Mathematical definition:

$$\operatorname{xlogy}(x, y) = \begin{cases} \mathrm{NaN}, & y = \mathrm{NaN}, \\ 0, & x = 0, \\ x \cdot \log y, & x \neq 0. \end{cases}$$

The convention covers only x == 0: with x != 0 the plain definition applies, so xlogy(2, 0) is -inf and a negative other gives NaN.

Gradients are $\partial/\partial x = \log y$ and $\partial/\partial y = x / y$, except where x == 0 and $\log y$ is not finite (y <= 0 or other infinite): there both are 0, so the backward never forms 0 \cdot \infty. At x == 0 with 0 < y < \infty the input gradient stays $\log y$. Two cells differ from the reference framework on purpose: at (0, 0) its other gradient is 0 / 0 = NaN and Lucid's is 0, and at (0, +\infty) its input gradient is $+\infty$ and Lucid's is 0 (a composite cannot give a value of 0 a gradient of $+\infty$ without forming 0 \cdot \infty in the forward pass).