the integrator itself because we are not solving the equations of motion for the Hamiltonian $\mathcal{H}$ exactly. For symplectic integrators such as those discussed here, this error term is bound and we call it $E_{\rm bound}$. If the mass ratio of the planets to the star is $\epsilon$, then the order of this error term is roughly $O(\epsilon\, \mathit{dt}^{2})$ for integrators without symplectic corectors and $O(\epsilon \,\mathit{dt}^{K} ) + O(\epsilon^2 \,\mathit{dt}^2)$ for those with symplectic correctors (see Sect. \[sec:correctors\]). Note that $E_{\rm bound}$ is independent of time $t$.

Second, there is an error term associated with the finite precision of numbers represented on a computer. We can only represent a small subset of all real numbers exactly in floating-point precision. Thus after every operation such as an addition or multiplication, the computer rounds to a nearby floating-point number. For CPUs and compilers that follow the IEEE754 standard [@IEEE754], we are guaranteed to round to the nearest floating-point number. Thus, if all operations follow the IEEE754 standard, then as long as the algorithm itself is unbiased, we expect the error to grow as the square root of the number of operations, i.e. $E_{\rm rand} \sim \sqrt{N}\sim \sqrt{t}$, where $N$ is the 