  Chernoff bounds are a powerful application of the Markov inequality to
produce strong bounds on the tails of probability distributions. They are often
used to bound the tail probabilities of sums of Poisson trials, or in
regression to produce conservative confidence intervals for the parameters of
such trials. The bounds provide expressions for the tail probabilities that can
be inverted for a given probability/confidence to provide tail intervals. The
inversions involve the solution of transcendental equations and it is often
convenient to substitute approximations that can be exactly solved e.g. by the
quadratic equation. In this paper we introduce approximations for the Chernoff
bounds whose inversion can be exactly solved with a quadratic equation, but
which are closer approximations than those adopted previously.
