




from __future__ import annotations







def        lowerCAmelCase_			(  snake_case_,snake_case_,snake_case_,):
      if (stress, tangential_force, area).count(0  ) != 1:
            raise ValueError("""You cannot supply more or less than 2 values"""  )
      elif stress < 0:
            raise ValueError("""Stress cannot be negative"""  )
      elif tangential_force < 0:
            raise ValueError("""Tangential Force cannot be negative"""  )
      elif area < 0:
            raise ValueError("""Area cannot be negative"""  )
      elif stress == 0:
            return (
                "stress",
                tangential_force / area,
            )
      elif tangential_force == 0:
            return (
                "tangential_force",
                stress * area,
            )
      else:
            return (
                "area",
                tangential_force / stress,
            )


if __name__ == "__main__":
  import doctest

  doctest.testmod()










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







def        lowerCAmelCase_			(  snake_case_  ):

      # A local function to see if a dot lands in the circle.
      def is_in_circle(snake_case_,snake_case_  ) -> bool:
            _A							:  List[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
      _A							:  Optional[int]    =		mean(
          int(is_in_circle(uniform(-1.0,1.0  ),uniform(-1.0,1.0  )  )  )
          for _ in range(snake_case_  )  )
      # The ratio of the area for circle to square is pi/4.
      _A							:  List[str]    =		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        lowerCAmelCase_			(  snake_case_,snake_case_,snake_case_ = 0.0,snake_case_ = 1.0,):
      return mean(
          function_to_integrate(uniform(snake_case_,snake_case_  )  ) for _ in range(snake_case_  )  ) * (max_value - min_value)







def        lowerCAmelCase_			(  snake_case_,snake_case_ = 0.0,snake_case_ = 1.0  ):

      def identity_function(snake_case_  ) -> float:
            return x

      _A							:  Any    =		area_under_curve_estimator(
          snake_case_,snake_case_,snake_case_,snake_case_  )
      _A							:  Tuple    =		(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        lowerCAmelCase_			(  snake_case_  ):

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

      _A							:  Optional[int]    =		area_under_curve_estimator(
          snake_case_,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()

