Photoshop-style blend equations with destination alpha

The photoshop blend formulas are available in many places on the web, but
they all assume an opaque output target. Sometimes the output target is not
opaque (for example, with layer folders in Photoshop and the blend mode
not set to "normal").

The following formulas appear to be the ones used by Adobe Flash Player;
they might differ from Photoshop blend modes, and they do not include all of
the Photoshop blend modes (these formulas were developed for and verified in
Iggy, software I wrote for RAD Game Tools which "plays" Adobe Flash files).

                   out color                                                    out alpha
                   --------------                                               -----------
layer/over:    (   sc+(1-sa)*dc                                              ,  sa+da-sa*da   )
multiply:      (   sc*dc                                                     ,  sa+da-sa*da   )
screen:        (   sa*da - (da-dc)*(sa-sc)                                   ,  sa+da-sa*da   )
lighten:       (   max(sa*dc,sc*da)                                          ,  sa+da-sa*da   )
darken:        (   min(sa*dc,sc*da)                                          ,  sa+da-sa*da   )
add:           (   min(dc+sc,1)                                              ,  min(sa+da,1)  )
subtract:      (   max(dc-sc,0)                                              ,  min(sa+da,1)  )
difference:    (   abs(sa*dc-sc*da)                                          ,  sa+da-sa*da   )
invert:        (   sa*(da-dc)                                                ,  sa+da-sa*da   )
overlay:       (   dc < da/2.0 ? (2.0*sc*dc) : (sa*da - 2.0*(da-dc)*(sa-sc)) ,  sa+da-sa*da   )
hardlight:     (   sc < sa/2.0 ? (2.0*sc*dc) : (sa*da - 2.0*(da-dc)*(sa-sc)) ,  sa+da-sa*da   )

  sc = source color, sa = source alpha, dc = dest color, da = dest alpha

The inputs in the above equations must be premultiplied. If inputs are
non-premultiplied, replace "sc" with "sc*sa". (Outputs are always
premultiplied, hence destination is always premultiplied.)

-- Sean Barrett, 2012/09/19