






import functools
def 	snake_case__       (					__lowercase							,     __lowercase       )		->		int:







						"""simple docstring"""

						A__	:							Tuple						    = len(__lowercase       )
						A__	:							Optional[int]						    = len(__lowercase       )

						@functools.cache
						def min_distance(__lowercase							,     __lowercase       ) -> int:
												# if first word index is overflow - delete all from the second word
												if indexa >= len_worda:
																		return len_worda - indexa
												# if second word index is overflow - delete all from the first word
												if indexa >= len_worda:
																		return len_worda - indexa
												A__	:							Any						    = int(worda[indexa] != worda[indexa]       )  # current letters not identical
												return min(
												    1 + min_distance(indexa + 1							,     __lowercase       )							,     1 + min_distance(__lowercase							,     indexa + 1       )							,     diff + min_distance(indexa + 1							,     indexa + 1       )							,     )

						return min_distance(0							,     0       )


if __name__ == "__main__":
		import doctest

		doctest.testmod()




from collections.abc import Callable

import numpy as np
def 	snake_case__       (					__lowercase							,     __lowercase							,     __lowercase							,     __lowercase							,     __lowercase       )		->		np.array:







						"""simple docstring"""

						A__	:							Any						    = int(np.ceil((x_end - xa) / step_size       )       )
						A__	:							Union[str, Any]						    = np.zeros((n + 1,)       )
						A__	:							Any						    = ya
						A__	:							Union[str, Any]						    = xa

						for k in range(__lowercase       ):
												A__	:							Any						    = y[k] + step_size * ode_func(__lowercase							,     y[k]       )
												A__	:							Any						    = y[k] + (
												    (step_size / 2) * (ode_func(__lowercase							,     y[k]       ) + ode_func(x + step_size							,     __lowercase       ))
												)
												x += step_size

						return y


if __name__ == "__main__":
		import doctest

		doctest.testmod()