I'm not entirely certain about when it is most appropriate to use with and by, respectively. An example should clarify my problem:
We describe the input with an exponential function.
or
We describe the input by an exponential function.
My understanding is that in this type of situation, with is used if the object is more like a tool and by if it is the active agent. This would suggest that with is more appropriate in the above example but in the following example, by is better:
The input is best described by an exponential function.
Is this usage correct? Is there a clear grammatical or stylistic rule?
Best Answer
To my ear, your explanation is largely correct. In particular, this sentence sounds more natural than the alternative with by:
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:
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.