

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





def    _A							(						__snake_case  :Optional[Any]  )   -> Any:



   """simple docstring"""


   def is_in_circle(__snake_case  :Any   , __snake_case  :List[str]  ) -> bool:
      __SCREAMING_SNAKE_CASE       =				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
   __SCREAMING_SNAKE_CASE       =				mean(
       int(is_in_circle(uniform(-1.0   , 1.0  )   , uniform(-1.0   , 1.0  )  )  )
       for _ in range(__snake_case  )  )
   # The ratio of the area for circle to square is pi/4.
   __SCREAMING_SNAKE_CASE       =				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    _A							(						__snake_case  :int   , __snake_case  :Tuple   , __snake_case  :Dict = 0.0   , __snake_case  :str = 1.0   , )   -> float:



   """simple docstring"""


   return mean(
       function_to_integrate(uniform(__snake_case   , __snake_case  )  ) for _ in range(__snake_case  )  ) * (max_value - min_value)





def    _A							(						__snake_case  :Union[str, Any]   , __snake_case  :Optional[int] = 0.0   , __snake_case  :List[str] = 1.0  )   -> None:



   """simple docstring"""


   def identity_function(__snake_case  :Union[str, Any]  ) -> float:
      return x

   __SCREAMING_SNAKE_CASE       =				area_under_curve_estimator(
       __snake_case   , __snake_case   , __snake_case   , __snake_case  )
   __SCREAMING_SNAKE_CASE       =				(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    _A							(						__snake_case  :Any  )   -> None:



   """simple docstring"""


   def function_to_integrate(__snake_case  :Union[str, Any]  ) -> float:
      return sqrt(4.0 - x * x  )

   __SCREAMING_SNAKE_CASE       =				area_under_curve_estimator(
       __snake_case   , __snake_case   , 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 __future__ import annotations

from random import choice


def    _snake_case  (	A			)		->       int:
							return choice(A			)


def    _snake_case  (	A					,       A			)		->       int:
							lowerCAmelCase__			  =      random_pivot(A			)

							# partition based on pivot
							# linear time
							lowerCAmelCase__			  =      [e for e in lst if e < pivot]
							lowerCAmelCase__			  =      [e for e in lst if e > pivot]

							# if we get lucky, pivot might be the element we want.
							# we can easily see this:
							# small (elements smaller than k)
							# + pivot (kth element)
							# + big (elements larger than k)
							if len(A			) == k - 1:
														return pivot
							# pivot is in elements bigger than k
							elif len(A			) < k - 1:
														return kth_number(A					,       k - len(A			) - 1			)
							# pivot is in elements smaller than k
							else:
														return kth_number(A					,       A			)


if __name__ == "__main__":
		import doctest

		doctest.testmod()