




from __future__ import annotations







class 		lowercase							:

     def __init__(						self	,  _a       )		->							None:
           _A							:  Any    =		order

           # a_{0} ... a_{k}
           _A							:  Any    =		[1.0] + [0.0] * order
           # b_{0} ... b_{k}
           _A							:  Optional[Any]    =		[1.0] + [0.0] * order

           # x[n-1] ... x[n-k]
           _A							:  int    =		[0.0] * self.order
           # y[n-1] ... y[n-k]
           _A							:  Union[str, Any]    =		[0.0] * self.order

     def       a__			(						self	,  _a	,  _a       )		->							None:
           if len(_a       ) < self.order:
                 _A							:  Optional[Any]    =		[1.0, *a_coeffs]

           if len(_a       ) != self.order + 1:
                 _A							:  Dict    =		(
                     F'''Expected a_coeffs to have {self.order + 1} elements '''
                     F'''for {self.order}-order filter, got {len(_a       )}'''
                 )
                 raise ValueError(_a       )

           if len(_a       ) != self.order + 1:
                 _A							:  List[str]    =		(
                     F'''Expected b_coeffs to have {self.order + 1} elements '''
                     F'''for {self.order}-order filter, got {len(_a       )}'''
                 )
                 raise ValueError(_a       )

           _A							:  Union[str, Any]    =		a_coeffs
           _A							:  Tuple    =		b_coeffs



     def       a__			(						self	,  _a       )		->							float:
           _A							:  Union[str, Any]    =		0.0

           # Start at index 1 and do index 0 at the end.
           for i in range(1	,  self.order + 1       ):
                 result += (
                     self.b_coeffs[i] * self.input_history[i - 1]
                     - self.a_coeffs[i] * self.output_history[i - 1]
                 )

           _A							:  Any    =		(result + self.b_coeffs[0] * sample) / self.a_coeffs[0]

           _A							:  Union[str, Any]    =		self.input_history[:-1]
           _A							:  Optional[Any]    =		self.output_history[:-1]

           _A							:  int    =		sample
           _A							:  str    =		result

           return result







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()

