

'''simple docstring'''





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





def a(lowercase__	):





	'''simple docstring'''

	# A local function to see if a dot lands in the circle.
	def is_in_circle(lowercase__			, lowercase__	) -> bool:
		snake_case_      		= 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
	snake_case_      		= mean(
	    int(is_in_circle(uniform(-1.0			, 1.0	)			, uniform(-1.0			, 1.0	)	)	)
	    for _ in range(lowercase__	)	)
	# The ratio of the area for circle to square is pi/4.
	snake_case_      		= 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(lowercase__			, lowercase__			, lowercase__ = 0.0			, lowercase__ = 1.0			, ):





	'''simple docstring'''
	return mean(
	    function_to_integrate(uniform(lowercase__			, lowercase__	)	) for _ in range(lowercase__	)	) * (max_value - min_value)





def a(lowercase__			, lowercase__ = 0.0			, lowercase__ = 1.0	):





	'''simple docstring'''

	def identity_function(lowercase__	) -> float:
		return x

	snake_case_      		= area_under_curve_estimator(
	    lowercase__			, lowercase__			, lowercase__			, lowercase__	)
	snake_case_      		= (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(lowercase__	):





	'''simple docstring'''

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

	snake_case_      		= area_under_curve_estimator(
	    lowercase__			, lowercase__			, 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()



def a(lowercase__			, lowercase__	):





	'''simple docstring'''
	if not isinstance(lowercase__			, lowercase__	):
		raise ValueError('iterations must be defined as integers'	)
	if not isinstance(lowercase__			, lowercase__	) or not number >= 1:
		raise ValueError(
		    'starting number must be\n                         and integer and be more than 0'	)
	if not iterations >= 1:
		raise ValueError('Iterations must be done more than 0 times to play FizzBuzz'	)

	snake_case_      		= ''
	while number <= iterations:
		if number % 3 == 0:
			out += "Fizz"
		if number % 5 == 0:
			out += "Buzz"
		if 0 not in (number % 3, number % 5):
			out += str(lowercase__	)

		# print(out)
		number += 1
		out += " "
	return out


if __name__ == "__main__":
		import doctest

		doctest.testmod()
