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    <p>Read paper titled "What every programmer should know about floating point" (google for it). Then you'll see that most of the floating values are represented approximately in computer hardware.</p> <p>To make calculations without this drawback, use symbolic calculation. But this is not as efficient as using floating point.</p> <p>To make floating point results consistent, use rounding to nearest power of 10, e.g., 0.1, 0.01, etc. To understand when you should stop apporximations, use some kind of threshold to watch for during approximation steps. E.g., if performing next approximation step yields only .001% change to already computed value, there's no sense to continue approximations.</p> <p><strong>Update</strong> I had my numerical computation classes long ago, but I can vaguely recall that substracting close numbers is very bad, because if numbers are very close, most reliable digits are cancelled out and you have unreliable digits. This is exactly what happens when you decreasing <code>h</code>. What is suggested in these situations is substitute substraction with some other operations. For example, you can switch to some kind of series to which your `f(x) expands.</p> <p>I don't quite understand your 2nd question, because answer depends on your requirements -- "as many as you wish".</p> <p>By the way, you might have better luck with finding answers to your questions at math.stackexchange.com.</p> <p>Additionally, visit link provided by <code>thrashgod</code>: <a href="http://en.wikipedia.org/wiki/Numerical_differentiation#Practical_considerations_using_floating_point_arithmetic" rel="nofollow">Numerical differentiation</a></p>
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