


'''simple docstring'''



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




def 			_a			(		_lowercase      :			int						):





     '''simple docstring'''






     def is_in_circle(_lowercase      :			float    ,      _lowercase      :			float						) -> bool:
          __UpperCAmelCase			:				str													=    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			:				Any													=    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.
     __UpperCAmelCase			:				List[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 			_a			(		_lowercase      :			int    ,      _lowercase      :			Callable[[float], float]    ,      _lowercase      :			float = 0.0    ,      _lowercase      :			float = 1.0    ,      ):





     '''simple docstring'''






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




def 			_a			(		_lowercase      :			int    ,      _lowercase      :			float = 0.0    ,      _lowercase      :			float = 1.0						):





     '''simple docstring'''






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

     __UpperCAmelCase			:				Tuple													=    area_under_curve_estimator(
         _lowercase    ,      _lowercase    ,      _lowercase    ,      _lowercase						)
     __UpperCAmelCase			:				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			(		_lowercase      :			int						):





     '''simple docstring'''






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

     __UpperCAmelCase			:				Union[str, Any]													=    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()



'''simple docstring'''



def 			_a			(		_lowercase      :			int = 4000000						):





     '''simple docstring'''






     __UpperCAmelCase			:				Tuple													=    []
     __UpperCAmelCase							,     __UpperCAmelCase			:				int													=    0, 1
     while b <= n:
          if b % 2 == 0:
               even_fibs.append(_lowercase						)
          __UpperCAmelCase							,     __UpperCAmelCase			:				Dict													=    b, a + b
     return sum(_lowercase						)


if __name__ == "__main__":
      print(f"""{solution() = }""")