Bug introduced after 5.2, fixed in 8.0, reintroduced in 9.0 and persisting through 12.3

Is this a bug?

If I do

FullSimplify[n E^(0``10 n)]

then it returns


which is obviously incorrect.

(I've simplified the example, following a comment.)

[Mathematica Version]

  • 3
    $\begingroup$ A wild guess: Since the -1.`10 in a, b, and nb are independent, the quotient has a great deal of uncertainty in nb/a, represented by 0``9.698970004336022. Consider Plot[MinMax[1.`10.*2.`10.^(Interval[0``9.698970004336022] n) n ]/ n, {n, -10^10, 10^10}]. But I don't understand the 1/2 for a point estimate of the coefficient, or why b/a is presented as an exact result. Note that a/a is not 1 exactly. $\endgroup$
    – Michael E2
    Mar 10, 2017 at 11:49
  • 1
    $\begingroup$ Simpler example: FullSimplify[n E^(0``100. n) ], Series[n E^(0``100. n) , {n, 0, 0}]. The accuracy 100. does not seem to matter, big or small, positive or negative. $\endgroup$
    – Michael E2
    Mar 10, 2017 at 12:47
  • $\begingroup$ I don't see how this isn't a bug. n/2 just isn't correct. Can you please contact Wolfram Support and let us know what they said? $\endgroup$
    – Szabolcs
    Mar 12, 2017 at 10:39
  • $\begingroup$ With versions 5.2 and 8.0.4 I get E^(0.*10^-10 n) n as the output (with InputForm being E^(0``10.*n)*n). So the bug was introduced in version 9.0. $\endgroup$ Mar 12, 2017 at 14:58
  • $\begingroup$ Sorry to say, but with version 7.0.1 on Win7x64 I get n/2. Incredible. $\endgroup$
    – innaiz
    Mar 12, 2017 at 17:19

1 Answer 1


I think I've traced down the problem. It hinges on two things. An identity:

Cosh[x] == Sinh[2 x]/(2 Sinh[x]) // Simplify
(*  True  *)

And a questionable auto-simplification:

Csch[0``10. n] Sinh[2 0``10. n]
(*  1  *)

{Csch[0``10. n], Sinh[2 0``10. n]} // FullForm
(*  List[Csch[Times[0``10.,n]],Sinh[Times[0``9.698970004336019,n]]]  *)

(The coefficients are equal, so I guess that's why they are treated as identical.)

Here are the steps in which these problems arise:

n Exp@(0``10 n) // ExpToTrig
(*  n (Cosh[0.*10^-10 n] + Sinh[0.*10^-10 n])  *)

n (Cosh[0``10. n] + Sinh[0``10. n]) //. SimplifyDump`CosToSinRules
(*  n (1/2 + Sinh[0.*10^-10 n])  *)

Sinh[0``10. n] // FullSimplify
(*  0  *)

The simplified expression is already wrong in the second step, in which the identity is applied. The last step is questionable, too, but understandable. It's what leads to n/2 as the answer.

I, for one, feel I understand why this bug has persisted.

Maybe the best workaround is to flush all "underflowed" arbitrary-precision numbers to exact 0:

FullSimplify[n E^(0``10 n) /. z_ /; z == 0 :> 0]
(*  n  *)

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.