Richardson's theorem for constants 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? 
 A: The page you cite eventually (at least as of 30 July 2015) links to an answer to your question about decidability.


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*An algebraic expression is decidable.  This is also the Tarski-Seidenberg theorem.

*Algebraic expressions plus $\exp$ are decidable if Schanuel's conjecture holds.  Otherwise, this is unknown.  See Tarski's exponential function problem.

*Throw in sine and determining whether the constant expression vanishes is undecidable.  (This can be linked to the undecidability of models of the integers (a la Godel's Incompleteness Theorems) via the zeroes of $\sin (\pi k)$ for any integer $k$.  See Hilbert's Tenth Problem and Matiyasevich's Theorem.)  Sine isn't critical; any (real-)periodic function would do.


[Edit:]  Following thinking about comments:  In fact, any function with a (real-)periodic level set (with positive minimal period) would do.  Every non-empty level set of all six trig functions qualify.  The easy case, for a warm-up, is to consider a function with a periodic discrete level set.
