

from __future__ import annotations

def       a__					(		UpperCAmelCase							:      str    ,			UpperCAmelCase							:      list[str] | None = None    ,			UpperCAmelCase							:      dict[str, float] | None = None    ,			UpperCAmelCase							:      bool = False    ,			)  ->						tuple[int, float, str]:
  UpperCAmelCase	:							List[Any]						=			cipher_alphabet or [chr(UpperCAmelCase      ) 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	:							int						=			{
        '''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	:							Tuple						=			frequencies_dict

  if not case_sensitive:
    UpperCAmelCase	:							int						=			ciphertext.lower()

  # Chi squared statistic values
  UpperCAmelCase	:							dict[int, tuple[float, str]]						=			{}

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

    # decrypt the message with the shift
    for letter in ciphertext:
      try:
        # Try to index the letter in the alphabet
        UpperCAmelCase	:							List[Any]						=			(alphabet_letters.index(letter.lower()      ) - shift) % len(
            UpperCAmelCase      )
        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	:							Dict						=			0.0

    # Loop through each letter in the decoded message with the shift
    for letter in decrypted_with_shift:
      if case_sensitive:
        UpperCAmelCase	:							Union[str, Any]						=			letter.lower()
        if letter in frequencies:
          # Get the amount of times the letter occurs in the message
          UpperCAmelCase	:							List[Any]						=			decrypted_with_shift.lower().count(UpperCAmelCase      )

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

          # Complete the chi squared statistic formula
          UpperCAmelCase	:							Union[str, Any]						=			((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	:							Dict						=			decrypted_with_shift.count(UpperCAmelCase      )

          # Get the excepcted amount of times the letter should appear based
          # on letter frequencies
          UpperCAmelCase	:							List[str]						=			frequencies[letter] * occurrences

          # Complete the chi squared statistic formula
          UpperCAmelCase	:							List[str]						=			((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	:							Any						=			(
        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(UpperCAmelCase							:      int      ) -> tuple[float, str]:
    return chi_squared_statistic_values[key]

  UpperCAmelCase	:							int						=			min(
      UpperCAmelCase    ,			key=UpperCAmelCase    ,			)

  # Get all the data from the most likely cipher (key, decoded message)
  (
      UpperCAmelCase						
  )	:							List[Any]						=			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__					(		UpperCAmelCase							:      int      )  ->						Dict:

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

def       a__					(		UpperCAmelCase							:      int    ,			UpperCAmelCase							:      float = 0.0    ,			UpperCAmelCase							:      float = 1.0      )  ->						None:

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

  UpperCAmelCase	:							int						=			area_under_curve_estimator(
      UpperCAmelCase    ,			UpperCAmelCase    ,			UpperCAmelCase    ,			UpperCAmelCase      )
  UpperCAmelCase	:							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       a__					(		UpperCAmelCase							:      int      )  ->						None:

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

  UpperCAmelCase	:							Optional[int]						=			area_under_curve_estimator(
      UpperCAmelCase    ,			UpperCAmelCase    ,			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()





