I'm not sure if what I ask is feasible nor into what theory or formalism to look in order to solve or find a satisfactory ansatz for it (or the tags to add here -- any corrections are welcome), but consider the simple expression
\begin{equation} A(c,d,\alpha,\beta,x,y):=cd(\alpha x+\beta y). \end{equation}
Is there a way to approximate it (with regard to some hypothetical precision or constraints) as a function which depends on the variables $c,d,\alpha,\beta,x,y$ only through the combinations $q=c^2(\alpha^2+\beta^2)$ and $r=d^2(x^2+y^2)$?
That is, is there a way to explicitly find $f$ such that
\begin{equation} cd(\alpha x+\beta y) \approx f(q,r) \end{equation} ?
There is no necessity for arbitrarily high precision (for instance, a proximity of 50% to the real value of $A$ would be acceptable -- actually, the order of magnitude is all I need), but, obviously, the method has to make 'analytical sense' and be well defined.
In principle, it can be assumed that $-1<c,d,\alpha,\beta,x,y<1$.
$q=c^2(\alpha^2+\beta^2)$ and $r=d^2(x^2+y^2)$
Want $cd(\alpha x+\beta y) \approx f(q,r)$
Note that $qr=c^2d^2(\alpha^2+\beta^2)(x^2+y^2)$
$qr=c^2d^2(\alpha^2x^2+\alpha^2y^2+\beta^2x^2+\beta^2y^2)$
$qr=c^2d^2(\alpha^2x^2+2\alpha x \beta y + \beta^2y^2+\alpha^2y^2-2\alpha x \beta y+\beta^2x^2)$
$qr=c^2d^2(\alpha x+\beta y)^2 + c^2d^2 (\alpha y-\beta x)^2$