





import math as m
def   _A (	SCREAMING_SNAKE_CASE  :     Optional[Any] , SCREAMING_SNAKE_CASE  :     int , SCREAMING_SNAKE_CASE  :     Union[str, Any]=0.0_0_1		):





  """simple docstring"""



  return (f(a + h		) - f(a - h		)) / (2 * h)
def   _A (	SCREAMING_SNAKE_CASE  :     int , SCREAMING_SNAKE_CASE  :     Dict=0 , SCREAMING_SNAKE_CASE  :     List[Any]=100 , SCREAMING_SNAKE_CASE  :     str=0.0_0_0_1 , SCREAMING_SNAKE_CASE  :     Optional[int]=1e-6 , SCREAMING_SNAKE_CASE  :     Any=False		):





  """simple docstring"""



  a__			:      int            =xa  # set the initial guess
  a__			:      List[str]            =[a]
  a__			:      Any            =abs(f(SCREAMING_SNAKE_CASE		)		)
  a__			:      List[str]            =lambda SCREAMING_SNAKE_CASE		: calc_derivative(SCREAMING_SNAKE_CASE , SCREAMING_SNAKE_CASE , h=SCREAMING_SNAKE_CASE		)  # noqa: E731  Derivative of f(x)
  for _ in range(SCREAMING_SNAKE_CASE		):
    if fa(SCREAMING_SNAKE_CASE		) == 0:
      raise ValueError("No converging solution found"		)
    a__			:      Any            =a - f(SCREAMING_SNAKE_CASE		) / fa(SCREAMING_SNAKE_CASE		)  # Calculate the next estimate
    if logsteps:
      steps.append(SCREAMING_SNAKE_CASE		)
    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[int]	  = lambda x: m.tanh(x) ** 2 - m.exp(3 * x)  # noqa: E731
     UpperCAmelCase  ,			UpperCAmelCase  ,			UpperCAmelCase			:			Any	  = 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}}}""")













def   _A (	SCREAMING_SNAKE_CASE  :     int		):





  """simple docstring"""



  if not isinstance(SCREAMING_SNAKE_CASE , SCREAMING_SNAKE_CASE		) or number < 0:
    raise ValueError("Input must be a non-negative integer"		)

  a__			:      List[str]            =number & ~(number - 1)
  a__			:      Optional[int]            =0
  while intermediate:
    intermediate >>= 1
    index += 1
  return index - 1


if __name__ == "__main__":
     import doctest

     doctest.testmod(verbose=True)






