


'''simple docstring'''


from random import randint, random






def  __lowerCAmelCase (__lowerCAmelCase      , __lowerCAmelCase      , __lowerCAmelCase      , __lowerCAmelCase = False      , __lowerCAmelCase = False      , __lowerCAmelCase = 5      , ):
		_UpperCAmelCase		:       Union[str, Any]								=      [[-1] * number_of_cells]  # Create a highway without any car
		_UpperCAmelCase		:       int								=      0
		_UpperCAmelCase		:       Any								=      max(__lowerCAmelCase      , 0   )
		while i < number_of_cells:
				_UpperCAmelCase		:       str								=      (
				    randint(0      , __lowerCAmelCase   ) if random_speed else initial_speed
				)  # Place the cars
				i += (
				    randint(1      , max_speed * 2   ) if random_frequency else frequency
				)  # Arbitrary number, may need tuning
		return highway






def  __lowerCAmelCase (__lowerCAmelCase      , __lowerCAmelCase   ):
		_UpperCAmelCase		:       int								=      0
		_UpperCAmelCase		:       Tuple								=      highway_now[car_index + 1 :]
		for cell in range(len(__lowerCAmelCase   )   ):  # May need a better name for this
				if cells[cell] != -1:  # If the cell is not empty then
						return distance  # we have the distance we wanted
				distance += 1
		# Here if the car is near the end of the highway
		return distance + get_distance(__lowerCAmelCase      , -1   )






def  __lowerCAmelCase (__lowerCAmelCase      , __lowerCAmelCase      , __lowerCAmelCase   ):
		_UpperCAmelCase		:       str								=      len(__lowerCAmelCase   )
		# Beforce calculations, the highway is empty
		_UpperCAmelCase		:       Dict								=      [-1] * number_of_cells

		for car_index in range(__lowerCAmelCase   ):
				if highway_now[car_index] != -1:
						# Add 1 to the current speed of the car and cap the speed
						_UpperCAmelCase		:       Dict								=      min(highway_now[car_index] + 1      , __lowerCAmelCase   )
						# Number of empty cell before the next car
						_UpperCAmelCase		:       Union[str, Any]								=      get_distance(__lowerCAmelCase      , __lowerCAmelCase   ) - 1
						# We can't have the car causing an accident
						_UpperCAmelCase		:       int								=      min(next_highway[car_index]      , __lowerCAmelCase   )
						if random() < probability:
								# Randomly, a driver will slow down
								_UpperCAmelCase		:       Any								=      max(next_highway[car_index] - 1      , 0   )
		return next_highway






def  __lowerCAmelCase (__lowerCAmelCase      , __lowerCAmelCase      , __lowerCAmelCase      , __lowerCAmelCase   ):
		_UpperCAmelCase		:       int								=      len(highway[0]   )

		for i in range(__lowerCAmelCase   ):
				_UpperCAmelCase		:       Optional[Any]								=      update(highway[i]      , __lowerCAmelCase      , __lowerCAmelCase   )
				_UpperCAmelCase		:       Dict								=      [-1] * number_of_cells

				for car_index in range(__lowerCAmelCase   ):
						_UpperCAmelCase		:       Optional[Any]								=      next_speeds_calculated[car_index]
						if speed != -1:
								# Change the position based on the speed (with % to create the loop)
								_UpperCAmelCase		:       Tuple								=      (car_index + speed) % number_of_cells
								# Commit the change of position
								_UpperCAmelCase		:       Any								=      speed
				highway.append(__lowerCAmelCase   )

		return highway


if __name__ == "__main__":
						import doctest

						doctest.testmod()







'''simple docstring'''


from math import factorial






class 	lowerCAmelCase__			:




			def __init__(		self  :     int	,  lowerCamelCase__  :     Union[str, Any]	,  lowerCamelCase__  :     str )   ->Dict:



					'''simple docstring'''







					_UpperCAmelCase		:       Optional[Any]								=      real
					if isinstance(lowerCamelCase__	,  lowerCamelCase__ ):
							_UpperCAmelCase		:       Any								=      [1] * rank
					else:
							_UpperCAmelCase		:       Dict								=      rank




			def __repr__(		self  :     str )   ->List[str]:



					'''simple docstring'''







					return (
					    F"""{self.real}+"""
					    F"""{'+'.join(str(lowerCamelCase__ )+'E'+str(n+1 )for n,dual in enumerate(self.duals ) )}"""
					)




			def 							lowerCAmelCase__    (		self  :     Dict )   ->Tuple:



					'''simple docstring'''







					_UpperCAmelCase		:       Union[str, Any]								=      self.duals.copy()
					while cur[-1] == 0:
							cur.pop(-1 )
					return Dual(self.real	,  lowerCamelCase__ )




			def __add__(		self  :     Dict	,  lowerCamelCase__  :     List[Any] )   ->Any:



