
"""simple docstring"""





from __future__ import annotations

from fractions import Fraction
from math import gcd, sqrt



def  lowerCamelCase__						(     __snake_case      )				->						bool:


					"""simple docstring"""





					_UpperCamelCase									=    int(number**0.5      )
					return number == sq * sq



def  lowerCamelCase__						(     __snake_case,	__snake_case,	__snake_case,	__snake_case,	__snake_case,	__snake_case      )				->						tuple[int, int]:


					"""simple docstring"""





					_UpperCamelCase									=    x_num * y_den * z_den + y_num * x_den * z_den + z_num * x_den * y_den
					_UpperCamelCase									=    x_den * y_den * z_den
					_UpperCamelCase									=    gcd(__snake_case,	__snake_case      )
					top //= hcf
					bottom //= hcf
					return top, bottom



def  lowerCamelCase__						(     __snake_case = 35      )				->						int:


					"""simple docstring"""





					_UpperCamelCase									=    set()
					_UpperCamelCase									=    42
					_UpperCamelCase									=    Fraction(0      )
					_UpperCamelCase									=    42

					for x_num in range(1,	order + 1      ):
										for x_den in range(x_num + 1,	order + 1      ):
															for y_num in range(1,	order + 1      ):
																				for y_den in range(y_num + 1,	order + 1      ):
																									# n=1
																									_UpperCamelCase									=    x_num * y_den + x_den * y_num
																									_UpperCamelCase									=    x_den * y_den
																									_UpperCamelCase									=    gcd(__snake_case,	__snake_case      )
																									z_num //= hcf
																									z_den //= hcf
																									if 0 < z_num < z_den <= order:
																														_UpperCamelCase									=    add_three(
																														    __snake_case,	__snake_case,	__snake_case,	__snake_case,	__snake_case,	__snake_case      )
																														unique_s.add(__snake_case      )

																									# n=2
																									_UpperCamelCase									=    (
																									    x_num * x_num * y_den * y_den + x_den * x_den * y_num * y_num
																									)
																									_UpperCamelCase									=    x_den * x_den * y_den * y_den
																									if is_sq(__snake_case      ) and is_sq(__snake_case      ):
																														_UpperCamelCase									=    int(sqrt(__snake_case      )      )
																														_UpperCamelCase									=    int(sqrt(__snake_case      )      )
																														_UpperCamelCase									=    gcd(__snake_case,	__snake_case      )
																														z_num //= hcf
																														z_den //= hcf
																														if 0 < z_num < z_den <= order:
																																			_UpperCamelCase									=    add_three(
																																			    __snake_case,	__snake_case,	__snake_case,	__snake_case,	__snake_case,	__snake_case      )
																																			unique_s.add(__snake_case      )

                    # n=-1
																									_UpperCamelCase									=    x_num * y_num
																									_UpperCamelCase									=    x_den * y_num + x_num * y_den
																									_UpperCamelCase									=    gcd(__snake_case,	__snake_case      )
																									z_num //= hcf
																									z_den //= hcf
																									if 0 < z_num < z_den <= order:
																														_UpperCamelCase									=    add_three(
																														    __snake_case,	__snake_case,	__snake_case,	__snake_case,	__snake_case,	__snake_case      )
																														unique_s.add(__snake_case      )

																									# n=2
																									_UpperCamelCase									=    x_num * x_num * y_num * y_num
																									_UpperCamelCase									=    (
																									    x_den * x_den * y_num * y_num + x_num * x_num * y_den * y_den
																									)
																									if is_sq(__snake_case      ) and is_sq(__snake_case      ):
																														_UpperCamelCase									=    int(sqrt(__snake_case      )      )
																														_UpperCamelCase									=    int(sqrt(__snake_case      )      )
																														_UpperCamelCase									=    gcd(__snake_case,	__snake_case      )
																														z_num //= hcf
																														z_den //= hcf
																														if 0 < z_num < z_den <= order:
																																			_UpperCamelCase									=    add_three(
																																			    __snake_case,	__snake_case,	__snake_case,	__snake_case,	__snake_case,	__snake_case      )
																																			unique_s.add(__snake_case      )

					for num, den in unique_s:
										total += Fraction(__snake_case,	__snake_case      )

					return total.denominator + total.numerator


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













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





