

"""simple docstring"""






import mpmath  # for roots of unity
import numpy as np






class      _UpperCAmelCase      :
			def __init__(		self   ,		_A=None   ,		_A=None  )     ->     Tuple:






				'''simple docstring'''







				_UpperCAmelCase     :				Optional[Any]          =       list(poly_a or [0]  )[:]
				_UpperCAmelCase     :				Any          =       list(poly_b or [0]  )[:]

				# Remove leading zero coefficients
				while self.polyA[-1] == 0:
					self.polyA.pop()
				_UpperCAmelCase     :				str          =       len(self.polyA  )

				while self.polyB[-1] == 0:
					self.polyB.pop()
				_UpperCAmelCase     :				Any          =       len(self.polyB  )

				# Add 0 to make lengths equal a power of 2
				_UpperCAmelCase     :				List[Any]          =       int(
				    2 ** np.ceil(np.loga(len(self.polyA  ) + len(self.polyB  ) - 1  )  )  )

				while len(self.polyA  ) < self.c_max_length:
					self.polyA.append(0  )
				while len(self.polyB  ) < self.c_max_length:
					self.polyB.append(0  )
				# A complex root used for the fourier transform
				_UpperCAmelCase     :				Optional[Any]          =       complex(mpmath.root(x=1   ,		n=self.c_max_length   ,		k=1  )  )

				# The product
				_UpperCAmelCase     :				Tuple          =       self.__multiply()
			def 				__snake_case			(		self   ,		_A  )     ->     List[Any]:






				'''simple docstring'''







				_UpperCAmelCase     :				List[Any]          =       [[x] for x in self.polyA] if which == '''A''' else [[x] for x in self.polyB]
				# Corner case
				if len(lowercase_  ) <= 1:
					return dft[0]
				#
				_UpperCAmelCase     :				Optional[Any]          =       self.c_max_length // 2
				while next_ncol > 0:
					_UpperCAmelCase     :				str          =       [[] for i in range(lowercase_  )]
					_UpperCAmelCase     :				str          =       self.root**next_ncol

					# First half of next step
					_UpperCAmelCase     :				str          =       1
					for j in range(self.c_max_length // (next_ncol * 2)  ):
						for i in range(lowercase_  ):
							new_dft[i].append(dft[i][j] + current_root * dft[i + next_ncol][j]  )
						current_root *= root
					# Second half of next step
					_UpperCAmelCase     :				Union[str, Any]          =       1
					for j in range(self.c_max_length // (next_ncol * 2)  ):
						for i in range(lowercase_  ):
							new_dft[i].append(dft[i][j] - current_root * dft[i + next_ncol][j]  )
						current_root *= root
					# Update
					_UpperCAmelCase     :				Optional[int]          =       new_dft
					_UpperCAmelCase     :				int          =       next_ncol // 2
				return dft[0]
			def 				__snake_case			(		self  )     ->     int:






				'''simple docstring'''







				_UpperCAmelCase     :				Dict          =       self.__dft("""A"""  )
				_UpperCAmelCase     :				Dict          =       self.__dft("""B"""  )
				_UpperCAmelCase     :				Tuple          =       [[dft_a[i] * dft_b[i] for i in range(self.c_max_length  )]]
				del dft_a
				del dft_b

				# Corner Case
				if len(inverce_c[0]  ) <= 1:
					return inverce_c[0]
				# Inverse DFT
				_UpperCAmelCase     :				Any          =       2
				while next_ncol <= self.c_max_length:
					_UpperCAmelCase     :				Any          =       [[] for i in range(lowercase_  )]
					_UpperCAmelCase     :				Optional[Any]          =       self.root ** (next_ncol // 2)
					_UpperCAmelCase     :				Dict          =       1
					# First half of next step
					for j in range(self.c_max_length // next_ncol  ):
						for i in range(next_ncol // 2  ):
							# Even positions
							new_inverse_c[i].append(
							    (
							        inverce_c[i][j]
							        + inverce_c[i][j + self.c_max_length // next_ncol]
							    )
							    / 2  )
							# Odd positions
							new_inverse_c[i + next_ncol // 2].append(
							    (
							        inverce_c[i][j]
							        - inverce_c[i][j + self.c_max_length // next_ncol]
							    )
							    / (2 * current_root)  )
						current_root *= root
					# Update
					_UpperCAmelCase     :				List[Any]          =       new_inverse_c
					next_ncol *= 2
				# Unpack
				_UpperCAmelCase     :				Union[str, Any]          =       [round(x[0].real   ,		8  ) + round(x[0].imag   ,		8  ) * 1J for x in inverce_c]

