





"""simple docstring"""




from collections.abc import Callable

import numpy as np







def        __lowerCamelCase (							__a       :Union[str, Any]   ,					__a       :Optional[int]   ,					__a       :List[Any]   ,					__a       :Tuple   ,					__a       :Union[str, Any]   )		->    Union[str, Any]:






		"""simple docstring"""

		A__  				=       int(np.ceil((x_end - xa) / step_size   )   )
		A__  				=       np.zeros((n + 1,)   )
		A__  				=       ya
		A__  				=       xa

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

		return y


if __name__ == "__main__":
					import doctest

					doctest.testmod()










def        __lowerCamelCase (							__a       :int = 3   ,					__a       :int = 7   ,					__a       :int = 1_0_0_0_0_0_0   )		->    int:






		"""simple docstring"""

		A__  				=       0
		A__  				=       1

		for current_denominator in range(1   ,					limit + 1   ):
				A__  				=       current_denominator * numerator // denominator
				if current_denominator % denominator == 0:
						current_numerator -= 1
				if current_numerator * max_denominator > current_denominator * max_numerator:
						A__  				=       current_numerator
						A__  				=       current_denominator
		return max_numerator


if __name__ == "__main__":
					print(solution(numerator=3, denominator=7, limit=1_0_0_0_0_0_0))



