





'''simple docstring'''

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





def 						UpperCamelCase				(							_lowerCamelCase					:					int       ):

    # A local function to see if a dot lands in the circle.
    def is_in_circle(_lowerCamelCase					:					float						,       _lowerCamelCase					:					float       ) -> bool:
        A__				=			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__				=			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.
    A__				=			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 						UpperCamelCase				(							_lowerCamelCase					:					int						,       _lowerCamelCase					:					Callable[[float], float]						,       _lowerCamelCase					:					float = 0.0						,       _lowerCamelCase					:					float = 1.0						,       ):
    return mean(
        function_to_integrate(uniform(_lowerCamelCase						,       _lowerCamelCase       )       ) for _ in range(_lowerCamelCase       )       ) * (max_value - min_value)





def 						UpperCamelCase				(							_lowerCamelCase					:					int						,       _lowerCamelCase					:					float = 0.0						,       _lowerCamelCase					:					float = 1.0       ):

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

    A__				=			area_under_curve_estimator(
        _lowerCamelCase						,       _lowerCamelCase						,       _lowerCamelCase						,       _lowerCamelCase       )
    A__				=			(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 						UpperCamelCase				(							_lowerCamelCase					:					int       ):

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

    A__				=			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 __future__ import annotations






class    UpperCAmelCase	:
 def __init__(				self      :Optional[int]						,       lowercase_      :int							)->							None:
     A__				=			order

     # a_{0} ... a_{k}
     A__				=			[1.0] + [0.0] * order
     # b_{0} ... b_{k}
     A__				=			[1.0] + [0.0] * order

     # x[n-1] ... x[n-k]
     A__				=			[0.0] * self.order
     # y[n-1] ... y[n-k]
     A__				=			[0.0] * self.order
 def      UpperCAmelCase_							(				self      :List[str]						,       lowercase_      :list[float]						,       lowercase_      :list[float]							)->							None:
     if len(lowercase_							) < self.order:
         A__				=			[1.0, *a_coeffs]

     if len(lowercase_							) != self.order + 1:
         A__				=			(
             F"Expected a_coeffs to have {self.order + 1} elements "
             F"for {self.order}-order filter, got {len(lowercase_							)}"
         )
         raise ValueError(lowercase_							)

     if len(lowercase_							) != self.order + 1:
         A__				=			(
             F"Expected b_coeffs to have {self.order + 1} elements "
             F"for {self.order}-order filter, got {len(lowercase_							)}"
         )
         raise ValueError(lowercase_							)

     A__				=			a_coeffs
     A__				=			b_coeffs



 def      UpperCAmelCase_							(				self      :Optional[int]						,       lowercase_      :float							)->							float:
     A__				=			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__				=			(result + self.b_coeffs[0] * sample) / self.a_coeffs[0]

     A__				=			self.input_history[:-1]
     A__				=			self.output_history[:-1]

     A__				=			sample
     A__				=			result

     return result



