Given a smooth hypersurface $H$ in $\mathbb{C}^n$, a theorem of Hironaka promises that one can find a strict normal crossings compactification $\bar{H}$ inside of a projective variety $X$. For me, this result is the definition of a "black box fact". What are some techniques for computing such a compactification in practice? 


More practically, I am interested in obtaining a strict normal crossings compactification of the Koras-Russell cubic which is the hypersurface in $\mathbb{C}^4$ cut out by the equation: 


$$ x + x^2y + z^2 + t^3 = 0 $$


I should confess that I haven't really tried in this example. I could just compactify the hypersurface in say $\mathbb{C}P^n$ (or some weighted projective space) and then start blowing things up and if I am lucky enough eventually arrive at something smooth. But I'm wondering how professional algebraic geometers go about doing these things in a more systematic way. 



Making Hironaka's theorem explicit for hypersurfaces
By now there are more tractable proofs of resolution of singularities than Hironaka's, so it no longer has to be a black box. A relatively elementary approach is described in Kollár's [Lectures on Resolution of Singularities](http://books.google.com/books/about/Lectures_on_Resolution_of_Singularities.html?id=Oygejj1QFhgC). There are many references there to algorithmic resolutions. In particular, in the case of hypersurfaces there is even a Maple program to do it. Check out Bodnár-Schicho's paper [Automated Resolution of Singularities for Hypersurfaces](http://www.sciencedirect.com/science/article/pii/S0747717199904145) and the references in there. (I'm not trying to be comprehensive in this answer to provide all the references, because there are lots of them. But probably most references that are at least 6-8 years old are referenced in the above two sources.)


