


"""simple docstring"""





def    _snake_case  (  _snake_case :	list       ,      _snake_case :	list	):
      _validate_point(_snake_case	)
      _validate_point(_snake_case	)
      if len(_snake_case	) != len(_snake_case	):
            raise ValueError('''Both points must be in the same n-dimensional space'''	)

      return float(sum(abs(a - b	) for a, b in zip(_snake_case       ,      _snake_case	)	)	)


def    _snake_case  (  _snake_case :	list[float]	):
      if point:
            if isinstance(_snake_case       ,      _snake_case	):
                  for item in point:
                        if not isinstance(_snake_case       ,      (int, float)	):
                              lowerCAmelCase    :		Union[str, Any]				=      (
                                  '''Expected a list of numbers as input, found '''
                                  f'''{type(_snake_case	).__name__}'''
                              )
                              raise TypeError(_snake_case	)
            else:
                  lowerCAmelCase    :		List[str]				=      f'''Expected a list of numbers as input, found {type(_snake_case	).__name__}'''
                  raise TypeError(_snake_case	)
      else:
            raise ValueError('''Missing an input'''	)


def    _snake_case  (  _snake_case :	list       ,      _snake_case :	list	):
      _validate_point(_snake_case	)
      _validate_point(_snake_case	)
      if len(_snake_case	) != len(_snake_case	):
            raise ValueError('''Both points must be in the same n-dimensional space'''	)

      return float(sum(abs(x - y	) for x, y in zip(_snake_case       ,      _snake_case	)	)	)


if __name__ == "__main__":
 import doctest

 doctest.testmod()






from __future__ import annotations

from math import ceil, floor, sqrt



def 				UpperCamelCase_(      lowerCamelCase_ = 200_0000 )    ->						int:
    _lowercase       :							list[int]      		=		[0]
    _lowercase       :							int

    for idx in range(1							, ceil(sqrt(target * 2 ) * 1.1 ) ):
        triangle_numbers.append(triangle_numbers[-1] + idx )

    # we want this to be as close as possible to target
    _lowercase       :							int      		=		0
    # the area corresponding to the grid that gives the product closest to target
    _lowercase       :							int      		=		0
    # an estimate of b, using the quadratic formula
    _lowercase       :							float
    # the largest integer less than b_estimate
    _lowercase       :							int
    # the largest integer less than b_estimate
    _lowercase       :							int
    # the triangle number corresponding to b_floor
    _lowercase       :							int
    # the triangle number corresponding to b_ceil
    _lowercase       :							int

    for idx_a, triangle_a in enumerate(triangle_numbers[1:]							, 1 ):
        _lowercase       :							Optional[int]      		=		(-1 + sqrt(1 + 8 * target / triangle_a )) / 2
        _lowercase       :							List[str]      		=		floor(lowerCamelCase_ )
        _lowercase       :							Dict      		=		ceil(lowerCamelCase_ )
        _lowercase       :							List[str]      		=		triangle_numbers[b_floor]
        _lowercase       :							List[str]      		=		triangle_numbers[b_ceil]

        if abs(target - triangle_b_first_guess * triangle_a ) < abs(
            target - best_product ):
            _lowercase       :							Union[str, Any]      		=		triangle_b_first_guess * triangle_a
            _lowercase       :							Union[str, Any]      		=		idx_a * b_floor

        if abs(target - triangle_b_second_guess * triangle_a ) < abs(
            target - best_product ):
            _lowercase       :							Any      		=		triangle_b_second_guess * triangle_a
            _lowercase       :							Optional[Any]      		=		idx_a * b_ceil

    return area


if __name__ == "__main__":
 print(F"{solution() = }")


