"Describe with" vs. "describe by"

Question

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?

Answer 1

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.

Answer 2

I think I've got an answer to this problem, but I am not 100% confident about this, so unless someone votes my solution up, do not accept it.

I believe if something is described by something else, the thing in question is all there is for the description to have been made.

On the other hand, if something is described with something else, the thing in question is a part of the solution, which helps the overall description.

To make the explanation easier, let's try some examples:

We describe the input with an exponential function.

In my opinion, this sentence means that the input was described by a longer list of functions, rules, definitions, which also contained an exponential function.

We describe the input by an exponential function.

This sentence means that the input was defined exactly and only by the exponential function, nothing more, nothing less.

To make it concrete, let's try a situation:

John, please, how many pills do I need to buy?

Answer A: 7 days multiplied by 3 pills a day = 21 pills.

Answer B: Well, that depends. If you're taking other pills, you can take only 2 per day, that would make it 7 days multiplied by 2 pills a day, 14 pills. It might also be possible we'll prolong our stay, in which case you'd need 14 days multiplied by 3 pills a day, that makes it 42 pills in total. You'd better get 42 pills to cover the worst case scenario.

In answer A, John resolved the problem by using an equation = the solution was described by the equation, just as the solution was described by John.

In answer B, John resolved the problem with a longer explanation, a part of which was an equation = the solution was described with the equation.

Again, I'm not 100% sure about this, it's still a long shot. I encourage other people to think about this and upvote/downvote this answer according to how they feel about it.

Answer 3

There are some differences between the two prepositions. For instance, an section of land with a lake is not the same as a section of land by a lake.

With implies that these is a direct connection, whereas by implies that while the items are associated, they are separate items.

In your example, if you have a choice of inputs, and you are choosing the input that contains an exponential function, you should use with. If you are using an exponential function (by itself) as an example input to the function, use by.