  Conformal prediction constructs a confidence region for an unobserved
response of a feature vector based on previous identically distributed and
exchangeable observations of responses and features. It has a coverage
guarantee at any nominal level without additional assumptions on their
distribution. However, it requires a refitting procedure for all replacement
candidates of the target response. In regression settings, this corresponds to
an infinite number of model fit. Apart from relatively simple estimators that
can be written as pieces of linear function of the response, efficiently
computing such sets is difficult and is still considered as an open problem. We
exploit the fact that, \emph{often}, conformal prediction sets are intervals
whose boundaries can be efficiently approximated by classical root-finding
software. We investigate how this approach can overcome many limitations of
formerly used strategies and achieves calculations that have been unattainable
so far. We discuss its complexity as well as its drawbacks and evaluate its
efficiency through numerical experiments.
