What does it mean to "bust out" a program on a computer? for example:
"bust out" the calculator.
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This is sense 2 from the Merriam-Webster definition:
to take (something) from the place where it is stored so that it can be used : to break out
This is extended metaphorically to opening a programme on a computer. To "bust out the calculator" is to take the calculator program 'from' where it is stored so it can be used - that is, the software in question is run. Though in those examples, they likely meant the actual physical device, not the program.
This is not a phrase with a single invariable meaning. Context would be needed to determine what is meant. It might mean to delete the program. It might mean to run the program, or to install the program.
This is slang, and not widely used slang in this context. I would not advise using it, precisely because it is not clear.
I've been a professional software developer since 1985 (and a student before that), and I don't recall ever having heard this phrase used of a computer program.
In the sense used in the now supplied context, "bust out" means "break out" or more simply "get out" or "deploy". "Bust out the calculator" means just "have the calculator ready for use" or "prepare to use the calculator." I still think that it is sub-optimal writing.
It could easily refer to either a calculator program on a computer, or to a separate physical calculator.
It's not referring to a program. It's referring to a physical device.
bust out (something) or bust (something) out : to take (something) from the place where it is stored so that it can be used : to break out He busted out the champagne.