Newcomb (1898) gives the following formula for calculating the tropical year:
$$365.24219879 - 0.00000614T$$
And the following for calculating the sidereal year:
$$365.25636042 + 0.00000011T$$
Where T is the number of Julian centuries (36525 ephemeris days) since J1900.0.
Since then, more accurate orbital elements for the planets have been calculated, like Simon et al. (1994). Several better formulae for the tropical year have been published since, such as in Borkowski (1991), which gives:
$$365.242189669781 - 0.000006161870T - 0.000000000644T^2$$
With T as Julian centuries since J2000.0. An even more accurate formula can be found in McCarthy & Seidelmann (2018, p. 267), which claims to have in turn calculated it from Laskar (1986):
$$365.2421896698 - 0.00000615359T - 0.000000000729T^2 + 0.000000000264T^3$$
With T as Julian centuries since J2000.0. A slightly different version can be found in Meeus & Savoie (1992):
$$365.242189623 - 0.000061522T - 0.0000000609T^2 + 0.00000026525T^3$$
Where this time, T is Julian millennia (365250 ephemeris days) since J2000.0.
However: I cannot find a single similar modern formula for the sidereal year. It seems almost incomprehensible to me that nobody would have calculated one in 100+ years.
Does such a formula exist in literature? If not, how could it be calculated?
Borkowski, K. M. (1991). The tropical year and solar calendar. Journal of the Royal Astronomical Society of Canada, 85, 121.
Laskar, J. (1986). Secular terms of classical planetary theories using the results of general theory. Astronomy and astrophysics, 157, 59-70.
McCarthy, D. D., & Seidelmann, P. K. (2018). Time: from Earth rotation to atomic physics. Cambridge University Press.
Meeus, J., & Savoie, D. (1992). The history of the tropical year. Journal of the British Astronomical Association, 102(1), 40-42.
Newcomb, S. (1898). Astronomical Papers prepared for the use of the American Ephemeris and Nautical Almanac (Vol. 6). Bureau of Navigation, Navy Department.
Simon, J. L., Bretagnon, P., Chapront, J., Chapront-Touze, M., Francou, G., & Laskar, J. (1994). Numerical expressions for precession formulae and mean elements for the Moon and the planets. Astronomy and Astrophysics, 282, 663-683.