
import math as m

def a__   (	UpperCAmelCase__,					UpperCAmelCase__,					UpperCAmelCase__=0.001 ):






  '''simple docstring'''





  return (f(a + h ) - f(a - h )) / (2 * h)

def a__   (	UpperCAmelCase__,					UpperCAmelCase__=0,					UpperCAmelCase__=100,					UpperCAmelCase__=0.0001,					UpperCAmelCase__=1e-6,					UpperCAmelCase__=False ):






  '''simple docstring'''





  __magic_name__    					=	xa  # set the initial guess
  __magic_name__    					=	[a]
  __magic_name__    					=	abs(f(_A ) )
  __magic_name__    					=	lambda UpperCAmelCase__ : 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""" )
    __magic_name__    					=	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

    __lowerCAmelCase      :					Dict   	=       lambda x: m.tanh(x) ** 2 - m.exp(3 * x)  # noqa: E731
    __lowerCAmelCase	,				__lowerCAmelCase	,				__lowerCAmelCase      :					List[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}}}''')



from itertools import compress, repeat
from math import ceil, sqrt

def a__   (	A_ ):






  '''simple docstring'''





  if num <= 2:
    return []
  if num == 3:
    return [2]

  # Odd sieve for numbers in range [3, num - 1]
  __magic_name__    					=	bytearray(b"""\x01""" ) * ((num >> 1) - 1)

  for i in range(3,					int(sqrt(A_ ) ) + 1,					2 ):
    if sieve[(i >> 1) - 1]:
      __magic_name__    					=	i**2
      __magic_name__    					=	repeat(
          0,					ceil((num - i_squared) / (i << 1) ) )

  return [2] + list(compress(range(3,					A_,					2 ),					A_ ) )


if __name__ == "__main__":
    import doctest

    doctest.testmod()

