

'''simple docstring'''

from collections.abc import Callable
from math import pi, sqrt
from random import uniform
from statistics import mean




def 				UpperCamelCase(					UpperCAmelCase_     ):

 # A local function to see if a dot lands in the circle.
 def is_in_circle(UpperCAmelCase_  ,	UpperCAmelCase_     ) -> bool:
  UpperCAmelCase	:     Union[str, Any]						 =			sqrt((x**2) + (y**2)     )
  # Our circle has a radius of 1, so a distance
  # greater than 1 would land outside the circle.
  return distance_from_centre <= 1

 # The proportion of guesses that landed in the circle
 UpperCAmelCase	:     List[Any]						 =			mean(
     int(is_in_circle(uniform(-1.0  ,	1.0     )  ,	uniform(-1.0  ,	1.0     )     )     )
     for _ in range(UpperCAmelCase_     )     )
 # The ratio of the area for circle to square is pi/4.
 UpperCAmelCase	:     List[str]						 =			proportion * 4
 print(F"""The estimated value of pi is {pi_estimate}"""     )
 print(F"""The numpy value of pi is {pi}"""     )
 print(F"""The total error is {abs(pi - pi_estimate     )}"""     )




def 				UpperCamelCase(					UpperCAmelCase_  ,	UpperCAmelCase_  ,	UpperCAmelCase_ = 0.0  ,	UpperCAmelCase_ = 1.0  ,	):
 return mean(
     function_to_integrate(uniform(UpperCAmelCase_  ,	UpperCAmelCase_     )     ) for _ in range(UpperCAmelCase_     )     ) * (max_value - min_value)




def 				UpperCamelCase(					UpperCAmelCase_  ,	UpperCAmelCase_ = 0.0  ,	UpperCAmelCase_ = 1.0     ):

 def identity_function(UpperCAmelCase_     ) -> float:
  return x

 UpperCAmelCase	:     Any						 =			area_under_curve_estimator(
     UpperCAmelCase_  ,	UpperCAmelCase_  ,	UpperCAmelCase_  ,	UpperCAmelCase_     )
 UpperCAmelCase	:     List[Any]						 =			(max_value * max_value - min_value * min_value) / 2

 print('******************'     )
 print(F"""Estimating area under y=x where x varies from {min_value} to {max_value}"""     )
 print(F"""Estimated value is {estimated_value}"""     )
 print(F"""Expected value is {expected_value}"""     )
 print(F"""Total error is {abs(estimated_value - expected_value     )}"""     )
 print('******************'     )




def 				UpperCamelCase(					UpperCAmelCase_     ):

 def function_to_integrate(UpperCAmelCase_     ) -> float:
  return sqrt(4.0 - x * x     )

 UpperCAmelCase	:     Optional[int]						 =			area_under_curve_estimator(
     UpperCAmelCase_  ,	UpperCAmelCase_  ,	0.0  ,	2.0     )

 print('******************'     )
 print('Estimating pi using area_under_curve_estimator'     )
 print(F"""Estimated value is {estimated_value}"""     )
 print(F"""Expected value is {pi}"""     )
 print(F"""Total error is {abs(estimated_value - pi     )}"""     )
 print('******************'     )


if __name__ == "__main__":
 import doctest

 doctest.testmod()







'''simple docstring'''

from collections.abc import Iterator, MutableMapping
from dataclasses import dataclass
from typing import Generic, TypeVar

lowercase__         =      TypeVar("KEY")
lowercase__         =      TypeVar("VAL")




@dataclass(frozen=_snake_case	,       slots=_snake_case							)
class 	A_			(    Generic[KEY, VAL]							):






     '''simple docstring'''



     UpperCAmelCase_						:     KEY
     UpperCAmelCase_						:     VAL




class 	A_			(    _Item							):






     '''simple docstring'''



     def __init__(		self			:					Any	)						->     None:
      super().__init__(lowercase_					,      lowercase_	)




     def __bool__(		self			:					List[str]	)						->     bool:
      return False



lowercase__         =      _DeletedItem()




class 	A_			(    MutableMapping[KEY, VAL]							):






