



"""simple docstring"""


class 			_a :
			"""simple docstring"""



			def __init__(		self     :    Tuple				,   __UpperCamelCase     :    list[int]   )->None:
							_UpperCAmelCase					=    len(__UpperCamelCase   )
							_UpperCAmelCase					=    [0] * len_array

							if len_array > 0:
											_UpperCAmelCase					=    array[0]

							for i in range(1				,   __UpperCamelCase   ):
											_UpperCAmelCase					=    self.prefix_sum[i - 1] + array[i]



			def  lowercase__						(		self     :    str				,   __UpperCamelCase     :    int				,   __UpperCamelCase     :    int   )->int:
							if start == 0:
											return self.prefix_sum[end]

							return self.prefix_sum[end] - self.prefix_sum[start - 1]



			def  lowercase__						(		self     :    Any				,   __UpperCamelCase     :    int   )->bool:
							_UpperCAmelCase					=    {0}
							for sum_item in self.prefix_sum:
											if sum_item - target_sum in sums:
															return True

											sums.add(__UpperCamelCase   )

							return False


if __name__ == "__main__":
					import doctest

					doctest.testmod()








"""simple docstring"""


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







def  lowercase	(      _SCREAMING_SNAKE_CASE   :						int				):






				'''simple docstring'''



				def is_in_circle(_SCREAMING_SNAKE_CASE   :						float					,     _SCREAMING_SNAKE_CASE   :						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(_SCREAMING_SNAKE_CASE				)				)
				# 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  lowercase	(      _SCREAMING_SNAKE_CASE   :						int					,     _SCREAMING_SNAKE_CASE   :						Callable[[float], float]					,     _SCREAMING_SNAKE_CASE   :						float = 0.0					,     _SCREAMING_SNAKE_CASE   :						float = 1.0					,     ):






				'''simple docstring'''



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







def  lowercase	(      _SCREAMING_SNAKE_CASE   :						int					,     _SCREAMING_SNAKE_CASE   :						float = 0.0					,     _SCREAMING_SNAKE_CASE   :						float = 1.0				):






				'''simple docstring'''



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

				_UpperCAmelCase					=    area_under_curve_estimator(
				    _SCREAMING_SNAKE_CASE					,     _SCREAMING_SNAKE_CASE					,     _SCREAMING_SNAKE_CASE					,     _SCREAMING_SNAKE_CASE				)
				_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  lowercase	(      _SCREAMING_SNAKE_CASE   :						int				):






				'''simple docstring'''



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

				_UpperCAmelCase					=    area_under_curve_estimator(
				    _SCREAMING_SNAKE_CASE					,     _SCREAMING_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()



