Given a real number uniquely defined by a finite system of equations and inequalities with rational coefficients involving the standard elementary functions only. Is it decidable whether this number is rational?

Edit: The system of equations and inequalities may have $n$ real-valued variables, and the real number to be defined is $\xi$. By decomposition using intermediate variables, we may suppose that each inequality is of the form $x\_i\,\omega\, c$ where $\omega\in\{<,>,\le,\ge\}$ and $c$ is an integer, and each equation is of the form $x\_i=\xi$, or $x\_i=c$ where $c$ is an integer, or $x\_i=x\_j\circ x\_k$ with $\circ\in\{+,-,\*,/\}$, or $x\_i=\phi(x\_k)$ where $\phi\in E$ with a specified set $E$ of elementary functions, such as $E=\{\exp,\log\}$. The answer to the question might depend on the choice of $E$.

I assume that $\xi$ is uniquely determined (hence defined) by this system of equations and inequalities. One may even assume that one has a proof for this, should this help.

The algorithm may be defined according to any of various traditional models, specifying a sensible set of primitive operations (including the rational operations [e.g., on bits or on reals] and branching on an expression). I am more interested in the applicable techniques than in an answer for a particular model.

Note that the Wikipedia article on the [constant problem](http://en.wikipedia.org/wiki/Constant_problem) 
claims (by reference to 
D.H. Bailey, Mathematics of Computation 50 (1988), 275–281)
a result from which decidability would follow, but this claim is spurious.