


"""simple docstring"""






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


def __lowercase     (      lowerCamelCase_  :      int  ):

				# A local function to see if a dot lands in the circle.
				def is_in_circle(lowerCamelCase_  :      float						,					lowerCamelCase_  :      float  ) -> 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(lowerCamelCase_  )  )
				# 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 __lowercase     (      lowerCamelCase_  :      int						,					lowerCamelCase_  :      Callable[[float], float]						,					lowerCamelCase_  :      float = 0.0						,					lowerCamelCase_  :      float = 1.0						,					):
				return mean(
				    function_to_integrate(uniform(lowerCamelCase_						,					lowerCamelCase_  )  ) for _ in range(lowerCamelCase_  )  ) * (max_value - min_value)


def __lowercase     (      lowerCamelCase_  :      int						,					lowerCamelCase_  :      float = 0.0						,					lowerCamelCase_  :      float = 1.0  ):

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

				SCREAMING_SNAKE_CASE__     							=				area_under_curve_estimator(
				    lowerCamelCase_						,					lowerCamelCase_						,					lowerCamelCase_						,					lowerCamelCase_  )
				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 __lowercase     (      lowerCamelCase_  :      int  ):

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

				SCREAMING_SNAKE_CASE__     							=				area_under_curve_estimator(
				    lowerCamelCase_						,					lowerCamelCase_						,					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 math import pi


def __lowercase     (      lowerCamelCase_  :      int						,					lowerCamelCase_  :      int  ):
				return 2 * pi * radius * (angle / 360)


if __name__ == "__main__":
	print(arc_length(90, 10))
