





from __future__ import annotations

import math
from collections.abc import Callable






def lowerCamelCase__    (							snake_case_	:  Callable[[int | float], int | float]       ,      snake_case_	:  int | float       ,      snake_case_	:  int | float       ,      snake_case_	:  int = 100       ,      )      ->				float:
   __snake_case							       =   x_start
   __snake_case							       =   fnc(snake_case_       )
   __snake_case							       =   0.0

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

      # Increment step
      __snake_case							       =   xa
      __snake_case							       =   fxa

   return length


if __name__ == "__main__":






    def lowerCamelCase__    (							snake_case_	:  Optional[Any]       )      ->				int:
       return math.sin(10 * x       )

    print('f(x) = sin(10 * x)')
    print('The length of the curve from x = -10 to x = 10 is:')
    snake_case_				=			10
    while i <= 100000:
        print(F'With {i} steps: {line_length(f, -10, 10, i)}')
        i *= 10





'''simple docstring'''






def 				UpperCamelCase_			(    A__					:   int					):
  '''simple docstring'''





  lowerCAmelCase_			:      Union[str, Any]			=   (1 + 24 * n) ** 0.5
  return ((1 + root) / 6) % 1 == 0







def 				UpperCamelCase_			(    A__					:   int = 50_00					):
  '''simple docstring'''





  lowerCAmelCase_			:      Optional[int]			=   [(i * (3 * i - 1)) // 2 for i in range(1							,    A__					)]
  for i, pentagonal_i in enumerate(A__					):
    for j in range(A__							,    len(A__					)					):
      lowerCAmelCase_			:      int			=   pentagonal_nums[j]
      lowerCAmelCase_			:      Union[str, Any]			=   pentagonal_i + pentagonal_j
      lowerCAmelCase_			:      List[Any]			=   pentagonal_j - pentagonal_i
      if is_pentagonal(A__					) and is_pentagonal(A__					):
        return b

  return -1


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



