  We construct a fast exact algorithm for the simulation of the first-passage
time, jointly with the undershoot and overshoot, of a tempered stable
subordinator over an arbitrary non-increasing absolutely continuous function.
We prove that the running time of our algorithm has finite exponential moments
and provide bounds on its expected running time with explicit dependence on the
characteristics of the process and the initial value of the function. The
expected running time grows at most cubically in the stability parameter (as it
approaches either $0$ or $1$) and is linear in the tempering parameter and the
initial value of the function. Numerical performance, based on the
implementation in the dedicated GitHub repository, exhibits a good agreement
with our theoretical bounds. We provide numerical examples to illustrate the
performance of our algorithm in Monte Carlo estimation.
