





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



def      __lowerCamelCase   (							a_     :		int							)			->      Tuple:

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



def      __lowerCamelCase   (							a_     :		int ,   a_     :		float = 0.0 ,   a_     :		float = 1.0							)			->      None:

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

      __SCREAMING_SNAKE_CASE   :str						=  area_under_curve_estimator(
          a_ ,   a_ ,   a_ ,   a_							)
      __SCREAMING_SNAKE_CASE   :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      __lowerCamelCase   (							a_     :		int							)			->      None:

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

      __SCREAMING_SNAKE_CASE   :Tuple						=  area_under_curve_estimator(
          a_ ,   a_ ,   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      __lowerCamelCase   (							a_     :		int ,   a_     :		str							)			->      Optional[int]:
      __SCREAMING_SNAKE_CASE   :Optional[int]						=  [1]
      for i in range(2 ,   a_							):
            factorials.append(factorials[-1] * i							)
      assert 0 <= k < factorials[-1] * n, "k out of bounds"

      __SCREAMING_SNAKE_CASE   :List[str]						=  []
      __SCREAMING_SNAKE_CASE   :Optional[Any]						=  list(range(a_							)							)

      # Find permutation
      while factorials:
            __SCREAMING_SNAKE_CASE   :Optional[int]						=  factorials.pop()
            __SCREAMING_SNAKE_CASE				,__SCREAMING_SNAKE_CASE   :Dict						=  divmod(a_ ,   a_							)
            permutation.append(elements[number]							)
            elements.remove(elements[number]							)
      permutation.append(elements[0]							)

      return permutation


if __name__ == "__main__":
  import doctest

  doctest.testmod()