We present numerically stable formulas for the analytical solution in the closed form of the
so-called X-IVAS scheme in 3D. The X-IVAS scheme is a material point based explicit exponential
integrator. An intermediate step in the X-IVAS scheme is the solution of tangent curves for
piecewise linear vector fields defined on simplicial meshes. This is what we refer to as particle
tracing of streamlines and independent formulas for the same can be easily distilled from the ones
presented for the X-IVAS scheme. The formulas involve functions of matrices which are defined using
the corresponding Newton interpolating polynomial. The evaluation of these formulas is stable, i.e.
a certain number of significant digits in the computed values are guaranteed to be exact. Using the
double-precision floating-point arithmetic specified by the IEEE 754 standard, we obtain at least
10 significant decimal digits in the worst case scenarios. These scenarios involve fourth-order
divided differences of the exponential function. Additionally, an optimal series
approximation of divided
differences is presented which is an essential part of the exposition.