code: from itertools import count
  from math import asin, pi, sqrt
  
  
  def 					__lowerCAmelCase			(  _A					):
  
  
  
  							"""simple docstring"""
  
  
  							return (
  							    (1 - 2 * point) * sqrt(point - point**2					) + 2 * point + asin(sqrt(1 - point					)					)
  							) / 4
  
  
  def 					__lowerCAmelCase			(  _A					):
  
  
  
  							"""simple docstring"""
  
  
  							_lowercase				=	(circles_number + 1 - sqrt(2 * circles_number					)) / (
  							    2 * (circles_number**2 + 1)
  							)
  							_lowercase				=	circles_number * intersection_y
  
  							_lowercase				=	intersection_x * intersection_y / 2
  							_lowercase				=	circle_bottom_arc_integral(
  							    1 / 2					) - circle_bottom_arc_integral(_A					)
  
  							return triangle_area + concave_region_area
  
  
  def 					__lowerCAmelCase			(  _A = 1 / 1_000					):
  
  
  
  							"""simple docstring"""
  
  
  							_lowercase				=	(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() = }''')
  
  
  
style_context: import math
  from timeit import timeit
  
  
  def 					__lowerCAmelCase			(  _A					):
  
  
  
  							"""simple docstring"""
  
  
  							_lowercase				=	0
  							_lowercase				=	abs(_A					)
  							while True:
  														_lowercase				=	n // 10
  														digits += 1
  														if n == 0:
  																					break
  							return digits
  
  
  def 					__lowerCAmelCase			(  _A					):
  
  
  
  							"""simple docstring"""
  
  
  							return 1 if n == 0 else math.floor(math.log(abs(_A					) ,10					) + 1					)
  
  
  def 					__lowerCAmelCase			(  _A					):
  
  
  
  							"""simple docstring"""
  
  
  							return len(str(abs(_A					)					)					)
  
  
  def 					__lowerCAmelCase			(  ):
  
  
  
  							"""simple docstring"""
  
  
  							from collections.abc import Callable
  
  							def benchmark_a_function(_A ,_A					) -> None:
  														_lowercase				=	f'''{func.__name__}({value})'''
  														_lowercase				=	timeit(f'''__main__.{call}''' ,setup="""import __main__"""					)
  														print(f'''{call}: {func(_A					)} -- {timing} seconds'''					)
  
  							for value in (262_144, 1_125_899_906_842_624, 1_267_650_600_228_229_401_496_703_205_376):
  														for func in (num_digits, num_digits_fast, num_digits_faster):
  																					benchmark_a_function(_A ,_A					)
  														print()
  
  
  if __name__ == "__main__":
  				import doctest
  
  				doctest.testmod()
  				benchmark()
  
  
  