from math import *

def f(x):
	return sin(x)
	
def df_approx(f, x, h):
	return (f(x+h)-f(x))/h
    #return (f(x+h)-f(x-h))/(2*h)


	
a = 1/2	
for n in range(1, 19):
	h = 10**(-n)
	df_app = df_approx(f, a, h)
	df = cos(a)
	print("h: %e, df_approx: %13.10f, abs_err: %e" % (h, df_app, abs(df - df_app)))