  I present general analytic expressions for distance calculations (comoving
distance, time coordinate, and absorption distance) in the standard
$\Lambda$CDM cosmology, allowing for the presence of radiation and for non-zero
curvature. The solutions utilise the symmetric Carlson basis of elliptic
integrals, which can be evaluated with fast numerical algorithms that allow
trivial parallelisation on GPUs and automatic differentiation without the need
for additional special functions. I introduce a PyTorch-based implementation in
the phytorch.cosmology package and briefly examine its accuracy and speed in
comparison with numerical integration and other known expressions (for special
cases). Finally, I demonstrate an application to high-dimensional Bayesian
analysis that utilises automatic differentiation through the distance
calculations to efficiently derive posteriors for cosmological parameters from
up to $10^6$ mock type Ia supernovae using variational inference.
