An application I'm working on has required me to find simple approximations to a rather complicated bivariate function $g(x,y)$ that also takes a long time to evaluate on the computer. Through sheer algebra and much sweat, I finally managed to derive two simpler approximations, which I shall call $g_1(x,y)$ and $g_2(x,y)$.
How do I (quantitatively) determine which approximation is a "better" approximation to the original function?
The original complicated bivariate function is physically sensible on a rectangular domain, and I thus thought to compare $g_1(x,y)$ and $g_2(x,y)$ by plotting, respectively, $\left|1-\frac{g_1(x,y)}{g(x,y)}\right|$ and $\left|1-\frac{g_2(x,y)}{g(x,y)}\right|$ (i.e. the relative error) over that domain. Two approaches I've thought of are 1. integrate the relative error function over the domain of definition (i.e. a double integral); and 2. find the maximum value of the relative error; after which I compare the results for the two relative error functions.
Are there other, better/more sensible ways of performing a quantitative comparison?