In a previous paper  we have identified a special class of linear hybrid automata, called Deterministic Transversal Linear Hybrid Automata, and shown that an ε-reach set up to a finite time, called a bounded ε-reach set, can be computed using infinite precision calculations. However, given the linearity of the system and the consequent presence of matrix exponentials, numerical errors are inevitable in this computation. In this paper we address the problem of determining a bounded ε-reach set using variable finite precision numerical approximations. We present an algorithm for computing it that uses only such numerical approximations. We further develop an architecture for such bounded ε-reach set computation which decouples the basic algorithm for an ε-reach set with given parameter values from the choice of several runtime adaptation needed by several parameters in the variable precision approximations.