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It seems that Mathematica can handle very big integers accurately, but cannot do the same for floating numbers.

See the following code:

aa = 10^200;
aa = aa + 1; 
N[aa*Sqrt[2] - Round[aa*Sqrt[2]], 20]

The error message is

N::meprec: Internal precision limit $MaxExtraPrecision = 50.` reached while evaluating -141421356237309504880168872420969807856967187537694807317667973799073247846210703885038753432764157273501384623091229702492483605585073721264412149709993583141322266592750559275579995050115278206057148+1000000000000000000000000000000000000000000000000000000000000000000000000000000000<<36>>00000000000000000000000000000000000000000000000000000000000000000000000000000000001 <<1>>.

How to handle this kind of problem?

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    $\begingroup$ This usually happens when the result of subtracting two numbers results in something that is relatively very tiny compared to the original numbers. Consider it a warning that subtractive cancellation is happening. Anyway: Block[{$MaxExtraPrecision = 250}, N[aa*Sqrt[2] - Round[aa*Sqrt[2]], 20]]. $\endgroup$ – J. M.'s technical difficulties Apr 10 '17 at 11:10
  • $\begingroup$ You mean it is just a warning message, not an error message? $\endgroup$ – S. Kohn Apr 10 '17 at 11:26
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    $\begingroup$ Yes, it's a warning, but a very important one. It says that the result may be rubbish. In fact, in this specific case it is rubbish. There is not a very clear distinction between "error" and "warning" messages in Mathematica. The practical question is: can you ignore it? You definitely should not. $\endgroup$ – Szabolcs Apr 10 '17 at 11:29
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    $\begingroup$ You should look up catastrophic cancellation. When such things happen, most systems just give you a wrong result, without any warning. Mathematica goes further in helping you and warns you that the result may be rubbish. $\endgroup$ – Szabolcs Apr 10 '17 at 11:30
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Oftentimes, to get a result with k digits of precision, the arithmetic needs to be done with more than k digits. Mathematica automatically increases the number of digits it uses to get a result with the requested precision. But there is a limit on how much it will increase it: $MaxExtraPrecision.

Why is there a limit? Consider examples like

N[
 Sin[Pi/9] - Root[-3 + 36 #1^2 - 96 #1^4 + 64 #1^6 &, 4],
 10
 ]

This triggers the same error. But no matter how much you increase the max extra precision, the error won't go away, because the exact result happens to be 0 here. Proving that it is zero is not trivial though, and Mathematica does not realize that it is so.


The solution, as J.M. said, is to allow for a higher precision increase during computations:

Block[{$MaxExtraPrecision = 250}, 
 N[aa*Sqrt[2] - Round[aa*Sqrt[2]], 20]]
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