A couple of points:
None of these formulae are going to be exact, they are approximations.
Gas chromatography is indeed in the best way to measure this and so we generally prefer approximations that can be performed with a number of hydrometer readings.
I believe the Cape Brewing approximation is better than the 131 approximating however I think Cape Brewing is doing it wrong. In the numerator you ought to be subtracting the real final extract and not the apparent final extract. Also this ought to be performed in degrees plato not SG terms.
The two approximations ought to agree quite closely if the Cape Brewing approximation (due to Balling) is being used correctly.
Here is an HBD post from George Fix on the Cape Brewing method.
HOMEBREW Digest #880 Wed 13 May 1992
Note that Fix claims that the Balling approximation is subject to error as large as 3%-5% (percent error, not in absolute terms) and the 131 approximation is a bit rougher.