  We illustrate that, like the truncation error, the round-off error has a
significant influence on the reliability of numerical simulations of chaotic
dynamic systems. Due to the butterfly-effect, all numerical approaches in
double precision cannot give a reliable long-term simulation of chaotic dynamic
systems. So, in order to avoid man-made uncertainty of numerical simulations of
chaos, we had to greatly decrease both of the truncation and round-off error to
a small enough level, plus a verification of solution reliability by means of
an additional computation using even smaller truncation and round-off errors.
