We present an update of our previous work, necessitated by availability of a significantly improved dataset. The work is a model-independent analysis of the cosmological supernova (Type Ia) data, where function families are fit to the data in form of luminosity distance as function of redshift, that is, $d_{L}(z)$; and subsequently time-derivatives of the scale function $a(t)$ are $analytically$ derived, but as functions of $z$, without making assumptions about of gravity or the contents of the universe... This gives, e.g. the redshift value at which the universe goes over from deceleration to acceleration, as $z_{t}=0.54 \pm 0.04$ for a flat universe. In the update, we switch to a more modern fit criterion and also take into account the uncertainty in the calibration of the SNIa luminosities. If a theory of gravity $is$ assumed, our results allow determination of the density of the universe as function of $z$, from which conclusions about the contents of the universe can be drawn. We update the previous work's result where this was done for Einstein gravity, finding a lower-limit on the dark energy fraction, $\Omega_{DE}>0.46$; and here we do this also for Starobinsky gravity, where we can find a Starobinsky parameter that can eliminate the need for dark energy. (read more)