






"""simple docstring"""

def 		__lowerCamelCase	(  __UpperCamelCase       )						->    str:

		"""simple docstring"""







		if isinstance(a_      ,       a_       ):
				raise TypeError("\'float\' object cannot be interpreted as an integer"       )
		if isinstance(a_      ,       a_       ):
				raise TypeError("\'str\' object cannot be interpreted as an integer"       )

		if num == 0:
				return "0b0"

		lowerCAmelCase_   :							Optional[Any]									=			False

		if num < 0:
				lowerCAmelCase_   :							str									=			True
				lowerCAmelCase_   :							int									=			-num

		lowerCAmelCase_   :							list[int]									=			[]
		while num > 0:
				binary.insert(0      ,       num % 2       )
				num >>= 1

		if negative:
				return "-0b" + "".join(str(a_       ) for e in binary       )

		return "0b" + "".join(str(a_       ) for e in binary       )


if __name__ == "__main__":
					import doctest

					doctest.testmod()







"""simple docstring"""
from collections.abc import Callable

import numpy as np



def      __lowerCamelCase   (							a_     :		Callable ,   a_     :		float ,   a_     :		float ,   a_     :		float ,   a_     :		float							)			->      np.ndarray:
      __SCREAMING_SNAKE_CASE   :List[Any]						=  int(np.ceil((x_end - xa) / step_size							)							)
      __SCREAMING_SNAKE_CASE   :Optional[Any]						=  np.zeros((n + 1,)							)
      __SCREAMING_SNAKE_CASE   :int						=  ya
      __SCREAMING_SNAKE_CASE   :str						=  xa

      for k in range(a_							):
            __SCREAMING_SNAKE_CASE   :Optional[int]						=  y[k] + step_size * ode_func(a_ ,   y[k]							)
            x += step_size

      return y


if __name__ == "__main__":
  import doctest

  doctest.testmod()