


'''simple docstring'''


def       snake_case_       (  SCREAMING_SNAKE_CASE__					):







				"""simple docstring"""





				if len(SCREAMING_SNAKE_CASE__					) == 0 or not all(len(SCREAMING_SNAKE_CASE__					) == 2 for pair in coordinates					):
								raise ValueError("""The program cannot work out a fitting polynomial."""					)

				if len({tuple(SCREAMING_SNAKE_CASE__					) for pair in coordinates}					) != len(SCREAMING_SNAKE_CASE__					):
								raise ValueError("""The program cannot work out a fitting polynomial."""					)

				_SCREAMING_SNAKE_CASE			:	List[str]							=				{x for x, _ in coordinates}
				if len(SCREAMING_SNAKE_CASE__					) == 1:
								return f"""x={coordinates[0][0]}"""

				if len(SCREAMING_SNAKE_CASE__					) != len(SCREAMING_SNAKE_CASE__					):
								raise ValueError("""The program cannot work out a fitting polynomial."""					)

				_SCREAMING_SNAKE_CASE			:	Union[str, Any]							=				len(SCREAMING_SNAKE_CASE__					)

				_SCREAMING_SNAKE_CASE			:	Tuple							=				0
				_SCREAMING_SNAKE_CASE			:	list[list[float]]							=				[]
				# put the x and x to the power values in a matrix
				while count_of_line < x:
								_SCREAMING_SNAKE_CASE			:	Optional[int]							=				0
								_SCREAMING_SNAKE_CASE			:	List[Any]							=				coordinates[count_of_line][0]
								_SCREAMING_SNAKE_CASE			:	list[float]							=				[]
								while count_in_line < x:
												count_line.append(a ** (x - (count_in_line + 1))					)
												count_in_line += 1
								matrix.append(SCREAMING_SNAKE_CASE__					)
								count_of_line += 1

				_SCREAMING_SNAKE_CASE			:	Union[str, Any]							=				0
				# put the y values into a vector
				_SCREAMING_SNAKE_CASE			:	list[float]							=				[]
				while count_of_line < x:
								vector.append(coordinates[count_of_line][1]					)
								count_of_line += 1

				_SCREAMING_SNAKE_CASE			:	Optional[int]							=				0

				while count < x:
								_SCREAMING_SNAKE_CASE			:	Dict							=				0
								while zahlen < x:
												if count == zahlen:
																zahlen += 1
												if zahlen == x:
																break
												_SCREAMING_SNAKE_CASE			:	int							=				matrix[zahlen][count] / matrix[count][count]
												for counting_columns, item in enumerate(matrix[count]					):
																# manipulating all the values in the matrix
																matrix[zahlen][counting_columns] -= item * bruch
												# manipulating the values in the vector
												vector[zahlen] -= vector[count] * bruch
												zahlen += 1
								count += 1

				_SCREAMING_SNAKE_CASE			:	Optional[Any]							=				0
				# make solutions
				_SCREAMING_SNAKE_CASE			:	list[str]							=				[]
				while count < x:
								solution.append(str(vector[count] / matrix[count][count]					)					)
								count += 1

				_SCREAMING_SNAKE_CASE			:	Any							=				0
				_SCREAMING_SNAKE_CASE			:	str							=				"""f(x)="""

				while count < x:
								_SCREAMING_SNAKE_CASE			:	list[str]							=				solution[count].split("""E"""					)
								if len(SCREAMING_SNAKE_CASE__					) > 1:
												_SCREAMING_SNAKE_CASE			:	Any							=				f"""{remove_e[0]}*10^{remove_e[1]}"""
								solved += f"""x^{x - (count + 1)}*{solution[count]}"""
								if count + 1 != x:
												solved += "+"
								count += 1

				return solved


if __name__ == "__main__":
				print(points_to_polynomial([]))
				print(points_to_polynomial([[]]))
				print(points_to_polynomial([[1, 0], [2, 0], [3, 0]]))
				print(points_to_polynomial([[1, 1], [2, 1], [3, 1]]))
				print(points_to_polynomial([[1, 3], [2, 3], [3, 3]]))
				print(points_to_polynomial([[1, 1], [2, 2], [3, 3]]))
				print(points_to_polynomial([[1, 1], [2, 4], [3, 9]]))
				print(points_to_polynomial([[1, 3], [2, 6], [3, 11]]))
				print(points_to_polynomial([[1, -3], [2, -6], [3, -11]]))
				print(points_to_polynomial([[1, 5], [2, 2], [3, 9]]))








'''simple docstring'''


def       snake_case_       (  SCREAMING_SNAKE_CASE__							,				SCREAMING_SNAKE_CASE__					):







				"""simple docstring"""





				if len(str(SCREAMING_SNAKE_CASE__					)					) == 1 or len(str(SCREAMING_SNAKE_CASE__					)					) == 1:
								return a * b

				_SCREAMING_SNAKE_CASE			:	Optional[Any]							=				max(len(str(SCREAMING_SNAKE_CASE__					)					)							,				len(str(SCREAMING_SNAKE_CASE__					)					)					)
				_SCREAMING_SNAKE_CASE			:	Tuple							=				ma // 2

				_SCREAMING_SNAKE_CASE		,	_SCREAMING_SNAKE_CASE			:	Optional[int]							=				divmod(SCREAMING_SNAKE_CASE__							,				10**ma					)
				_SCREAMING_SNAKE_CASE		,	_SCREAMING_SNAKE_CASE			:	Optional[int]							=				divmod(SCREAMING_SNAKE_CASE__							,				10**ma					)

				_SCREAMING_SNAKE_CASE			:	Dict							=				karatsuba(SCREAMING_SNAKE_CASE__							,				SCREAMING_SNAKE_CASE__					)
				_SCREAMING_SNAKE_CASE			:	List[Any]							=				karatsuba((aa + aa)							,				(ba + ba)					)
				_SCREAMING_SNAKE_CASE			:	List[Any]							=				karatsuba(SCREAMING_SNAKE_CASE__							,				SCREAMING_SNAKE_CASE__					)

				return (z * 10 ** (2 * ma)) + ((y - z - x) * 10 ** (ma)) + (x)







def       snake_case_       (  ):







				"""simple docstring"""





				print(karatsuba(1_5463							,				2_3489					)					)


if __name__ == "__main__":
				main()




