



'''simple docstring'''





from __future__ import annotations

from collections.abc import Iterable






class  lowerCamelCase_   :
  """simple docstring"""




  def __init__(			self				:     int      ,     _a				:     Dict      ,     _a				:     Optional[Any]       ) -> str:
    __lowerCamelCase	,__lowerCamelCase	:	List[Any]   	=			float(_a       ), float(_a       )




  def __eq__(			self				:     Tuple      ,     _a				:     List[Any]       ) -> Optional[int]:
    return self.x == other.x and self.y == other.y




  def __ne__(			self				:     Tuple      ,     _a				:     Optional[int]       ) -> List[str]:
    return not self == other




  def __gt__(			self				:     Tuple      ,     _a				:     str       ) -> str:
    if self.x > other.x:
      return True
    elif self.x == other.x:
      return self.y > other.y
    return False




  def __lt__(			self				:     Union[str, Any]      ,     _a				:     List[Any]       ) -> Optional[Any]:
    return not self > other




  def __ge__(			self				:     Tuple      ,     _a				:     Optional[Any]       ) -> Optional[int]:
    if self.x > other.x:
      return True
    elif self.x == other.x:
      return self.y >= other.y
    return False




  def __le__(			self				:     Any      ,     _a				:     List[str]       ) -> Optional[int]:
    if self.x < other.x:
      return True
    elif self.x == other.x:
      return self.y <= other.y
    return False




  def __repr__(			self				:     int       ) -> Optional[int]:
    return f'({self.x}, {self.y})'




  def __hash__(			self				:     Optional[Any]       ) -> Union[str, Any]:
    return hash(self.x       )




def 	a_					(							_lowerCAmelCase	,)				->				list[Point]:
  __lowerCamelCase	:	list[Point]   	=			[]
  if list_of_tuples:
    for p in list_of_tuples:
      if isinstance(_lowerCAmelCase	,_lowerCAmelCase	):
        points.append(_lowerCAmelCase	)
      else:
        try:
          points.append(Point(p[0]	,p[1]	)	)
        except (IndexError, TypeError):
          print(
              F'Ignoring deformed point {p}. All points'
              ' must have at least 2 coordinates.'	)
  return points

def 	a_					(							_lowerCAmelCase	)				->				list[Point]:

  if not hasattr(_lowerCAmelCase	,'__iter__'	):
    __lowerCamelCase	:	str   	=			F'Expecting an iterable object but got an non-iterable type {points}'
    raise ValueError(_lowerCAmelCase	)

  if not points:
    __lowerCamelCase	:	Any   	=			F'Expecting a list of points but got {points}'
    raise ValueError(_lowerCAmelCase	)

  return _construct_points(_lowerCAmelCase	)

def 	a_					(							_lowerCAmelCase	,_lowerCAmelCase	,_lowerCAmelCase	)				->				float:
  __lowerCamelCase	:	Optional[int]   	=			(a.x * b.y + b.x * c.y + c.x * a.y) - (a.y * b.x + b.y * c.x + c.y * a.x)
  return det

def 	a_					(							_lowerCAmelCase	)				->				list[Point]:
  __lowerCamelCase	:	Optional[Any]   	=			sorted(_validate_input(_lowerCAmelCase	)	)
  __lowerCamelCase	:	Optional[int]   	=			len(_lowerCAmelCase	)
  __lowerCamelCase	:	Union[str, Any]   	=			set()

  for i in range(n - 1	):
    for j in range(i + 1	,_lowerCAmelCase	):
      __lowerCamelCase	:	List[Any]   	=			False
      __lowerCamelCase	:	Union[str, Any]   	=			True
      for k in range(_lowerCAmelCase	):
        if k != i and k != j:
          __lowerCamelCase	:	Any   	=			_det(points[i]	,points[j]	,points[k]	)

          if det_k > 0:
            __lowerCamelCase	:	List[Any]   	=			True
          elif det_k < 0:
            __lowerCamelCase	:	Any   	=			True
          else:
            # point[i], point[j], point[k] all lie on a straight line
            # if point[k] is to the left of point[i] or it's to the
            # right of point[j], then point[i], point[j] cannot be
            # part of the convex hull of A
            if points[k] < points[i] or points[k] > points[j]:
              __lowerCamelCase	:	Dict   	=			False
              break

        if points_left_of_ij and points_right_of_ij:
          __lowerCamelCase	:	int   	=			False
          break

      if ij_part_of_convex_hull:
        convex_set.update([points[i], points[j]]	)

  return sorted(_lowerCAmelCase	)

def 	a_					(							_lowerCAmelCase	)				->				list[Point]:
  __lowerCamelCase	:	Dict   	=			sorted(_validate_input(_lowerCAmelCase	)	)
  __lowerCamelCase	:	Optional[Any]   	=			len(_lowerCAmelCase	)

  # divide all the points into an upper hull and a lower hull
  # the left most point and the right most point are definitely
  # members of the convex hull by definition.
  # use these two anchors to divide all the points into two hulls,
  # an upper hull and a lower hull.

