`a -> B/A`

b -> C/A

c -> D/A

d -> E/A

Expanding the formula into a standard form, of course, is another matter. I used to amuse myself doing these by hand when I was 13, but now I am too busy for it.

So even without consulting Mathematica, it is trivial to see that exists such a formula--but when you asked "is there a formula," maybe you meant "has the formula ever actually been written, and simplified." My answer is "probably; I could easily imagine someone like me doing this." Calcyman's formulas don't seem to be simplified thoroughly, though; they still contain square roots and cube roots of non-polynomial rational expressions, as well as sums of fractions having different denominators. I think that if the formula were to be entirely simplified, no term but A would be taken to a power higher than the fourth. (In the expanded cubic formula, no term but A has an exponent higher than 3.) If provable, this would work as a good test to see whether an evaluation by hand was correct.