'''simple docstring'''




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

def _lowercase	(      lowerCamelCase__						)					->       Dict:





       """simple docstring"""


       def is_in_circle(lowerCamelCase__							,    lowerCamelCase__						) -> bool:
              __UpperCAmelCase  :				List[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  :				Dict       				=		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.
       __UpperCAmelCase  :				Optional[Any]       				=		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 _lowercase	(      lowerCamelCase__							,    lowerCamelCase__							,    lowerCamelCase__ = 0.0							,    lowerCamelCase__ = 1.0							,    )					->       float:





       """simple docstring"""


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

def _lowercase	(      lowerCamelCase__							,    lowerCamelCase__ = 0.0							,    lowerCamelCase__ = 1.0						)					->       None:





       """simple docstring"""


       def identity_function(lowerCamelCase__						) -> float:
              return x

       __UpperCAmelCase  :				int       				=		area_under_curve_estimator(
           lowerCamelCase__							,    lowerCamelCase__							,    lowerCamelCase__							,    lowerCamelCase__						)
       __UpperCAmelCase  :				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 _lowercase	(      lowerCamelCase__						)					->       None:





       """simple docstring"""


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

       __UpperCAmelCase  :				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()







'''simple docstring'''




from maths.is_square_free import is_square_free
from maths.prime_factors import prime_factors

def _lowercase	(      lowerCamelCase__						)					->       int:





       """simple docstring"""


       __UpperCAmelCase  :				Any       				=		prime_factors(lowerCamelCase__						)
       if is_square_free(lowerCamelCase__						):
              return -1 if len(lowerCamelCase__						) % 2 else 1
       return 0


if __name__ == "__main__":
   import doctest

   doctest.testmod()