  # all points to the left (above) the line joining the extreme points belong to the
  # upper hull
  # all points to the right (below) the line joining the extreme points below to the
  # lower hull
  # ignore all points on the line joining the extreme points since they cannot be
  # part of the convex hull

  __lowerCamelCase	:	str   	=			points[0]
  __lowerCamelCase	:	Optional[Any]   	=			points[n - 1]

  __lowerCamelCase	:	Tuple   	=			{left_most_point, right_most_point}
  __lowerCamelCase	:	int   	=			[]
  __lowerCamelCase	:	Tuple   	=			[]

  for i in range(1	,n - 1	):
    __lowerCamelCase	:	Dict   	=			_det(_lowerCAmelCase	,_lowerCAmelCase	,points[i]	)

    if det > 0:
      upper_hull.append(points[i]	)
    elif det < 0:
      lower_hull.append(points[i]	)

  _construct_hull(_lowerCAmelCase	,_lowerCAmelCase	,_lowerCAmelCase	,_lowerCAmelCase	)
  _construct_hull(_lowerCAmelCase	,_lowerCAmelCase	,_lowerCAmelCase	,_lowerCAmelCase	)

  return sorted(_lowerCAmelCase	)

def 	a_					(							_lowerCAmelCase	,_lowerCAmelCase	,_lowerCAmelCase	,_lowerCAmelCase	)				->				None:
  if points:
    __lowerCamelCase	:	Optional[int]   	=			None
    __lowerCamelCase	:	Optional[Any]   	=			float('-inf'	)
    __lowerCamelCase	:	Optional[int]   	=			[]

    for p in points:
      __lowerCamelCase	:	List[str]   	=			_det(_lowerCAmelCase	,_lowerCAmelCase	,_lowerCAmelCase	)

      if det > 0:
        candidate_points.append(_lowerCAmelCase	)

        if det > extreme_point_distance:
          __lowerCamelCase	:	Union[str, Any]   	=			det
          __lowerCamelCase	:	Dict   	=			p

    if extreme_point:
      _construct_hull(_lowerCAmelCase	,_lowerCAmelCase	,_lowerCAmelCase	,_lowerCAmelCase	)
      convex_set.add(_lowerCAmelCase	)
      _construct_hull(_lowerCAmelCase	,_lowerCAmelCase	,_lowerCAmelCase	,_lowerCAmelCase	)

def 	a_					(							_lowerCAmelCase	)				->				list[Point]:
  __lowerCamelCase	:	Optional[int]   	=			sorted(_validate_input(_lowerCAmelCase	)	)
  __lowerCamelCase	:	List[Any]   	=			len(_lowerCAmelCase	)

  __lowerCamelCase	:	Tuple   	=			points[:2]
  for i in range(2	,_lowerCAmelCase	):
    __lowerCamelCase	:	Tuple   	=			_det(convex_hull[1]	,convex_hull[0]	,points[i]	)
    if det > 0:
      convex_hull.insert(0	,points[i]	)
      break
    elif det < 0:
      convex_hull.append(points[i]	)
      break
    else:
      __lowerCamelCase	:	Optional[int]   	=			points[i]
  i += 1

  for j in range(_lowerCAmelCase	,_lowerCAmelCase	):
    if (
        _det(convex_hull[0]	,convex_hull[-1]	,points[j]	) > 0
        and _det(convex_hull[-1]	,convex_hull[0]	,points[1]	) < 0
    ):
      # The point lies within the convex hull
      continue

    convex_hull.insert(0	,points[j]	)
    convex_hull.append(points[j]	)
    while _det(convex_hull[0]	,convex_hull[1]	,convex_hull[2]	) >= 0:
      del convex_hull[1]
    while _det(convex_hull[-1]	,convex_hull[-2]	,convex_hull[-3]	) <= 0:
      del convex_hull[-2]

    # `convex_hull` is contains the convex hull in circular order
  return sorted(convex_hull[1:] if len(_lowerCAmelCase	) > 3 else convex_hull	)

def 	a_					(							)				->				Optional[int]:
  __lowerCamelCase	:	List[Any]   	=			[
      (0, 3),
      (2, 2),
      (1, 1),
      (2, 1),
      (3, 0),
      (0, 0),
      (3, 3),
      (2, -1),
      (2, -4),
      (1, -3),
  ]
  # the convex set of points is
  # [(0, 0), (0, 3), (1, -3), (2, -4), (3, 0), (3, 3)]
  __lowerCamelCase	:	Any   	=			convex_hull_bf(_lowerCAmelCase	)

  __lowerCamelCase	:	Optional[int]   	=			convex_hull_recursive(_lowerCAmelCase	)
  assert results_bf == results_recursive

  __lowerCamelCase	:	int   	=			convex_hull_melkman(_lowerCAmelCase	)
  assert results_bf == results_melkman

  print(_lowerCAmelCase	)


if __name__ == "__main__":
   main()











'''simple docstring'''





import random






class  lowerCamelCase_   :
  """simple docstring"""




  def __init__(			self				:     Union[str, Any]      ,     _a				:     float      ,     _a				:     float       ) -> None:
    __lowerCamelCase	:	Optional[int]   	=			x
    __lowerCamelCase	:	Union[str, Any]   	=			y




  def 						_lowercase					(			self				:     Tuple       ) -> bool:
    return (self.x**2 + self.y**2) <= 1




  @classmethod
  def 						_lowercase					(			cls				:     List[Any]       ) -> Optional[int]:
    return cls(x=random.random()      ,     y=random.random()       )




def 	a_					(							_lowerCAmelCase	)				->				float:
  if number_of_simulations < 1:
    raise ValueError('At least one simulation is necessary to estimate PI.'	)

  __lowerCamelCase	:	Optional[int]   	=			0
  for _ in range(_lowerCAmelCase	):
    __lowerCamelCase	:	Dict   	=			Point.random_unit_square()

    if random_point.is_in_unit_circle():
      number_in_unit_circle += 1

  return 4 * number_in_unit_circle / number_of_simulations


if __name__ == "__main__":
   # import doctest

   # doctest.testmod()
   from math import pi

   _UpperCamelCase							    =						'Please enter the desired number of Monte Carlo simulations: '
   _UpperCamelCase							    =						estimate_pi(int(input(prompt).strip()))
   print(f'''An estimate of PI is {my_pi} with an error of {abs(my_pi - pi)}''')






