



"""simple docstring"""







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

def       _lowerCamelCase				(							UpperCAmelCase_ :       int     )	-> Optional[Any]:






	"""simple docstring"""





	def is_in_circle(UpperCAmelCase_ :       float,	UpperCAmelCase_ :       float     ) -> bool:
		A__											=					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__											=					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.
	A__											=					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       _lowerCamelCase				(							UpperCAmelCase_ :       int,	UpperCAmelCase_ :       Callable[[float], float],	UpperCAmelCase_ :       float = 0.0,	UpperCAmelCase_ :       float = 1.0,	)	-> float:






	"""simple docstring"""





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

def       _lowerCamelCase				(							UpperCAmelCase_ :       int,	UpperCAmelCase_ :       float = 0.0,	UpperCAmelCase_ :       float = 1.0     )	-> None:






	"""simple docstring"""





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

	A__											=					area_under_curve_estimator(
	    UpperCAmelCase_,	UpperCAmelCase_,	UpperCAmelCase_,	UpperCAmelCase_     )
	A__											=					(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       _lowerCamelCase				(							UpperCAmelCase_ :       int     )	-> None:






	"""simple docstring"""





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

	A__											=					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()











def 					SCREAMING_SNAKE_CASE__				(							UpperCamelCase__:				float       , UpperCamelCase__:				float     ):
		if mass < 0:
				raise ValueError("""The mass of a body cannot be negative"""     )
		return 0.5 * mass * abs(UpperCamelCase__     ) * abs(UpperCamelCase__     )


if __name__ == "__main__":
	import doctest

	doctest.testmod(verbose=True)