


'''simple docstring'''



def      _UpperCamelCase   (		UpperCamelCase__			,   UpperCamelCase__       ):
			_validate_point(UpperCamelCase__       )
			_validate_point(UpperCamelCase__       )
			if len(UpperCamelCase__       ) != len(UpperCamelCase__       ):
						raise ValueError("""Both points must be in the same n-dimensional space"""       )

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







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







def      _UpperCamelCase   (		UpperCamelCase__			,   UpperCamelCase__       ):
			_validate_point(UpperCamelCase__       )
			_validate_point(UpperCamelCase__       )
			if len(UpperCamelCase__       ) != len(UpperCamelCase__       ):
						raise ValueError("""Both points must be in the same n-dimensional space"""       )

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


if __name__ == "__main__":
		import doctest

		doctest.testmod()






from __future__ import annotations

import math
from collections.abc import Callable


def       UpperCamelCase__(			UpperCamelCase__     :			Callable[[int | float], int | float]					,	UpperCamelCase__     :			int | float					,	UpperCamelCase__     :			int | float					,	UpperCamelCase__     :			int = 1_00					,	)->float:
       A__		   =		x_start
       A__		   =		fnc(UpperCamelCase__    )
       A__		   =		0.0

       for _ in range(UpperCamelCase__    ):
              # Approximates curve as a sequence of linear lines and sums their length
              A__		   =		(x_end - x_start) / steps + xa
              A__		   =		fnc(UpperCamelCase__    )
              length += math.hypot(xa - xa					,	fxa - fxa    )

              # Increment step
              A__		   =		xa
              A__		   =		fxa

       return length


if __name__ == "__main__":


       def       UpperCamelCase__(			UpperCamelCase__     :			Union[str, Any]    )->Dict:
              return math.sin(10 * x    )

       print('f(x) = sin(10 * x)')
       print('The length of the curve from x = -10 to x = 10 is:')
       a__: str							   =						10
       while i <= 100_000:
              print(F"With {i} steps: {line_length(f, -10, 10, i)}")
              i *= 10





