http://www.bing.com/videos/search?q=3770+x+compared+to+ps4+jaguar&view=detail&mid=9AE449CF9936C0076FDE9AE449CF9936C0076FDE&first=0&FORM=NVPFVR&qpvt=3770+x+compared+to+ps4+jaguar
Here is a nice video comparison, and a text play by play of what the PS4 adds on benchmarking Vs pc and what it lacks.
And here is a link re guarding your Chipset and GDDR vram right from forbes
http://www.forbes.com/sites/erikkain/2013/02/26/ps4-vs-pc-where-the-wild-things-are/
Ill put my 3770X and 7970 OC up against a PS4 or Xbox one any day. last time I checked my graphics card had 3 gigs of DDR 5 memory, may not be on my mother board but im sure ill be fine with the 16 gigs of 1600 MHz DDR3. The comparison I gave is what anolyst compare it to. And that's not a budget pc its considered a mid range. I have two high end PC's ones an older chipset 990X 12 core with a 590 GTX and my current system is lower end of high end . I prefer my later system as I built it myself , I wasted money on a prebuilt the first one.
http://www.engadget.com/2013/02/25/playstation-4-processor/ there is another one for you, don't get all hot and bothered over that 500 dollar price tag on the PS4 . it is a low end Gaming Rig comparatively to a PC, its cheaper for a reason. Not saying they are bad , ill likely buy one .but there is some facts for your specs. that 7850 is about 170 bucks now and is a 1 gig card. and that 8 core processor of the jaguar will not be running anywhere near the 3 ghz range like a mid range CPU due to heat issues in a confined area, the rumor is they will be running between 1.2 and 1.5 ghz. As you said they don't process the same way. so I doubt they will need all that speed as they are really just running much simpler OS and focusing on the game.