


from collections.abc import Callable
from math import pi, sqrt
from random import uniform
from statistics import mean
def   _UpperCamelCase				(						lowerCAmelCase_   )						->Dict:

   # A local function to see if a dot lands in the circle.
   def is_in_circle(lowerCAmelCase_  ,						lowerCAmelCase_   ) -> bool:
      UpperCAmelCase							=     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							=     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.
   UpperCAmelCase							=     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_  ,						lowerCAmelCase_  ,						lowerCAmelCase_ = 0.0  ,						lowerCAmelCase_ = 1.0  ,						)						->float:
   return mean(
       function_to_integrate(uniform(lowerCAmelCase_  ,						lowerCAmelCase_   )   ) for _ in range(lowerCAmelCase_   )   ) * (max_value - min_value)
def   _UpperCamelCase				(						lowerCAmelCase_  ,						lowerCAmelCase_ = 0.0  ,						lowerCAmelCase_ = 1.0   )						->None:

   def identity_function(lowerCAmelCase_   ) -> float:
      return x

   UpperCAmelCase							=     area_under_curve_estimator(
       lowerCAmelCase_  ,						lowerCAmelCase_  ,						lowerCAmelCase_  ,						lowerCAmelCase_   )
   UpperCAmelCase							=     (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_   )						->None:

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

   UpperCAmelCase							=     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()







from __future__ import annotations
def   _UpperCamelCase				(						lowerCAmelCase_   )						->None:
   create_state_space_tree(lowerCAmelCase_  ,						[]  ,						0  ,						[0 for i in range(len(lowerCAmelCase_   )   )]   )
def   _UpperCamelCase				(						lowerCAmelCase_  ,						lowerCAmelCase_  ,						lowerCAmelCase_  ,						lowerCAmelCase_  ,						)						->None:

   if index == len(lowerCAmelCase_   ):
      print(lowerCAmelCase_   )
      return

   for i in range(len(lowerCAmelCase_   )   ):
      if not index_used[i]:
         current_sequence.append(sequence[i]   )
         UpperCAmelCase							=     True
         create_state_space_tree(lowerCAmelCase_  ,						lowerCAmelCase_  ,						index + 1  ,						lowerCAmelCase_   )
         current_sequence.pop()
         UpperCAmelCase							=     False
__a     =			[3, 1, 2, 4]
generate_all_permutations(sequence)

__a     =			["A", "B", "C"]







generate_all_permutations(sequence_a)



