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




def     UpperCamelCase						(	__lowercase						:			int     ):
		'''simple docstring'''




		def is_in_circle(__lowercase						:			float						,__lowercase						:			float     ) -> bool:
				A_		:    Tuple         =				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
		A_		:    Dict         =				mean(
		    int(is_in_circle(uniform(-1.0						,1.0     )						,uniform(-1.0						,1.0     )     )     )
		    for _ in range(__lowercase     )     )
		# The ratio of the area for circle to square is pi/4.
		A_		:    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						(	__lowercase						:			int						,__lowercase						:			Callable[[float], float]						,__lowercase						:			float = 0.0						,__lowercase						:			float = 1.0						,):
		'''simple docstring'''




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




def     UpperCamelCase						(	__lowercase						:			int						,__lowercase						:			float = 0.0						,__lowercase						:			float = 1.0     ):
		'''simple docstring'''




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

		A_		:    List[Any]         =				area_under_curve_estimator(
		    __lowercase						,__lowercase						,__lowercase						,__lowercase     )
		A_		:    str         =				(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						(	__lowercase						:			int     ):
		'''simple docstring'''




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

		A_		:    List[str]         =				area_under_curve_estimator(
		    __lowercase						,__lowercase						,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()







import math




def     UpperCamelCase						(	__lowercase						:			int = 1_00     ):
		'''simple docstring'''




		A_		:    List[Any]         =				sum(i * i for i in range(1						,n + 1     )     )
		A_		:    int         =				int(math.pow(sum(range(1						,n + 1     )     )						,2     )     )
		return square_of_sum - sum_of_squares


if __name__ == "__main__":
	print(F"""{solution() = }""")






