Well, since you said all suggestions are welcome, have you considered trying to tunnel under the slab? You can dig as far as you can go, then use high pressure water through a capped and drilled pvc pipe to get you where you need to go. A digging bar would be helpful as well. You can rent a concrete hole saw to bore a hole for the pipe. I've used this method of excavation to run sprinkler lines under a 4' slab in the past.
Ding it, I got all excited after reading the OP's question. "Oh, I know this one! Me, me, call on me!!" And then Rancho steals my thunder!!!
Dig, or water jet under, don't cut the slab. If they laid the slab on sand, as some are done, you'll have an especially easy time of it. Dig if it's soft enough, use water if not. Alternately, Home Depot sells a 3" auger. I used a couple of those to get under a slab, because I didn't want to fill the trench I had dug with water (from the water drilling). I went 2' in from one side, and then 2' in from the other and met in the middle closely enough to shove a couple of 2" PVC pipes through. Water jetting for 2" would have used a significantly larger volume of water, and I would have had to stop often to bail out the trench. I only saw $60 augers just now at Home Depot. But I remember paying more like under $20. I had to buy two, because the first one broke. You get what you pay for. You can always repurpose a good one later, to justify the expense of a better one. Attach it to a cordless drill and do some weeding or planting in the garden. It's a nice tool to have if you go that route.
The trick in your case will be leaving yourself enough room to work in the elbow, either from the hole you drill through the slab, or the hole you drill through the dirt. It'll be tight, but I definitely wouldn't cut the concrete. You'll be cutting through a metal grid and/or rebar and weakening the slab...
Man, would my kids love a slide!