Learn English – Usage of “before” in “I watched the sequel before the original movie”

To my ear, your explanation is largely correct. In particular, this sentence sounds more natural than the alternative with by:

We describe the input with an exponential function.

The primary reason here is that the agent and the subject of the sentence is we, and the phrase "an exponential function" is an instrument deployed by the subject. However, this case is different:

The input is best described by an exponential function.

You were correct in using by here. However, in this case it's crucial that this is passive voice (with the passive verb is described), and the passive voice always uses by. Questions of instrumentality vs. agency don't enter into it.

That would be grammatically correct, but not idiomatic.

The phrase "as is usually done" is used much more often. See the ngram.