  The Floater--Hormann family of rational interpolants do not have spurious
poles or unattainable points, are efficient to calculate, and have arbitrarily
high approximation orders. One concern when using them is that the
amplification of rounding errors increases with approximation order, and can
make balancing the interpolation error and rounding error difficult. This
article proposes to modify the Floater--Hormann interpolants by including
additional local polynomial interpolants at the ends of the interval. This
appears to improve the conditioning of the interpolants and allow higher
approximation orders to be used in practice.
