import math 

def newton( fn,df, start1, err):
   a=float(start1)
   fa=fn(a)
   fpa=df(a)
   while True:
     c=a-fa/fpa
     b=a
     if math.fabs(b-a)<err: break
     a=c
     fa=fn(a)
     fpa=deriv(fn,a)
   return c 


def deriv2(f,x,h):
  return (f(x+h)-f(x-h))/2/h

def deriv(f,x):
  h=max(math.fabs(x),math.fabs(f(x)))/2**25
  if h==0: h=1e-30
  h=((x+h)-(x-h))/2
  d1=deriv2(f,x,h)
  h=h/2
  h=((x+h)-(x-h))/2
  d2=deriv2(f,x,h)
  return (4*d2-d1)/3



Y=2
def f(x):
   return x**2 -Y

def df(x):
    return deriv(f,x)

print newton(f,df,1,1e-10)