Is Harry Potter's magic random, "mathematical" or does it have its own personality?

After reading Harry Potter series, I can't really figure out what nature of magic did J.K. Rowling try to picture in their books. Sometimes the spells seem to be basically "mathematical" in their nature, performed very literally up to the point when it's unfavourable for the caster. In other cases, such as teleportation, the effects seem to be "whimsy" as if there was some external entity that decided whether they would work or not.

Are there any rules on this in the Harry Potter universe? Is magic an exact science here in any sense?

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Did the gatekeeper and the keymaster get intimate in Ghostbusters?

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