We need a terminology fix here.
A half
butterfly is a simply a save where one leg or the other is extended,
FLAT ON THE ICE.
A half pad
extension is simply a save where one leg or the other is
ELEVATED OFF THE ICE.
The key points are elevation vs. flat on the ice. See the post by Scottyboy or proper definition using pictures.
I rarely get riled about things like this but please understand the difference before commenting. Please. Pretty please.
Without this basic understanding every facet of this discussion is useless.
With all due respect to Goalieworld20: I think you have been playing for a total of 3, maybe 4 years according to your profile. As someone has already noted, there are NO absolutes in goaltending, aside from a need to have a good awareness, good angle, good depth and be square to the puck. Everything else - they are simply details.
Regardless of the 'mathematical' formulas you have witnessed (please post them for us, I really want to see them) nothing is set in stone. The game is too dynamic for a one solution approach.
I spend my life (and this is a sad commentary

) essentially devoted to analyzing and understanding this wonderful position/sport. I read resource after resource, analyze multiple sources of material, talk to coaches, talk to NHL goalies and shoot hours of video which I then watch frame-by-(often) painful-frame to see what works. If it was as simple as saying do this, this and this I would be a gazillionaire right now (like the cookie cutter guys!).
Let me save the drama: it's not that simple. I wish it were.
I will repeat a simple truism:
every situation is unique and must be treated accordingly. The photo posted earlier by Goalieworld20, showing a half-pad extension save was textbook: no screen, no chance of a tip, a read & react save and no rebound. It does not get any better than that, regardless of what anyone tries to 'prove'.
I think I've said my piece here. I'm always open to new ideas but this is so old it must seem new to the rookies. God bless 'em for keeping us sharp.
Wellsie