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






def     lowerCamelCase__				(   a ) ->  float:
   return (
       (1 - 2 * point) * sqrt(point - point**2 ) + 2 * point + asin(sqrt(1 - point ) )
   ) / 4






def     lowerCamelCase__				(   a ) ->  float:
   _A:    int						     =			(circles_number + 1 - sqrt(2 * circles_number )) / (
       2 * (circles_number**2 + 1)
   )
   _A:    Any						     =			circles_number * intersection_y

   _A:    int						     =			intersection_x * intersection_y / 2
   _A:    List[str]						     =			circle_bottom_arc_integral(
       1 / 2 ) - circle_bottom_arc_integral(a )

   return triangle_area + concave_region_area






def     lowerCamelCase__				(   a = 1 / 10_00 ) ->  int:
   _A:    Optional[Any]						     =			(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() = }""")





import math as m






def     lowerCamelCase__				(   a						,						a						,						a=0.001 ) ->  Optional[int]:
   return (f(a + h ) - f(a - h )) / (2 * h)






def     lowerCamelCase__				(   a						,						a=0						,						a=1_00						,						a=0.0001						,						a=1E-6						,						a=False ) ->  List[Any]:
   _A:    Optional[Any]						     =			xa  # set the initial guess
   _A:    Dict						     =			[a]
   _A:    Tuple						     =			abs(f(a ) )
   _A:    Tuple						     =			lambda a : calc_derivative(a						,						a						,						h=a )  # noqa: E731  Derivative of f(x)
   for _ in range(a ):
      if fa(a ) == 0:
         raise ValueError('''No converging solution found''' )
      _A:    Any						     =			a - f(a ) / fa(a )  # Calculate the next estimate
      if logsteps:
         steps.append(a )
      if error < maxerror:
         break
   else:
      raise ValueError('''Iteration limit reached, no converging solution found''' )
   if logsteps:
      # If logstep is true, then log intermediate steps
      return a, error, steps
   return a, error


if __name__ == "__main__":
   from matplotlib import pyplot as plt

   UpperCAmelCase__						:	Optional[Any]	     = lambda x: m.tanh(x) ** 2 - m.exp(3 * x)  # noqa: E731
   UpperCAmelCase__     ,UpperCAmelCase__     ,UpperCAmelCase__						:	str	     = newton_raphson(
       f, xa=10, maxiter=1000, step=1E-6, logsteps=True
   )
   plt.plot([abs(f(x)) for x in steps])
   plt.xlabel('step')
   plt.ylabel('error')
   plt.show()
   print(F"""solution = {{{solution:f}}}, error = {{{error:f}}}""")



