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It's known that there is no algorithm for deciding for any elementary function is it identically zero or not (http://en.wikipedia.org/wiki/Richardson%27s_theorem ).

But if I consider only constants - is there some algorithm for deciding for any constant expression composed from elementary functions (e. g. $\ln (\sin 1 - \tan (\pi^2))$), is it equal to zero or not?

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For constants? Just evaluate them. You can use some heuristics, like $e^x \neq 0 \forall x$. –  Newb Dec 24 '13 at 19:39
@Newb, computers use only finitely much memory. Real numbers don't. –  Karolis JuodelÄ— Dec 24 '13 at 19:47
Try deciding whether $\tan p - q = 0, p, q \in \mathbb{Z}$. I'm sure that this number can be made arbitrarily small. How will a computer decide if you can't? –  Karolis JuodelÄ— Dec 24 '13 at 19:50
Also there are expressions like $\sqrt[3]{2 + \sqrt{5}} + \sqrt[3]{2 - \sqrt{5}} - 1$ (it's actually zero). –  ptashek Dec 24 '13 at 19:55
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