
def 	A_ (							snake_case							:   int ,      snake_case							:   list )      ->    Optional[Any]:



						'''simple docstring'''

						_enforce_args(snake_case ,      snake_case )
						if n == 0:
												return 0
						__UpperCamelCase       =			float('''-inf''' )
						for i in range(1 ,      n + 1 ):
												__UpperCamelCase       =			max(
												    snake_case ,      prices[i - 1] + naive_cut_rod_recursive(n - i ,      snake_case ) )

						return max_revue




def 	A_ (							snake_case							:   int ,      snake_case							:   list )      ->    Union[str, Any]:



						'''simple docstring'''

						_enforce_args(snake_case ,      snake_case )
						__UpperCamelCase       =			[float('''-inf''' ) for _ in range(n + 1 )]
						return _top_down_cut_rod_recursive(snake_case ,      snake_case ,      snake_case )




def 	A_ (							snake_case							:   int ,      snake_case							:   list ,      snake_case							:   list )      ->    Union[str, Any]:



						'''simple docstring'''

						if max_rev[n] >= 0:
												return max_rev[n]
						elif n == 0:
												return 0
						else:
												__UpperCamelCase       =			float('''-inf''' )
												for i in range(1 ,      n + 1 ):
																		__UpperCamelCase       =			max(
																		    snake_case ,      prices[i - 1] + _top_down_cut_rod_recursive(n - i ,      snake_case ,      snake_case ) ,      )

												__UpperCamelCase       =			max_revenue

						return max_rev[n]




def 	A_ (							snake_case							:   int ,      snake_case							:   list )      ->    List[str]:



						'''simple docstring'''

						_enforce_args(snake_case ,      snake_case )

						# length(max_rev) = n + 1, to accommodate for the revenue obtainable from a rod of
						# length 0.
						__UpperCamelCase       =			[float('''-inf''' ) for _ in range(n + 1 )]
						__UpperCamelCase       =			0

						for i in range(1 ,      n + 1 ):
												__UpperCamelCase       =			max_rev[i]
												for j in range(1 ,      i + 1 ):
																		__UpperCamelCase       =			max(snake_case ,      prices[j - 1] + max_rev[i - j] )

												__UpperCamelCase       =			max_revenue_i

						return max_rev[n]




def 	A_ (							snake_case							:   int ,      snake_case							:   list )      ->    Optional[Any]:



						'''simple docstring'''

						if n < 0:
												__UpperCamelCase       =			f"n must be greater than or equal to 0. Got n = {n}"
												raise ValueError(snake_case )

						if n > len(snake_case ):
												__UpperCamelCase       =			(
												    '''Each integral piece of rod must have a corresponding price. '''
												    f"Got n = {n} but length of prices = {len(snake_case )}"
												)
												raise ValueError(snake_case )




def 	A_ (							)      ->    Union[str, Any]:



						'''simple docstring'''

						__UpperCamelCase       =			[6, 10, 12, 15, 20, 23]
						__UpperCamelCase       =			len(snake_case )

						# the best revenue comes from cutting the rod into 6 pieces, each
						# of length 1 resulting in a revenue of 6 * 6 = 36.
						__UpperCamelCase       =			36

						__UpperCamelCase       =			top_down_cut_rod(snake_case ,      snake_case )
						__UpperCamelCase       =			bottom_up_cut_rod(snake_case ,      snake_case )
						__UpperCamelCase       =			naive_cut_rod_recursive(snake_case ,      snake_case )

						assert expected_max_revenue == max_rev_top_down
						assert max_rev_top_down == max_rev_bottom_up
						assert max_rev_bottom_up == max_rev_naive


if __name__ == "__main__":
				main()






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




def 	A_ (							snake_case							:   int )      ->    int:



						'''simple docstring'''

						def is_in_circle(snake_case							:   float ,      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(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 	A_ (							snake_case							:   int ,      snake_case							:   Callable[[float], float] ,      snake_case							:   float = 0.0 ,      snake_case							:   float = 1.0 ,      )      ->    float:



						'''simple docstring'''

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




def 	A_ (							snake_case							:   int ,      snake_case							:   float = 0.0 ,      snake_case							:   float = 1.0 )      ->    None:



						'''simple docstring'''

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

						__UpperCamelCase       =			area_under_curve_estimator(
						    snake_case ,      snake_case ,      snake_case ,      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 	A_ (							snake_case							:   int )      ->    None:



						'''simple docstring'''

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

						__UpperCamelCase       =			area_under_curve_estimator(
						    snake_case ,      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()




