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So, as the question asks, is we have only normalised floating-point values and normalised results, could you please explain how x + y = x?

I know it all relates to precision, but how can I explain that?

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    If `y/x` is tinier in magnitude than machine epsilon, well...2011-05-01
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    Can you please continue your explanation?2011-05-01
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    I would suggest that you look up what machine epsilon is first, and then we can talk.2011-05-01

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Suppose you have a floating-point system which keeps track of, say, two decimal digits. If $x=1.0 \times 10^0$ and $y=3.4 \times 10^{-10}$, then the exact value of $x+y$ is $1.00000000034$, which would be represented in your system by the closest floating-point number, which is... Well, what is it?

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    please see [link](http://math.stackexchange.com/questions/757598/floating-point-operations-do-not-satisfy-the-well-known-laws-for-arithmetic-oper). How to prove general problem that mentioned in this link?2014-04-18
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    @MathMan: I already saw that question, but unfortunately I must say that it was too poorly formatted for me even to bother to figure out exactly what it was about...2014-04-18
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    how to prove it in general for any $x,y$ with $|y|?2014-04-18