					'''simple docstring'''







					if not isinstance(lowerCamelCase__	,  lowerCamelCase__ ):
							return Dual(self.real + other	,  self.duals )
					_UpperCAmelCase		:       Optional[int]								=      self.duals.copy()
					_UpperCAmelCase		:       Optional[int]								=      other.duals.copy()
					if len(lowerCamelCase__ ) > len(lowerCamelCase__ ):
							o_dual.extend([1] * (len(lowerCamelCase__ ) - len(lowerCamelCase__ )) )
					elif len(lowerCamelCase__ ) < len(lowerCamelCase__ ):
							s_dual.extend([1] * (len(lowerCamelCase__ ) - len(lowerCamelCase__ )) )
					_UpperCAmelCase		:       Union[str, Any]								=      []
					for i in range(len(lowerCamelCase__ ) ):
							new_duals.append(s_dual[i] + o_dual[i] )
					return Dual(self.real + other.real	,  lowerCamelCase__ )

			lowerCAmelCase :	Tuple          =     __add__




			def __sub__(		self  :     List[Any]	,  lowerCamelCase__  :     Union[str, Any] )   ->Dict:



					'''simple docstring'''







					return self + other * -1




			def __mul__(		self  :     List[str]	,  lowerCamelCase__  :     Optional[Any] )   ->Union[str, Any]:



					'''simple docstring'''







					if not isinstance(lowerCamelCase__	,  lowerCamelCase__ ):
							_UpperCAmelCase		:       Optional[int]								=      []
							for i in self.duals:
									new_duals.append(i * other )
							return Dual(self.real * other	,  lowerCamelCase__ )
					_UpperCAmelCase		:       Union[str, Any]								=      [0] * (len(self.duals ) + len(other.duals ) + 1)
					for i, item in enumerate(self.duals ):
							for j, jtem in enumerate(other.duals ):
									new_duals[i + j + 1] += item * jtem
					for k in range(len(self.duals ) ):
							new_duals[k] += self.duals[k] * other.real
					for index in range(len(other.duals ) ):
							new_duals[index] += other.duals[index] * self.real
					return Dual(self.real * other.real	,  lowerCamelCase__ )

			lowerCAmelCase :	Union[str, Any]          =     __mul__




			def __truediv__(		self  :     Optional[Any]	,  lowerCamelCase__  :     List[Any] )   ->Union[str, Any]:



					'''simple docstring'''







					if not isinstance(lowerCamelCase__	,  lowerCamelCase__ ):
							_UpperCAmelCase		:       Union[str, Any]								=      []
							for i in self.duals:
									new_duals.append(i / other )
							return Dual(self.real / other	,  lowerCamelCase__ )
					raise ValueError




			def __floordiv__(		self  :     str	,  lowerCamelCase__  :     str )   ->List[str]:



					'''simple docstring'''







					if not isinstance(lowerCamelCase__	,  lowerCamelCase__ ):
							_UpperCAmelCase		:       Tuple								=      []
							for i in self.duals:
									new_duals.append(i // other )
							return Dual(self.real // other	,  lowerCamelCase__ )
					raise ValueError






			def __pow__(		self  :     Tuple	,  lowerCamelCase__  :     Optional[Any] )   ->Optional[int]:



					'''simple docstring'''







					if n < 0 or isinstance(lowerCamelCase__	,  lowerCamelCase__ ):
							raise ValueError("power must be a positive integer" )
					if n == 0:
							return 1
					if n == 1:
							return self
					_UpperCAmelCase		:       str								=      self
					for _ in range(n - 1 ):
							x *= self
					return x






def  __lowerCAmelCase (__lowerCAmelCase      , __lowerCAmelCase      , __lowerCAmelCase   ):
		if not callable(__lowerCAmelCase   ):
				raise ValueError("differentiate() requires a function as input for func"   )
		if not isinstance(__lowerCAmelCase      , (float, int)   ):
				raise ValueError("differentiate() requires a float as input for position"   )
		if not isinstance(__lowerCAmelCase      , __lowerCAmelCase   ):
				raise ValueError("differentiate() requires an int as input for order"   )
		_UpperCAmelCase		:       int								=      Dual(__lowerCAmelCase      , 1   )
		_UpperCAmelCase		:       Optional[int]								=      func(__lowerCAmelCase   )
		if order == 0:
				return result.real
		return result.duals[order - 1] * factorial(__lowerCAmelCase   )


if __name__ == "__main__":
						import doctest

						doctest.testmod()






						def  __lowerCAmelCase (__lowerCAmelCase   ):
								return y**2 * y**4





						print(differentiate(f, 9, 2))



