



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






def 			lowerCAmelCase__	(					lowerCamelCase_	:	int):





				'''simple docstring'''

				def is_in_circle(lowerCamelCase_	:	float ,lowerCamelCase_	:	float) -> bool:
								lowerCAmelCase__  :       Any				=  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
				lowerCAmelCase__  :       List[Any]				=  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.
				lowerCAmelCase__  :       int				=  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__	(					lowerCamelCase_	:	int ,lowerCamelCase_	:	Callable[[float], float] ,lowerCamelCase_	:	float = 0.0 ,lowerCamelCase_	:	float = 1.0 ,):





				'''simple docstring'''

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






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





				'''simple docstring'''

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

				lowerCAmelCase__  :       Dict				=  area_under_curve_estimator(
				    lowerCamelCase_ ,lowerCamelCase_ ,lowerCamelCase_ ,lowerCamelCase_)
				lowerCAmelCase__  :       int				=  (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__	(					lowerCamelCase_	:	int):





				'''simple docstring'''

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

				lowerCAmelCase__  :       Optional[Any]				=  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()






from __future__ import annotations

from collections.abc import Iterable, Iterator
from dataclasses import dataclass

__snake_case       :	Optional[int]            =(3, 9, -1_1, 0, 7, 5, 1, -1)
__snake_case       :	str            =(4, 6, 2, 0, 8, 1_0, 3, -2)



@dataclass
class  lowerCamelCase__    :




		'''simple docstring'''



		snake_case_			=42
		snake_case_			=42



class  lowerCamelCase__    :




		'''simple docstring'''



		def __init__(self	,__lowerCamelCase				)  ->    None:

						"""simple docstring"""






						lowerCAmelCase__  :       Node | None				=  None
						for i in sorted(__lowerCamelCase	,reverse=__lowerCamelCase				):
										lowerCAmelCase__  :       Dict				=  Node(__lowerCamelCase	,self.head				)



		def __iter__(self				)  ->    Iterator[int]:

						"""simple docstring"""






						lowerCAmelCase__  :       Optional[Any]				=  self.head
						while node:
										yield node.data
										lowerCAmelCase__  :       Dict				=  node.next_node



		def __len__(self				)  ->    int:

						"""simple docstring"""






						return sum(1 for _ in self				)



		def __str__(self				)  ->    str:

						"""simple docstring"""






						return " -> ".join([str(__lowerCamelCase				) for node in self]				)






def 			lowerCAmelCase__	(					lowerCamelCase_	:	SortedLinkedList ,lowerCamelCase_	:	SortedLinkedList):





				'''simple docstring'''

				return SortedLinkedList(list(lowerCamelCase_) + list(lowerCamelCase_))


if __name__ == "__main__":
							import doctest

							doctest.testmod()
							__snake_case       :	Any            =SortedLinkedList
							print(merge_lists(SSL(test_data_odd), SSL(test_data_even)))

