


from math import factorial


class 		_SCREAMING_SNAKE_CASE     :


     def __init__(  self							,					A_							,					A_    ):
       _UpperCAmelCase		:							int							=     real
       if isinstance(A_							,					A_    ):
         _UpperCAmelCase		:							List[str]							=     [1] * rank
       else:
         _UpperCAmelCase		:							Union[str, Any]							=     rank


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


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


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

     __SCREAMING_SNAKE_CASE     					=    __add__


     def __sub__(  self							,					A_    ):
       return self + other * -1


     def __mul__(  self							,					A_    ):
       if not isinstance(A_							,					A_    ):
         _UpperCAmelCase		:							Union[str, Any]							=     []
         for i in self.duals:
           new_duals.append(i * other    )
         return Dual(self.real * other							,					A_    )
       _UpperCAmelCase		:							Optional[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							,					A_    )

     __SCREAMING_SNAKE_CASE     					=    __mul__


     def __truediv__(  self							,					A_    ):
       if not isinstance(A_							,					A_    ):
         _UpperCAmelCase		:							Optional[int]							=     []
         for i in self.duals:
           new_duals.append(i / other    )
         return Dual(self.real / other							,					A_    )
       raise ValueError


     def __floordiv__(  self							,					A_    ):
       if not isinstance(A_							,					A_    ):
         _UpperCAmelCase		:							List[str]							=     []
         for i in self.duals:
           new_duals.append(i // other    )
         return Dual(self.real // other							,					A_    )
       raise ValueError




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




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


if __name__ == "__main__":
 import doctest

 doctest.testmod()




 def    a__    (		snake_case__					:							int     ):
   return y**2 * y**4





 print(differentiate(f, 9, 2))









from random import randint, random




def    a__    (		snake_case__					:							int						,   snake_case__					:							int						,   snake_case__					:							int						,   snake_case__					:							bool = False						,   snake_case__					:							bool = False						,   snake_case__					:							int = 5						,   ):
  _UpperCAmelCase		:							Optional[int]							=     [[-1] * number_of_cells]  # Create a highway without any car
  _UpperCAmelCase		:							Optional[Any]							=     0
  _UpperCAmelCase		:							Dict							=     max(snake_case__						,   0     )
  while i < number_of_cells:
    _UpperCAmelCase		:							int							=     (
        randint(0						,   snake_case__     ) 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    a__    (		snake_case__					:							list						,   snake_case__					:							int     ):
  _UpperCAmelCase		:							List[Any]							=     0
  _UpperCAmelCase		:							int							=     highway_now[car_index + 1 :]
  for cell in range(len(snake_case__     )     ):  # 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(snake_case__						,   -1     )




def    a__    (		snake_case__					:							list						,   snake_case__					:							float						,   snake_case__					:							int     ):
  _UpperCAmelCase		:							Optional[Any]							=     len(snake_case__     )
  # Beforce calculations, the highway is empty
  _UpperCAmelCase		:							Dict							=     [-1] * number_of_cells

  for car_index in range(snake_case__     ):
    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						,   snake_case__     )
      # Number of empty cell before the next car
      _UpperCAmelCase		:							List[str]							=     get_distance(snake_case__						,   snake_case__     ) - 1
      # We can't have the car causing an accident
      _UpperCAmelCase		:							List[str]							=     min(next_highway[car_index]						,   snake_case__     )
      if random() < probability:
        # Randomly, a driver will slow down
        _UpperCAmelCase		:							Dict							=     max(next_highway[car_index] - 1						,   0     )
  return next_highway




def    a__    (		snake_case__					:							list						,   snake_case__					:							int						,   snake_case__					:							float						,   snake_case__					:							int     ):
  _UpperCAmelCase		:							Union[str, Any]							=     len(highway[0]     )

  for i in range(snake_case__     ):
    _UpperCAmelCase		:							Tuple							=     update(highway[i]						,   snake_case__						,   snake_case__     )
    _UpperCAmelCase		:							int							=     [-1] * number_of_cells

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

  return highway


if __name__ == "__main__":
 import doctest

 doctest.testmod()





