


from collections.abc import Callable
from math import pi, sqrt
from random import uniform
from statistics import mean
def        UpperCamelCase_(				_A   :int  )->		List[Any]:

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

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

			UpperCamelCase__					=       area_under_curve_estimator(
			    _A				,     _A				,     _A				,     _A  )
			UpperCamelCase__					=       (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   :int  )->		None:

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

			UpperCamelCase__					=       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()






from __future__ import annotations
def        UpperCamelCase_(				_A   :list[int]				,     _A   :int  )->		list[int]:
			UpperCamelCase__					=       0
			UpperCamelCase__					=       len(_A  ) - 1

			while i < j:
						if nums[i] + nums[j] == target:
									return [i, j]
						elif nums[i] + nums[j] < target:
									UpperCamelCase__					=       i + 1
						else:
									UpperCamelCase__					=       j - 1

			return []


if __name__ == "__main__":
						import doctest

						doctest.testmod()
						print(f'''{two_pointer([2, 7, 1_1, 1_5], 9) = }''')


