


"""simple docstring"""
from __future__ import annotations

UpperCAmelCase   =	"""Alexander Joslin"""

def 							_snake_case			(      __snake_case       :			int      ):




			"""simple docstring"""







			if number <= 0:
						_lowerCamelCase  :				List[str]      =		F'n must be greater than 0. Got n = {number}'
						raise ValueError(__snake_case      )

			_lowerCamelCase  :				Optional[Any]      =		[number + 1] * (number + 1)

			# starting position
			_lowerCamelCase  :				Tuple      =		0
			for i in range(1	,   __snake_case      ):
						_lowerCamelCase  :				List[str]      =		min(table[i + 1]	,   table[i] + 1      )
						# check if out of bounds
						if i * 2 <= number:
									_lowerCamelCase  :				int      =		min(table[i * 2]	,   table[i] + 1      )
						# check if out of bounds
						if i * 3 <= number:
									_lowerCamelCase  :				int      =		min(table[i * 3]	,   table[i] + 1      )
			return table[number]


if __name__ == "__main__":
	import doctest

	doctest.testmod()





"""simple docstring"""
import timeit




class   lowercase__							:





   def __init__(       self			,      SCREAMING_SNAKE_CASE)     ->     Tuple:
      if isinstance(SCREAMING_SNAKE_CASE			,      SCREAMING_SNAKE_CASE):  # Initializes a matrix identical to the one provided.
         _lowerCamelCase  :				List[str]      =		arg
         _lowerCamelCase  :				List[str]      =		len(SCREAMING_SNAKE_CASE)
      else:  # Initializes a square matrix of the given size and set values to zero.
         _lowerCamelCase  :				List[Any]      =		arg
         _lowerCamelCase  :				Union[str, Any]      =		[[0 for _ in range(self.n)] for _ in range(self.n)]





   def __mul__(       self			,      SCREAMING_SNAKE_CASE)     ->     Optional[Any]:
      _lowerCamelCase  :				List[Any]      =		Matrix(self.n)
      for i in range(self.n):
         for j in range(self.n):
            for k in range(self.n):
               matrix.t[i][j] += self.t[i][k] * b.t[k][j]
      return matrix







def 							_snake_case			(      __snake_case       :			Union[str, Any]	,   __snake_case       :			List[Any]      ):




   """simple docstring"""







   _lowerCamelCase  :				List[Any]      =		Matrix([[1, 0], [0, 1]]      )
   while b > 0:
      if b & 1:
         matrix *= a
      a *= a
      b >>= 1
   return matrix

def 							_snake_case			(      __snake_case       :			List[str]	,   __snake_case       :			str	,   __snake_case       :			List[Any]      ):




   """simple docstring"""







   if n == 1:
      return fa
   elif n == 2:
      return fa
   _lowerCamelCase  :				Any      =		Matrix([[1, 1], [1, 0]]      )
   _lowerCamelCase  :				List[str]      =		modular_exponentiation(__snake_case	,   n - 2      )
   return fa * matrix.t[0][0] + fa * matrix.t[0][1]

def 							_snake_case			(      __snake_case       :			int	,   __snake_case       :			Optional[int]	,   __snake_case       :			List[Any]      ):




   """simple docstring"""







   if n == 1:
      return fa
   elif n == 2:
      return fa

   _lowerCamelCase  :				Optional[Any]      =		fa
   _lowerCamelCase  :				int      =		fa
   n -= 2

   while n > 0:
      _lowerCamelCase      ,						_lowerCamelCase  :				Tuple      =		fn_a + fn_a, fn_a
      n -= 1

   return fn_a

def 							_snake_case			(      ):




   """simple docstring"""







   _lowerCamelCase  :				List[str]      =		"""
from random import randint
from __main__ import fibonacci_with_matrix_exponentiation
"""
   _lowerCamelCase  :				int      =		"""fibonacci_with_matrix_exponentiation(randint(1,70000), 1, 1)"""
   _lowerCamelCase  :				Tuple      =		timeit.timeit(setup=__snake_case	,   stmt=__snake_case	,   number=100      )
   print("""With matrix exponentiation the average execution time is """	,   exec_time / 100      )
   return exec_time

def 							_snake_case			(      ):




   """simple docstring"""







   _lowerCamelCase  :				Dict      =		"""
from random import randint
from __main__ import simple_fibonacci
"""
   _lowerCamelCase  :				Optional[int]      =		"""simple_fibonacci(randint(1,70000), 1, 1)"""
   _lowerCamelCase  :				List[Any]      =		timeit.timeit(setup=__snake_case	,   stmt=__snake_case	,   number=100      )
   print(
       """Without matrix exponentiation the average execution time is """	,   exec_time / 100      )
   return exec_time

def 							_snake_case			(      ):




   """simple docstring"""







   matrix_exponentiation_time()
   simple_fibonacci_time()


if __name__ == "__main__":
 main()

