
from __future__ import annotations




def 	A_ (							snake_case							:   str ,      snake_case							:   list[str] | None = None ,      snake_case							:   dict[str, float] | None = None ,      snake_case							:   bool = False ,      )      ->    tuple[int, float, str]:



						'''simple docstring'''

						__UpperCamelCase       =			cipher_alphabet or [chr(snake_case ) for i in range(97 ,      123 )]

						# If the argument is None or the user provided an empty dictionary
						if not frequencies_dict:
												# Frequencies of letters in the english language (how much they show up)
												__UpperCamelCase       =			{
												    '''a''': 0.08497,
												    '''b''': 0.01492,
												    '''c''': 0.02202,
												    '''d''': 0.04253,
												    '''e''': 0.11162,
												    '''f''': 0.02228,
												    '''g''': 0.02015,
												    '''h''': 0.06094,
												    '''i''': 0.07546,
												    '''j''': 0.00153,
												    '''k''': 0.01292,
												    '''l''': 0.04025,
												    '''m''': 0.02406,
												    '''n''': 0.06749,
												    '''o''': 0.07507,
												    '''p''': 0.01929,
												    '''q''': 0.00095,
												    '''r''': 0.07587,
												    '''s''': 0.06327,
												    '''t''': 0.09356,
												    '''u''': 0.02758,
												    '''v''': 0.00978,
												    '''w''': 0.02560,
												    '''x''': 0.00150,
												    '''y''': 0.01994,
												    '''z''': 0.00077,
												}
						else:
												# Custom frequencies dictionary
												__UpperCamelCase       =			frequencies_dict

						if not case_sensitive:
												__UpperCamelCase       =			ciphertext.lower()

						# Chi squared statistic values
						__UpperCamelCase       =			{}

						# cycle through all of the shifts
						for shift in range(len(snake_case ) ):
												__UpperCamelCase       =			''''''

												# decrypt the message with the shift
												for letter in ciphertext:
																		try:
																								# Try to index the letter in the alphabet
																								__UpperCamelCase       =			(alphabet_letters.index(letter.lower() ) - shift) % len(
																								    snake_case )
																								decrypted_with_shift += (
																								    alphabet_letters[new_key].upper()
																								    if case_sensitive and letter.isupper()
																								    else alphabet_letters[new_key]
																								)
																		except ValueError:
																								# Append the character if it isn't in the alphabet
																								decrypted_with_shift += letter

												__UpperCamelCase       =			0.0

												# Loop through each letter in the decoded message with the shift
												for letter in decrypted_with_shift:
																		if case_sensitive:
																								__UpperCamelCase       =			letter.lower()
																								if letter in frequencies:
																														# Get the amount of times the letter occurs in the message
																														__UpperCamelCase       =			decrypted_with_shift.lower().count(snake_case )

																														# Get the excepcted amount of times the letter should appear based
																														# on letter frequencies
																														__UpperCamelCase       =			frequencies[letter] * occurrences

																														# Complete the chi squared statistic formula
																														__UpperCamelCase       =			((occurrences - expected) ** 2) / expected

																														# Add the margin of error to the total chi squared statistic
																														chi_squared_statistic += chi_letter_value
																		else:
																								if letter.lower() in frequencies:
																														# Get the amount of times the letter occurs in the message
																														__UpperCamelCase       =			decrypted_with_shift.count(snake_case )

																														# Get the excepcted amount of times the letter should appear based
																														# on letter frequencies
																														__UpperCamelCase       =			frequencies[letter] * occurrences

																														# Complete the chi squared statistic formula
																														__UpperCamelCase       =			((occurrences - expected) ** 2) / expected

																														# Add the margin of error to the total chi squared statistic
																														chi_squared_statistic += chi_letter_value

        # Add the data to the chi_squared_statistic_values dictionary
												__UpperCamelCase       =			(
												    chi_squared_statistic,
												    decrypted_with_shift,
												)

						# Get the most likely cipher by finding the cipher with the smallest chi squared
						# statistic
						def chi_squared_statistic_values_sorting_key(snake_case							:   int ) -> tuple[float, str]:
												return chi_squared_statistic_values[key]

						__UpperCamelCase       =			min(
						    snake_case ,      key=snake_case ,      )

						# Get all the data from the most likely cipher (key, decoded message)
						(
						    (
						    __UpperCamelCase
						)  ,							(
						    __UpperCamelCase
						)  ,							
						)       =			chi_squared_statistic_values[most_likely_cipher]

						# Return the data on the most likely shift
						return (
						    most_likely_cipher,
						    most_likely_cipher_chi_squared_value,
						    decoded_most_likely_cipher,
						)






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




def 	A_ (							snake_case							:   int )      ->    int:



						'''simple docstring'''

						def is_in_circle(snake_case							:   float ,      snake_case							:   float ) -> 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(snake_case ) )
						# 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 	A_ (							snake_case							:   int ,      snake_case							:   Callable[[float], float] ,      snake_case							:   float = 0.0 ,      snake_case							:   float = 1.0 ,      )      ->    float:



						'''simple docstring'''

						return mean(
						    function_to_integrate(uniform(snake_case ,      snake_case ) ) for _ in range(snake_case ) ) * (max_value - min_value)




def 	A_ (							snake_case							:   int ,      snake_case							:   float = 0.0 ,      snake_case							:   float = 1.0 )      ->    None:



						'''simple docstring'''

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

						__UpperCamelCase       =			area_under_curve_estimator(
						    snake_case ,      snake_case ,      snake_case ,      snake_case )
						__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 	A_ (							snake_case							:   int )      ->    None:



						'''simple docstring'''

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

						__UpperCamelCase       =			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()




