How to scale this expression

[Luca]

Hello everyone,

I have a bit of an issue regarding scaling of an expression. So, the
scenario is as follows.

I have a confidence value that can be associated with the solution
given by an optimization routine and it is as follows:

C = exp(-A)/(exp(-A) + exp(-B))

where A, B and C are some energy values returned by the optimization
routine and C represents the confidence or the probability assigned to
the solution A. Also, B is always greater than A.

After some simple manipulation, the expression becomes:

C = 1.0 / (1 + exp(A-B))

Now, in the beginning my issue was that the values A, B and C were
usually quite large (in tens of thousands). So this expression was
giving values of 0.5 when A and B were very close and when the
difference was something a bit larger (in absolute numbers), then the
expression would basically become 1.

So, I realized I needed to do some normalization and the first thing I
tried was divide everything by A. So, now the expression becomes:

C = 1.0 / (1 + exp(1-B/A))

Now typically B/A is something from 1 to 1.01. So, now I have a
similar problem: As exp(1-B/A) will basically be 0.5.

So, what I would like to do is introduce some scaling, normalization
on this expression that would help me basically capture the changes in
my data range. I would be grateful for any suggestions that anyone
might have.

Thanks,
Luca