     '''simple docstring'''



     def __init__(		self			:					Optional[int]					,      lowercase_			:					int = 8					,      lowercase_			:					float = 0.75	)						->     None:
      UpperCAmelCase	:     Dict						 =			initial_block_size
      UpperCAmelCase	:     list[_Item | None]						 =			[None] * initial_block_size
      assert 0.0 < capacity_factor < 1.0
      UpperCAmelCase	:     Any						 =			capacity_factor
      UpperCAmelCase	:     Union[str, Any]						 =			0


     def        UpperCAmelCase_	(		self			:					str					,      lowercase_			:					KEY	)						->     int:
      return hash(lowercase_	) % len(self._buckets	)


     def        UpperCAmelCase_	(		self			:					Any					,      lowercase_			:					int	)						->     int:
      return (ind + 1) % len(self._buckets	)


     def        UpperCAmelCase_	(		self			:					Dict					,      lowercase_			:					int					,      lowercase_			:					KEY					,      lowercase_			:					VAL	)						->     bool:
      UpperCAmelCase	:     List[Any]						 =			self._buckets[ind]
      if not stored:
       UpperCAmelCase	:     Dict						 =			_Item(lowercase_					,      lowercase_	)
       self._len += 1
       return True
      elif stored.key == key:
       UpperCAmelCase	:     Dict						 =			_Item(lowercase_					,      lowercase_	)
       return True
      else:
       return False


     def        UpperCAmelCase_	(		self			:					Any	)						->     bool:
      UpperCAmelCase	:     List[str]						 =			len(self._buckets	) * self._capacity_factor
      return len(self	) >= int(lowercase_	)


     def        UpperCAmelCase_	(		self			:					Optional[int]	)						->     bool:
      if len(self._buckets	) <= self._initial_block_size:
       return False
      UpperCAmelCase	:     Optional[int]						 =			len(self._buckets	) * self._capacity_factor / 2
      return len(self	) < limit


     def        UpperCAmelCase_	(		self			:					Dict					,      lowercase_			:					int	)						->     None:
      UpperCAmelCase	:     int						 =			self._buckets
      UpperCAmelCase	:     List[str]						 =			[None] * new_size
      UpperCAmelCase	:     Dict						 =			0
      for item in old_buckets:
       if item:
        self._add_item(item.key					,      item.val	)


     def        UpperCAmelCase_	(		self			:					Dict	)						->     None:
      self._resize(len(self._buckets	) * 2	)


     def        UpperCAmelCase_	(		self			:					Union[str, Any]	)						->     None:
      self._resize(len(self._buckets	) // 2	)


     def        UpperCAmelCase_	(		self			:					Union[str, Any]					,      lowercase_			:					KEY	)						->     Iterator[int]:
      UpperCAmelCase	:     Dict						 =			self._get_bucket_index(lowercase_	)
      for _ in range(len(self._buckets	)	):
       yield ind
       UpperCAmelCase	:     Union[str, Any]						 =			self._get_next_ind(lowercase_	)


     def        UpperCAmelCase_	(		self			:					Dict					,      lowercase_			:					KEY					,      lowercase_			:					VAL	)						->     None:
      for ind in self._iterate_buckets(lowercase_	):
       if self._try_set(lowercase_					,      lowercase_					,      lowercase_	):
        break


     def __setitem__(		self			:					Union[str, Any]					,      lowercase_			:					KEY					,      lowercase_			:					VAL	)						->     None:
      if self._is_full():
       self._size_up()

      self._add_item(lowercase_					,      lowercase_	)


     def __delitem__(		self			:					Tuple					,      lowercase_			:					KEY	)						->     None:
      for ind in self._iterate_buckets(lowercase_	):
       UpperCAmelCase	:     int						 =			self._buckets[ind]
       if item is None:
        raise KeyError(lowercase_	)
       if item is _deleted:
        continue
       if item.key == key:
        UpperCAmelCase	:     Union[str, Any]						 =			_deleted
        self._len -= 1
        break
      if self._is_sparse():
       self._size_down()


     def __getitem__(		self			:					Dict					,      lowercase_			:					KEY	)						->     VAL:
      for ind in self._iterate_buckets(lowercase_	):
       UpperCAmelCase	:     List[str]						 =			self._buckets[ind]
       if item is None:
        break
       if item is _deleted:
        continue
       if item.key == key:
        return item.val
      raise KeyError(lowercase_	)


     def __len__(		self			:					Optional[Any]	)						->     int:
      return self._len


     def __iter__(		self			:					int	)						->     Iterator[KEY]:
      yield from (item.key for item in self._buckets if item)




     def __repr__(		self			:					Optional[Any]	)						->     str:
      UpperCAmelCase	:     int						 =			' ,'.join(
          f"""{item.key}: {item.val}""" for item in self._buckets if item	)
      return f"""HashMap({val_string})"""




