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enter image description here

So I substitued with the following values in my function T(x,n) and it gave me this red box with the value in the picture. When I hover over it, it tells me that there are no significant digits to display. Is it safe to treat(or approximate) the value in the red box as 0? That is, is the number in the red box a number that is extremely small, almost zero or is it a huge number that Mathematica did not manage to compute its other digits?

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    $\begingroup$ "Is it safe to treat(or approximate) the value in the red box as 0" - in general, no. This means the safeguards in place for precision tracking failed, and you now have a potentially useless result. This calls for using higher precision or a better algorithm (preferably the latter). $\endgroup$ Jun 30, 2016 at 8:35
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    $\begingroup$ It could of course be a coincidence, but I can't help but notice that your value for x is approximately $\pi$, but evaluated to way more digits than you have significant numbers. This suggests to me a precision failure also. $\endgroup$
    – Feyre
    Jun 30, 2016 at 9:34

1 Answer 1

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The output means that the result accuracy of the result is around 435. The actual result can be inspect with either InputForm or FullForm. The number you see after the 0``... is the Accuracy of the result. As defined by Mathematica, this means that the computed result is zero with an uncertainty of roughly 10^435.

Mathematica graphics

In the example above, the computed uncertainty is 2 * 10^435, but the red box shows an approximate result. What may be confusing is that the input is calculated with the 10^435:

(1.`20*^455 + 10^435) // InputForm
(* 
  1.00000000000000000001`20.*^455
*)

However, because the Precision is 20, this number is treated as insignificant. It is discarded when all significant digits are lost due to cancellation. You get the same result if the 10^435 term is omitted.

uncertainty = 10^(-Accuracy[1.`20*^455 - 1.`20*^455])
(*
  2.000000000000*10^435
*)
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