i've also done some searching for short maneuvers for superflip,
although not to the extent that dik has. i was never really
satisfied with my efforts to exploit its symmetry and centrality.
however, i've recently had some new thoughts about this which
consider two cases:
suppose that there is a minimal sequence for superflip which
contains a half-turn. then, by applying R' U2 to superflip,
we've moved 3q (or 2f ) closer to start.
otherwise, every minimal sequence contains only 90 degree turns.
then either R' U' gets us 2q (or 2f ) closer to start,
or R' U gets us 2q (or 2f ) closer to start. (and probably
it would be nice to reduce this latter case to only one of R' U'
or R' U . can anyone do this?
this was how i found the 24q sequence for superflip. i figured
that case 1 was fairly likely, so i tested the position
superflip R' U2
kociemba's algorithm found in succession 11 + 22 = 33q, 11 + 20 = 31q, 12 + 17 = 29q, 13 + 14 = 27q, 13 + 12 = 25q, which gives superflip in 28q. there was no improvement for quite some time until depth 17 in stage 1 when it found 17 + 6 = 23q. it searched for several days more and finished depth 17 and depth 18 in stage 1. i was about to give up when i saw that it found 19 + 2 = 21q, to give superflip in 24q.
here's a nice (and quite surprising application) of the above
use of symmetry and centrality:
when searching for superflip in the face turn metric, it's
sufficient to search through depth 17 in stage 1!
suppose we have a 19f sequence for superflip. then, by considering
parity, some turn must be a half-turn. now we may suppose (as above)
that the last two face turns are U R2 , which is in stage 2!