Effective resolution of singularities

Consider a projective variety $X\subset {\mathbb{P}}^n$ (over an algebraically closed field of characteristic zero), together with a (reduced) simple normal crossings divisor $E\subset {\mathbb{P}}^n$, where the degrees of both $X$ and $E$ are at most $d$. We show there is a pair $(n^{\mathrm{\prime }},d^{\mathrm{\prime }})$ which can be explicitly computed in terms of $(n,d)$, such that $(X,E)$ has a log resolution of singularities $(X^{\mathrm{\prime }},E^{\mathrm{\prime }})$, where $(X^{\mathrm{\prime }},E^{\mathrm{\prime }})$ can be embedded in ${\mathbb{P}}^{n^{\mathrm{\prime }}}$ and both $X^{\mathrm{\prime }}$ and $E^{\mathrm{\prime }}$ have degrees at most $d^{\mathrm{\prime }}$ in ${\mathbb{P}}^{n^{\mathrm{\prime }}}$.