If I'm going to the optometrist to get a pair of new lenses with a new prescription for my glasses, do I say I'm going to renew, or I'm going to update my prescription?
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Update or perhaps refresh both seem like suitable words to me — suggesting that you are going to check your prescription, and change if necessary.
To me, renew suggests that you are going there to get another prescription which is the same (or extend the life of the old prescription) — which of course, would not usually be the case.
The bottom line is, if you said you were going to ‘update your prescription’, would most people understand what you were meaning? Yes, they would.
I'd just say that I was getting new glasses. That seems to cover everything (even though glass is no longer involved in their manufacture).
It's irrelevant whether you keep the same frames or not, unless you think of glasses as facial adornments, instead of seeing aids, in which case it's unlikely anyone would care about the details of your prescription change.
Update is definitely the right word. You might be thinking of the word "renew" because of its usage in the phrase "renewing a subscription," like for a magazine.