






"""simple docstring"""





from .hash_table import HashTable
from .number_theory.prime_numbers import is_prime, next_prime







class 						_a  (		lowercase_       ):





   '''simple docstring'''






   def __init__(       self					, *UpperCAmelCase_					, **UpperCAmelCase_)			->   Tuple:




          '''simple docstring'''




          super().__init__(*UpperCAmelCase_					, **UpperCAmelCase_)





   def __lowercase							(       self					, UpperCAmelCase_					, UpperCAmelCase_)			->   Dict:




          '''simple docstring'''




          lowercase__:							Union[str, Any]         =     (
              next_prime(value % self.size_table)
              if not is_prime(value % self.size_table)
              else value % self.size_table
          )  # gt = bigger than
          return next_prime_gt - (data % next_prime_gt)





   def __lowercase							(       self					, UpperCAmelCase_					, UpperCAmelCase_					, UpperCAmelCase_)			->   Union[str, Any]:




          '''simple docstring'''




          return (increment * self.__hash_function_a(UpperCAmelCase_					, UpperCAmelCase_)) % self.size_table





   def __lowercase							(       self					, UpperCAmelCase_					, UpperCAmelCase_=None)			->   Union[str, Any]:




          '''simple docstring'''




          lowercase__:							str         =     1
          lowercase__:							List[str]         =     self.hash_function(UpperCAmelCase_)

          while self.values[new_key] is not None and self.values[new_key] != key:
                 lowercase__:							int         =     (
                     self.__hash_double_function(UpperCAmelCase_					, UpperCAmelCase_					, UpperCAmelCase_)
                     if self.balanced_factor() >= self.lim_charge
                     else None
                 )
                 if new_key is None:
                        break
                 else:
                        i += 1

          return new_key













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

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

					lowerCamelCase__      =							{x for x, _ in coordinates}
					if len(a) == 1:
										return f"x={coordinates[0][0]}"

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

					lowerCamelCase__      =							len(a)

					lowerCamelCase__      =							0
					lowerCamelCase__      =							[]
					# put the x and x to the power values in a matrix
					while count_of_line < x:
										lowerCamelCase__      =							0
										lowerCamelCase__      =							coordinates[count_of_line][0]
										lowerCamelCase__      =							[]
										while count_in_line < x:
															count_line.append(a ** (x - (count_in_line + 1)))
															count_in_line += 1
										matrix.append(a)
										count_of_line += 1

					lowerCamelCase__      =							0
					# put the y values into a vector
					lowerCamelCase__      =							[]
					while count_of_line < x:
										vector.append(coordinates[count_of_line][1])
										count_of_line += 1

					lowerCamelCase__      =							0

					while count < x:
										lowerCamelCase__      =							0
										while zahlen < x:
															if count == zahlen:
																				zahlen += 1
															if zahlen == x:
																				break
															lowerCamelCase__      =							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

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

					lowerCamelCase__      =							0
					lowerCamelCase__      =							"f(x)="

					while count < x:
										lowerCamelCase__      =							solution[count].split("E")
										if len(a) > 1:
															lowerCamelCase__      =							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, 1_1]]))
						print(points_to_polynomial([[1, -3], [2, -6], [3, -1_1]]))
						print(points_to_polynomial([[1, 5], [2, 2], [3, 9]]))