				# Remove leading 0's
				while inverce_c[-1] == 0:
					inverce_c.pop()
				return inverce_c
			def __str__(		self  )     ->     List[str]:






				'''simple docstring'''







				_UpperCAmelCase     :				List[str]          =       '''A = ''' + ''' + '''.join(
				    f'''{coef}*x^{i}''' for coef, i in enumerate(self.polyA[: self.len_A]  )  )
				_UpperCAmelCase     :				int          =       '''B = ''' + ''' + '''.join(
				    f'''{coef}*x^{i}''' for coef, i in enumerate(self.polyB[: self.len_B]  )  )
				_UpperCAmelCase     :				Union[str, Any]          =       '''A*B = ''' + ''' + '''.join(
				    f'''{coef}*x^{i}''' for coef, i in enumerate(self.product  )  )

				return f'''{a}\n{b}\n{c}'''


# Unit tests
if __name__ == "__main__":
						import doctest

						doctest.testmod()








"""simple docstring"""





from __future__ import annotations

import math
from collections import Counter
from string import ascii_lowercase

def     __SCREAMING_SNAKE_CASE    ( A_					):
	lowerCAmelCase__							,lowerCAmelCase__      :		Dict    							=      analyze_text(A_					)
	lowerCAmelCase__      :		Any    							=      list(''' ''' + ascii_lowercase					)
	# what is our total sum of probabilities.
	lowerCAmelCase__      :		Optional[Any]    							=      sum(single_char_strings.values()					)

	# one length string
	lowerCAmelCase__      :		int    							=      0
	# for each alpha we go in our dict and if it is in it we calculate entropy
	for ch in my_alphas:
		if ch in single_char_strings:
			lowerCAmelCase__      :		Union[str, Any]    							=      single_char_strings[ch]
			lowerCAmelCase__      :		Dict    							=      my_str / all_sum
			my_fir_sum += prob * math.loga(A_					)  # entropy formula.

    # print entropy
	print(f'{round(-1 * my_fir_sum					):.1f}'					)

	# two len string
	lowerCAmelCase__      :		List[Any]    							=      sum(two_char_strings.values()					)
	lowerCAmelCase__      :		Optional[int]    							=      0
	# for each alpha (two in size) calculate entropy.
	for cha in my_alphas:
		for cha in my_alphas:
			lowerCAmelCase__      :		int    							=      cha + cha
			if sequence in two_char_strings:
				lowerCAmelCase__      :		Optional[Any]    							=      two_char_strings[sequence]
				lowerCAmelCase__      :		int    							=      int(A_					) / all_sum
				my_sec_sum += prob * math.loga(A_					)

    # print second entropy
	print(f'{round(-1 * my_sec_sum					):.1f}'					)

	# print the difference between them
	print(f'{round((-1 * my_sec_sum) - (-1 * my_fir_sum)					):.1f}'					)

def     __SCREAMING_SNAKE_CASE    ( A_					):
	lowerCAmelCase__      :		List[str]    							=      Counter()  # type: ignore
	lowerCAmelCase__      :		Optional[int]    							=      Counter()  # type: ignore
	single_char_strings[text[-1]] += 1

	# first case when we have space at start.
	two_char_strings[" " + text[0]] += 1
	for i in range(0		,	len(A_					) - 1					):
		single_char_strings[text[i]] += 1
		two_char_strings[text[i : i + 2]] += 1
	return single_char_strings, two_char_strings

def     __SCREAMING_SNAKE_CASE    ( ):
	import doctest

	doctest.testmod()
	# text = (
	#     "Had repulsive dashwoods suspicion sincerity but advantage now him. Remark "
	#     "easily garret nor nay. Civil those mrs enjoy shy fat merry. You greatest "
	#     "jointure saw horrible. He private he on be imagine suppose. Fertile "
	#     "beloved evident through no service elderly is. Blind there if every no so "
	#     "at. Own neglected you preferred way sincerity delivered his attempted. To "
	#     "of message cottage windows do besides against uncivil.  Delightful "
	#     "unreserved impossible few estimating men favourable see entreaties. She "
	#     "propriety immediate was improving. He or entrance humoured likewise "
	#     "moderate. Much nor game son say feel. Fat make met can must form into "
	#     "gate. Me we offending prevailed discovery. "
	# )

	# calculate_prob(text)


if __name__ == "__main__":
							main()




