  We present EnckeHH, a new, highly accurate code for orbital dynamics of
perturbed Keplerian systems such as planetary systems or galactic centre
systems. It solves Encke's equations of motion, which assume perturbed
Keplerian orbits. By incorporating numerical techniques, we have made the code
follow optimal roundoff error growth. In a $10^ {12}$ day integration of the
outer Solar System, EnckeHH was $3.5$ orders of magnitude more accurate than
IAS15 in a fixed time step test. Adaptive steps are recommended for IAS15.
Through study of efficiency plots, we show that EnckeHH reaches significantly
higher accuracy than the Rebound integrators IAS15 and WHCKL for fixed step
size.
