I used heinzb's strategy to get the Insane AI romp achievement, so I had to do it 10 times in a row. This gave me the opportunity to optimise the rush a bit more. My fastest time was 3:42, but that was with very poor execution so I think some more time can be shaved off that.

First, when heinzb writes "9th probe" I assume he means 8th probe: when the 8th probe pops out, the 9th starts building and the food counter says "9", but that's the eighth probe.

However, you can go much faster if you're satisfied putting only one cannon on the low ground, which is sufficient. Set your probes to mining and build a 7th probe. Grab one of the first returning probes and immediately send it to the enemy base. Build an 8th probe. Now build a pylon in the position suggested by heinzb. Build the forge, then chrono out two more probes. Wait for the forge to finish, then put down a single cannon.

Here is a second departure from heinzb's strat: do not wait for the cannon to almost warp in, but *almost* immediately run into the base and start a second cannon there. There will be no marines because we're quite early. However, the scvs will attack the second cannon. This is desired: as they are attacking, the first cannon should warp in and start firing on them. They will then run out of the base to suicide on the first cannon, which will make the computer gg much faster. If you are too quick putting down the second cannon the scvs might kill it, so you might have to wait for one or two seconds before putting it down - but no more.

Obviously spend all remaining minerals on putting down a pylon and further cannons asap.
Good luck!

edit: if you have good micro (I don't), you can also try to put a pylon down in the computer's mineral line. All scvs will start following your probe, but if you can avoid them this will allow you to warp in two cannons who can hit the nexus even faster. My best time with this strat is about the same as if you build the second cannon first.