  Superhard materials with good fracture toughness have found wide industrial
applications, which necessitates the development of accurate hardness and
fracture toughness models for efficient materials design. Although several
macroscopic models have been proposed, they are mostly semiempirical based on
prior knowledge or assumptions, and obtained by fitting limited experimental
data. Here, through an unbiased and explanatory symbolic regression technique,
we built a macroscopic hardness model and fracture toughness model, which only
require shear and bulk moduli as inputs. The developed hardness model was
trained on an extended dataset, which not only includes cubic systems, but also
contains non-cubic systems with anisotropic elastic properties. The obtained
models turned out to be simple, accurate, and transferable. Moreover, we
assessed the performance of three popular deep learning models for predicting
bulk and shear moduli, and found that the crystal graph convolutional neural
network and crystal explainable property predictor perform almost equally well,
both better than the atomistic line graph neural network. By combining the
machine-learned bulk and shear moduli with the hardness and fracture toughness
prediction models, potential superhard materials with good fracture toughness
can be efficiently screened out through high-throughput calculations.
