'''simple docstring'''

from itertools import count
from math import asin, pi, sqrt



def  _lowerCAmelCase				(				__a      )  ->      float:
     '''simple docstring'''







     return (
         (1 - 2 * point) * sqrt(point - point**2      ) + 2 * point + asin(sqrt(1 - point      )      )
     ) / 4



def  _lowerCAmelCase				(				__a      )  ->      float:
     '''simple docstring'''







     _UpperCamelCase  :Optional[int]       =(circles_number + 1 - sqrt(2 * circles_number      )) / (
         2 * (circles_number**2 + 1)
     )
     _UpperCamelCase  :Dict       =circles_number * intersection_y

     _UpperCamelCase  :Dict       =intersection_x * intersection_y / 2
     _UpperCamelCase  :Any       =circle_bottom_arc_integral(
         1 / 2      ) - circle_bottom_arc_integral(__a      )

     return triangle_area + concave_region_area



def  _lowerCAmelCase				(				__a = 1 / 10_00      )  ->      int:
     '''simple docstring'''







     _UpperCamelCase  :str       =(1 - pi / 4) / 4

     for n in count(1      ):
          if concave_triangle_area(__a      ) / l_section_area < fraction:
               return n

     return -1


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






from ..utils import DummyObject, requires_backends



class  SCREAMING_SNAKE_CASE_  ( metaclass=SCREAMING_SNAKE_CASE__			):


  """simple docstring"""





  _a								=							["""torch""", """transformers""", """k_diffusion"""]





  def __init__(  self	,  *A	,  **A						)		->   Optional[Any]:





       '''simple docstring'''







       requires_backends(self	,  ['''torch''', '''transformers''', '''k_diffusion''']						)





  @classmethod
  def 			__A      (  cls	,  *A	,  **A						)		->   Dict:





       '''simple docstring'''







       requires_backends(cls	,  ['''torch''', '''transformers''', '''k_diffusion''']						)







  @classmethod
  def 			__A      (  cls	,  *A	,  **A						)		->   Optional[Any]:





       '''simple docstring'''







       requires_backends(cls	,  ['''torch''', '''transformers''', '''k_diffusion''']						)