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Here is the situation;

There has been a value A, and now I'm replacing the value A with the new result of adding A and the new value B (i.e., A + B).

The simplest way that I can think of to describe this is "I'm accumulating B to A", but I can hardly find any of this usage on the internet. Could anyone recommend an idiomatic expression to describe this?

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You're perhaps overthinking this. You can simply say:

I'm adding B to A

To me this implies what you're trying to say above.

As trivia, this operation in programming is often known as addition assignment, and would be written A += B

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