  Let $P$ be a convex polygon in the plane, and let $T$ be a triangulation of
$P$. An edge $e$ in $T$ is called a diagonal if it is shared by two triangles
in $T$. A flip of a diagonal $e$ is the operation of removing $e$ and adding
the opposite diagonal of the resulting quadrilateral to obtain a new
triangulation of $P$ from $T$. The flip distance between two triangulations of
$P$ is the minimum number of flips needed to transform one triangulation into
the other. The Convex Flip Distance problem asks if the flip distance between
two given triangulations of $P$ is at most $k$, for some given parameter $k$.
  We present an FPT algorithm for the Convex Flip Distance problem that runs in
time $O(3.82^k)$ and uses polynomial space, where $k$ is the number of flips.
This algorithm significantly improves the previous best FPT algorithms for the
problem.
