denominator == _ONE
                          and self.numerator == other
                          if isinstance(other, Int)
                          else NotImplemented))

        def __float__(self) -> float:
            return self._numerator._value / self._denominator._value

        @_overload
        def __floordiv__(self, other: _Any) -> _Any:
            ...

        @_overload
        def __floordiv__(self, other: _Union['Fraction', Int]) -> Int:
            ...

        def __floordiv__(self, other: _Union['Fraction', Int, _Any]
                         ) -> _Union[Int, _Any]:
            return ((self.numerator * other.denominator)
                    // (self.denominator * other.numerator)
                    if isinstance(other, Fraction)
                    else (self.numerator // (self.denominator * other)
                          if isinstance(other, Int)
                          else NotImplemented))

        def __ge__(self, other: 'Fraction') -> bool:
            return (self.numerator * other.denominator
                    >= other.numerator * self.denominator
                    if isinstance(other, Fraction)
                    else (self.numerator >= other * self.denominator
                          if isinstance(other, Int)
                          else NotImplemented))

        def __getstate__(self) -> _Tuple[Int, Int]:
            return self._numerator, self._denominator

        def __gt__(self, other: 'Fraction') -> bool:
            return (self.numerator * other.denominator
                    > other.numerator * self.denominator
                    if isinstance(other, Fraction)
                    else (self.numerator > other * self.denominator
                          if isinstance(other, Int)
                          else NotImplemented))

        def __le__(self, other: 'Fraction') -> bool:
            return (self.numerator * other.denominator
                    <= other.numerator * self.denominator
                    if isinstance(other, Fraction)
                    else (self.numerator <= other * self.denominator
                          if isinstance(other, Int)
                          else NotImplemented))

        def __lt__(self, other: 'Fraction') -> bool:
            return (self.numerator * other.denominator
                    < other.numerator * self.denominator
                    if isinstance(other, Fraction)
                    else (self.numerator < other * self.denominator
                          if isinstance(other, Int)
                          else NotImplemented))

        @_overload
        def __mod__(self, other: _Any) -> _Any:
            ...

        @_overload
        def __mod__(self, other: _Union['Fraction', Int]) -> 'Fraction':
            ...

        def __mod__(self, other: _Union['Fraction', Int, _Any]
                    ) -> _Union['Fraction', _Any]:
            return (Fraction((self.numerator * other.denominator)
                             % (self.denominator * other.numerator),
                             self.denominator * other.denominator)
                    if isinstance(other, Fraction)
                    else (Fraction(self.numerator % (self.denominator * other),
                                   self.denominator)
                          if isinstance(other, Int)
                          else NotImplemented))

        @_overload
        def __mul__(self, other: _Any) -> _Any:
            ...

        @_overload
        def __mul__(self, other: _Union['Fraction', Int]) -> 'Fraction':
            ...

        def __mul__(self, other: _Union['Fraction', Int, _Any]
                    ) -> _Union['Fraction', _Any]:
            return (self._mul_by_fraction(other)
                    if isinstance(other, Fraction)
                    else (self._mul_by_int(other)
                          if isinstance(other, Int)
                          else NotImplemented))

        def __neg__(self) -> 'Fraction':
            return Fraction(-self.numerator, self.denominator,
                            _normalize=False)

        def __pow__(self, exponent: 'Int', divisor: None = None) -> 'Fraction':
            return (
                (Fraction(self.numerator ** exponent,
                          self.denominator ** exponent,
                          _normalize=False)
                 if exponent >= _ZERO
                 else Fraction(
                    *_normalize_components_sign(self.denominator ** -exponent,
                                                self.numerator ** -exponent),
                    _normalize=False))
                if isinstance(exponent, Int) and divisor is None
                else NotImplemented)

        def __repr__(self) -> str:
            return f'rithm.Fraction({self.numerator!r}, {self.denominator!r})'

        @_overload
        def __radd__(self, other: _Any) -> _Any:
            ...

        @_overload
        def __radd__(self, other: Int) -> 'Fraction':
            ...

        def __radd__(self, other: _Union[Int, _Any]
                     ) -> _Union['Fraction', _Any]:
            return (self._add_int(other)
                    if isinstance(other, Int)
                    else NotImplemented)

        @_overload
        def __rfloordiv__(self, other: _Any) -> _Any:
            ...

        @_overload
        def __rfloordiv__(self, other: Int) -> Int:
            ...

        def __rfloordiv__(self, other: _Union[Int, _Any]) -> _Union[Int, _Any]:
            return ((other * self.denominator) // self.numerator
                    if isinstance(other, Int)
                    else NotImplemented)

        @_overload
        def __rmod__(self, other: _Any) -> _Any:
            ...

        @_overload
        def __rmod__(self, other: Int) -> 'Fraction':
            ...

        def __rmod__(self, other: _Union[Int, _Any]
                     ) -> _Union['Fraction', _Any]:
            return (Fraction((other * self.denominator) % self.numerator,
                             self.denominator)
                    if isinstance(other, Int)
                    else NotImplemented)

        @_overload
        def __rmul__(self, other: _Any) -> _Any:
            ...

