


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







def   _a			( _SCREAMING_SNAKE_CASE			)      ->			Optional[Any]:

							# A local function to see if a dot lands in the circle.
							def is_in_circle(_SCREAMING_SNAKE_CASE			,	_SCREAMING_SNAKE_CASE			) -> 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(_SCREAMING_SNAKE_CASE			)			)
							# 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			( _SCREAMING_SNAKE_CASE			,	_SCREAMING_SNAKE_CASE			,	_SCREAMING_SNAKE_CASE = 0.0			,	_SCREAMING_SNAKE_CASE = 1.0			,	)      ->			float:
							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			)      ->			None:

							def identity_function(_SCREAMING_SNAKE_CASE			) -> float:
														return x

							snake_case_          =						area_under_curve_estimator(
							    _SCREAMING_SNAKE_CASE			,	_SCREAMING_SNAKE_CASE			,	_SCREAMING_SNAKE_CASE			,	_SCREAMING_SNAKE_CASE			)
							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			( _SCREAMING_SNAKE_CASE			)      ->			None:

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

							snake_case_          =						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()







"""simple docstring"""
def   _a			( _SCREAMING_SNAKE_CASE			,	_SCREAMING_SNAKE_CASE			)      ->			str:
							return "\n".join(
							    f"""{number} * {i} = {number * i}""" for i in range(1			,	number_of_terms + 1			)			)


if __name__ == "__main__":
				print(multiplication_table(number=5, number_of_terms=10))



