I have a family of irrational numbers, say $w_k$ , and I have the hypothese, that they all are of the form
$$ w_k = {a \beta + b\over c \beta +d } \qquad \qquad a,b,c,d \in \mathbb Z
$$
where $\beta$ is likely some power of $e$.
I could decode a first small subset of numbers $w_k$ successfully using $\beta = e =\small \exp(1)$ , and also another subset using $\beta = e^2 $ with the help of Wolfram-Alpha ("W/A").
Now I have another subset where very likely I need $\beta=e^{0.5}$ , and so on.
As W/A cannot help much with the latter and I'd like to increase the subset of decodings I'd like to understand how I could approach such a decoding myself.
I use Pari/GP and the lindep procedure cannot work efficiently because in the denominator I have to expect that mixed term.
An example which I have solved is
$w_1 = 3.90612672462231517556062815632 ... $ finding $w_1=- {49 e^2- 315 \over 41 e^2-315} $ using W/A,
and one which I could not yet solve is
$w_2 =4.32632954361426254079031222453 ... $ again very likely implying $\beta = e^2$.
Another unsolved is
$ w_3 = 6.79282828995516187191178543614 ...$ which very likely implies now $\beta = e^{0.5}$
Q: how could I approach such a detection routine, or at least a meaningful preprocessing to adapt that numbers to the available (Pari/GP, W/A) procedures?
Footnote: this question occurs in an attempt to generalize the table shown in MO (which uses $\beta=e$) to other families of fractions which I had not looked at before.
P.S.: I've no good idea for tagging, perhaps someone can help improve the tags