For example, robust linear regression from chapter 17, that fits 300 points over 4 parameters (easy, but far from trivial) runs in 180 seconds in JAGS and 485 in Stan, in parallel with 4 chains, taking 20,000 samples.
Bayadera takes 276,297,912 samples in 300 milliseconds, giving much fine-grained estimations.
So, depending on how you count the difference, it would be 500-1000 times faster for this particular analysis, while per-sample ratio is something like 7,000,000 (compared to JAGS).
Of course, JAGS and Stan are mature software packages, while Bayadera is still pre-release...
Is the speedup coming from a better implementation, or because GPUs are just way faster, or because it cuts statistical corners? If its cutting corners, are they sensible?
They may do a lot of work to make sure that MCMC is validly converging, and Bayadera also does its stuff on that front, but the truth is, and you'll find it in any book on MCMC (Gelman included) that you can never guarantee MCMC convergence.