Just slow down and look at it line by line. Take detours to make sure you understand every part.
A better approach might be to learn how cosine can be approximated with a Taylor expansion, then realize this is just an optimized version of that. (Even if it’s a different algorithm, it’s still an approximation algorithm, which is the point.)
Staring at math constants won’t get you that insight.
I can reason through the code and understand it.
The thing that makes me wonder if I'll ever be a real programmer when I grow up is that someone wrote this, probably on a Thursday after lunch (at least, that's how my brain imagines it).
They were able to conceptualize the math solution to a major optimization problem, deal with all sorts of precision issues with bit manipulation, and come up with a blazing fast bedrock algorithm that's dazzling both in it's simplicity and complexity. And then likely popped out to a pub for a pint and didn't think much about it again.
Intellectually, I understand that this person was specialized, and may struggle mightily navigating the complexity of modern web frameworks, something that I find straightforward because I do it every day.
But for some reason, this stuff always strikes me as being "real" programming, and what I'm doing as just plumbing, piping together the bits that the "real" programmers write.
This could have taken weeks of research and refinement. You can't tell from just looking at the final result.
Give yourself some credit. If you were tasked with producing a cosine implementation, you'd pull out some textbooks, read specs, research algorithms, try a few ideas, test, iterate. You could come up with this. The only reason you didn't is that you didn't need to make a cos function yet; you had other stuff to do.
My guess is that the author of this code was possessed by a spirit like this. Maybe I'm wrong; maybe he was just assigned by his manager to write a cosine function that met a certain performance threshold, and he immediately had a general idea of how to do it, and banged it out in an afternoon. But I doubt it.
My point is that, if you have felt this way about programming, you are doing "real" programming. It's not about whether you're writing a web framework or a cosine function. It's about the thrill of the chase.
Also, have you ever formally studied maths? People don't just come up with this kind of thing. They read about algorithms in textbooks. They might not have read this specific algorithm (which may be original), but they'll have read lots of similar ones.
You've spent your time one X, they've spent their time on Y. That's it. Most of these people are math people. They may have CS degrees, but they've spent the bulk of their time in math.
Actually, now that I recall, one of them was a nuclear physicist and the other was an industrial engineer - neither had CS degrees. I don't know about the others.