



from collections.abc import Callable
from math import pi, sqrt
from random import uniform
from statistics import mean
def 		UpperCamelCase     (       snake_case__	:   int     )     ->					int:

   # A local function to see if a dot lands in the circle.
   def is_in_circle(snake_case__	:   float	,					snake_case__	:   float     ) -> bool:
      UpperCamelCase   :  List[str]						     =     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   :  Optional[int]						     =     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.
   UpperCamelCase   :  int						     =     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     (       snake_case__	:   int	,					snake_case__	:   Callable[[float], float]	,					snake_case__	:   float = 0.0	,					snake_case__	:   float = 1.0	,					)     ->					float:
   return mean(
       function_to_integrate(uniform(snake_case__	,					snake_case__     )     ) for _ in range(snake_case__     )     ) * (max_value - min_value)
def 		UpperCamelCase     (       snake_case__	:   int	,					snake_case__	:   float = 0.0	,					snake_case__	:   float = 1.0     )     ->					None:

   def identity_function(snake_case__	:   float     ) -> float:
      return x

   UpperCamelCase   :  Tuple						     =     area_under_curve_estimator(
       snake_case__	,					snake_case__	,					snake_case__	,					snake_case__     )
   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     (       snake_case__	:   int     )     ->					None:

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

   UpperCamelCase   :  Optional[Any]						     =     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()








from graphs.minimum_spanning_tree_kruskal import kruskal
def 		UpperCamelCase     (       )     ->					Tuple:
   UpperCamelCase   :  List[str]						     =     9
   UpperCamelCase   :  Optional[Any]						     =     [
       [0, 1, 4],
       [0, 7, 8],
       [1, 2, 8],
       [7, 8, 7],
       [7, 6, 1],
       [2, 8, 2],
       [8, 6, 6],
       [2, 3, 7],
       [2, 5, 4],
       [6, 5, 2],
       [3, 5, 14],
       [3, 4, 9],
       [5, 4, 10],
       [1, 7, 11],
   ]

   UpperCamelCase   :  int						     =     kruskal(snake_case__	,					snake_case__     )

   UpperCamelCase   :  List[str]						     =     [
       [7, 6, 1],
       [2, 8, 2],
       [6, 5, 2],
       [0, 1, 4],
       [2, 5, 4],
       [2, 3, 7],
       [0, 7, 8],
       [3, 4, 9],
   ]

   assert sorted(snake_case__     ) == sorted(snake_case__     )



