Though "formal science" is not a familiar term to me, I take your word for it and hear what you are saying. However, I always considered mathematics to be it's own thing, different than science. Real science is physics and chem and then biology starts to merge more into a body of knowledge.
Anything that has "science" in the title isn't really. And in line with that, I consider "computer science" to be a misnomer as it is more accurately categorized as a branch of mathematics. And computer science is, of course, completely distinct from software development, which is a craft.
All that said, I would be hesitant to put psychology and psychiatry in the same boat. Neither one is a hard science, but psychiatry is much less likely to be anyone's fall-back major.
I won’t say what psychology or psychiatry are. Depending on who you ask, they’re empirical sciences, bodies of knowledge, or cargo cult medicine. Software development seems to be all over the place. Some of it clearly relies on computer science, some of it seems to be a body of knowledge.