

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



def 	UpperCAmelCase__   (				A__   )      ->      Any:




   """simple docstring"""

   # A local function to see if a dot lands in the circle.
   def is_in_circle(A__	,    A__   ) -> bool:
      lowerCamelCase__						=			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
   lowerCamelCase__						=			mean(
       int(is_in_circle(uniform(-1.0	,    1.0   )	,    uniform(-1.0	,    1.0   )   )   )
       for _ in range(A__   )   )
   # The ratio of the area for circle to square is pi/4.
   lowerCamelCase__						=			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__   (				A__	,    A__	,    A__ = 0.0	,    A__ = 1.0	,    )      ->      float:




   """simple docstring"""
   return mean(
       function_to_integrate(uniform(A__	,    A__   )   ) for _ in range(A__   )   ) * (max_value - min_value)



def 	UpperCAmelCase__   (				A__	,    A__ = 0.0	,    A__ = 1.0   )      ->      None:




   """simple docstring"""

   def identity_function(A__   ) -> float:
      return x

   lowerCamelCase__						=			area_under_curve_estimator(
       A__	,    A__	,    A__	,    A__   )
   lowerCamelCase__						=			(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__   (				A__   )      ->      None:




   """simple docstring"""

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

   lowerCamelCase__						=			area_under_curve_estimator(
       A__	,    A__	,    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"""




# Algorithm for the pigeonhole sorting





def 	UpperCAmelCase__   (				A__   )      ->      Dict:




   """simple docstring"""
   lowerCamelCase__						=			min(A__   )  # min() finds the minimum value
   lowerCamelCase__						=			max(A__   )  # max() finds the maximum value

   lowerCamelCase__						=			max_val - min_val + 1  # size is difference of max and min values plus one

   # list of pigeonholes of size equal to the variable size
   lowerCamelCase__						=			[0] * size

   # Populate the pigeonholes.
   for x in a:
      assert isinstance(A__	,    A__   ), "integers only please"
      holes[x - min_val] += 1

   # Putting the elements back into the array in an order.
   lowerCamelCase__						=			0
   for count in range(A__   ):
      while holes[count] > 0:
         holes[count] -= 1
         lowerCamelCase__						=			count + min_val
         i += 1



def 	UpperCAmelCase__   (				)      ->      Optional[int]:




   """simple docstring"""
   lowerCamelCase__						=			[8, 3, 2, 7, 4, 6, 8]
   pigeonhole_sort(A__   )
   print("Sorted order is:"	,    " ".join(A__   )   )


if __name__ == "__main__":
     main()




