Hoffmann et al. (2022) established a precise empirical equation for the Chinchilla scaling law to compute the test loss ($\mathcal{L}$) based on the model size ($N$) and the dataset size ($D$). The formulation is expressed as:

$\mathcal{L}(N,D) = \underbrace{\frac{406.4}{N^{0.34}}}_{\text{model scaling}} + \underbrace{\frac{410.7}{D^{0.28}}}_{\text{dataset scaling}} + \underbrace{1.69}_{\text{irreducible error}}$

This relationship divides the overall loss into three distinct components: a model scaling term, a dataset scaling term, and a baseline irreducible error of ${}1.69$.