        @_overload
        def __rmul__(self, other: Int) -> 'Fraction':
            ...

        def __rmul__(self, other: _Union[Int, _Any]
                     ) -> _Union['Fraction', _Any]:
            return (self._mul_by_int(other)
                    if isinstance(other, Int)
                    else NotImplemented)

        @_overload
        def __rsub__(self, other: _Any) -> _Any:
            ...

        @_overload
        def __rsub__(self, other: Int) -> 'Fraction':
            ...

        def __rsub__(self, other: _Union[Int, _Any]
                     ) -> _Union['Fraction', _Any]:
            return (
                Fraction(
                    *_normalize_components_moduli(other * self.denominator
                                                  - self.numerator,
                                                  self.denominator),
                    _normalize=False)
                if isinstance(other, Int)
                else NotImplemented)

        def __setstate__(self, state: _Tuple[Int, Int]) -> None:
            self._numerator, self._denominator = state

        def __str__(self) -> str:
            return (str(self.numerator)
                    if self.denominator == _ONE
                    else f'{self.numerator}/{self.denominator}')

        @_overload
        def __sub__(self, other: _Any) -> _Any:
            ...

        @_overload
        def __sub__(self, other: _Union['Fraction', Int]) -> 'Fraction':
            ...

        def __sub__(self, other: _Union['Fraction', Int, _Any]
                    ) -> _Union['Fraction', _Any]:
            return (
                Fraction(
                    *_normalize_components_moduli(
                        self.numerator * other.denominator
                        - other.numerator * self.denominator,
                        self.denominator * other.denominator),
                    _normalize=False)
                if isinstance(other, Fraction)
                else
                (Fraction(
                    *_normalize_components_moduli(self.numerator
                                                  - other * self.denominator,
                                                  self.denominator),
                    _normalize=False)
                 if isinstance(other, Int)
                 else NotImplemented))

        @_overload
        def __rtruediv__(self, other: _Any) -> _Any:
            ...

        @_overload
        def __rtruediv__(self, other: Int) -> 'Fraction':
            ...

        def __rtruediv__(self, other: _Union[Int, _Any]
                         ) -> _Union['Fraction', _Any]:
            return (self._rtruediv_by_int(other)
                    if isinstance(other, Int)
                    else NotImplemented)

        @_overload
        def __truediv__(self, other: _Any) -> _Any:
            ...

        @_overload
        def __truediv__(self, other: _Union['Fraction', Int]) -> 'Fraction':
            ...

        def __truediv__(self, other: _Union['Fraction', Int, _Any]
                        ) -> _Union['Fraction', _Any]:
            return (
                Fraction(*_normalize_components_sign(
                    *map(_mul,
                         _normalize_components_moduli(self.numerator,
                                                      other.numerator),
                         _normalize_components_moduli(other.denominator,
                                                      self.denominator))),
                         _normalize=False)
                if isinstance(other, Fraction)
                else (self._truediv_by_int(other)
                      if isinstance(other, Int)
                      else NotImplemented))

        def _add_fraction(self, other: 'Fraction') -> 'Fraction':
            return Fraction(
                *_normalize_components_moduli(
                    self.numerator * other.denominator
                    + other.numerator * self.denominator,
                    self.denominator * other.denominator),
                _normalize=False)

        def _add_int(self, other: Int) -> 'Fraction':
            return Fraction(
                *_normalize_components_moduli(self.numerator
                                              + other * self.denominator,
                                              self.denominator),
                _normalize=False)

        def _mul_by_fraction(self, other: 'Fraction') -> 'Fraction':
            numerator, other_denominator = _normalize_components_moduli(
                self.numerator, other.denominator)
            other_numerator, denominator = _normalize_components_moduli(
                other.numerator, self.denominator)
            return Fraction(numerator * other_numerator,
                            denominator * other_denominator,
                            _normalize=False)

        def _mul_by_int(self, other: Int) -> 'Fraction':
            other, denominator = _normalize_components_moduli(other,
                                                              self.denominator)
            return Fraction(self.numerator * other, denominator,
                            _normalize=False)

        def _rtruediv_by_int(self, other: Int) -> 'Fraction':
            other, numerator = _normalize_components_moduli(other,
                                                            self.numerator)
            return Fraction(
                *_normalize_components_sign(other * self.denominator,
                                            numerator),
                _normalize=False)

        def _truediv_by_int(self, other: Int) -> 'Fraction':
            numerator, other = _normalize_components_moduli(self.numerator,
                                                            other)
            return Fraction(
                *_normalize_components_sign(numerator,
                                            other * self.denominator),
                _normalize=False)


    def _normalize_components_moduli(numerator: Int,
                                     denominator: Int) -> _Tuple[Int, Int]:
        gcd = numerator.gcd(denominator)
        return numerator // gcd, denominator // gcd


    def _normalize_components_sign(numerator: Int,
                                   denominator: Int) -> _Tuple[Int, Int]:
        return ((-numerator, -denominator)
                if denominator < _ZERO
                else (numerator, denominator))

Arbitrary precision arithmetic.