RGB Blending

Let C by the destination background

Firstly, layer 0 (RGB0, A0) blends with the destination like this:
RGBBlend0 = C * (1 - A0) + RGB0 * A0

Then layer 1 (RGB1, A1) blends with the destination like this:
RGBBlend1 = RGBBlend0 * (1 - A1) + RGB1 * A1

-------------------------------------------------

Let's combine these two steps (substituting):
RGBBlend1 = C * (1-A0)*(1-A1) + RGB0*A0*(1-A1) + RGB1*A1

What we want is a layer that achieves this in one blend, ie (RGB, A) such that:
RGBBlend1 = C * (1 - A) + RGB * A

Equating C terms,
1-A = (1-A0)*(1-A1)

A = 1 - (1-A0)*(1-A1) = A0 + A1 - A0*A1	-- (equation Alpha)

Equating RGB terms,
RGB = ( RGB0*A0*(1-A1) + RGB1*A1 ) / A	-- (equation for RGB)

------------------------------------------------

With some rearrangement, we get:
RGB = RGB0 + (RGB1 - RGB0) * F(A0,A1)

where F(A0,A1) is a lookup table given by
F(A0,A1) = A1 / (A0 + A1 - A0*A1)