

from __future__ import annotations

from fractions import Fraction



def 			_A	(      SCREAMING_SNAKE_CASE	,SCREAMING_SNAKE_CASE      ):
	return (
	    num != den and num % 1_0 == den // 1_0 and (num // 1_0) / (den % 1_0) == num / den
	)



def 			_A	(      SCREAMING_SNAKE_CASE      ):
	UpperCAmelCase__:	Tuple									= []
	UpperCAmelCase__:	int									= 1_1
	UpperCAmelCase__:	List[str]									= int("1" + "0" * digit_len      )
	for num in range(SCREAMING_SNAKE_CASE	,SCREAMING_SNAKE_CASE      ):
		while den <= 9_9:
			if (num != den) and (num % 1_0 == den // 1_0) and (den % 1_0 != 0):
				if is_digit_cancelling(SCREAMING_SNAKE_CASE	,SCREAMING_SNAKE_CASE      ):
					solutions.append(f"{num}/{den}"      )
			den += 1
		num += 1
		UpperCAmelCase__:	Optional[int]									= 1_0
	return solutions



def 			_A	(      SCREAMING_SNAKE_CASE = 2      ):
	UpperCAmelCase__:	Dict									= 1.0
	for fraction in fraction_list(SCREAMING_SNAKE_CASE      ):
		UpperCAmelCase__:	List[str]									= Fraction(SCREAMING_SNAKE_CASE      )
		result *= frac.denominator / frac.numerator
	return int(SCREAMING_SNAKE_CASE      )


if __name__ == "__main__":
					print(solution())


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



def 			_A	(      SCREAMING_SNAKE_CASE      ):

	# A local function to see if a dot lands in the circle.
	def is_in_circle(SCREAMING_SNAKE_CASE	,SCREAMING_SNAKE_CASE      ) -> bool:
		UpperCAmelCase__:	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
	UpperCAmelCase__:	Optional[int]									= 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__:	Optional[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 			_A	(      SCREAMING_SNAKE_CASE	,SCREAMING_SNAKE_CASE	,SCREAMING_SNAKE_CASE = 0.0	,SCREAMING_SNAKE_CASE = 1.0	,):
	return mean(
	    function_to_integrate(uniform(SCREAMING_SNAKE_CASE	,SCREAMING_SNAKE_CASE      )      ) for _ in range(SCREAMING_SNAKE_CASE      )      ) * (max_value - min_value)



def 			_A	(      SCREAMING_SNAKE_CASE	,SCREAMING_SNAKE_CASE = 0.0	,SCREAMING_SNAKE_CASE = 1.0      ):

	def identity_function(SCREAMING_SNAKE_CASE      ) -> float:
		return x

	UpperCAmelCase__:	Tuple									= area_under_curve_estimator(
	    SCREAMING_SNAKE_CASE	,SCREAMING_SNAKE_CASE	,SCREAMING_SNAKE_CASE	,SCREAMING_SNAKE_CASE      )
	UpperCAmelCase__:	Optional[Any]									= (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	(      SCREAMING_SNAKE_CASE      ):

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

	UpperCAmelCase__:	Union[str, Any]									= 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()