We describe an algorithm (VQE)\ for a \emph{variant} of the real quantifier elimination problem (QE). The variant problem requires the input to satisfy a certain \emph{extra condition}, and allows the output to be \emph{almost} equivalent to the input. The motivation/rationale for studying such a variant QE problem is that many quantified formulas arising in applications do satisfy the extra conditions. Furthermore, in most applications, it is sufficient that the output formula is almost equivalent to the input formula. The main idea underlying the algorithm is to substitute the repeated projection step of CAD by a single projection without carrying out a parametric existential decision over the reals. We find that the algorithm can tackle important and challenging problems, such as numerical stability analysis of the widely-used MacCormack's scheme. The problem has been practically out of reach for standard QE algorithms in spite of many attempts to tackle it. However the current implementation of VQE can solve it in about 12 hours. This paper extends the results reported at the conference ISSAC~2009.