How do you reverse a gradient from any one color to any other color

If I have a gradient, know color a (primary) and b (secondary), how would I go about reversing said gradient so color a instead transitions into an equivalent alpha that when layered over a pure background of color b would producing the starting gradient.

To further elaborate, say I have this gradient:

And I want to split to so that I have this instead:

Where if I overlay it with this:

I get back to this:

I have been only able to get it work with the primaries at best. My math doesn't seem to work on anywhere else but the calculator (will post if requested), any ideas?