Mathematical definition with the limit convention:

$$\text{xlog1py}(x, y) = \begin{cases} \mathrm{NaN}, & y = \mathrm{NaN} \\[2pt] 0, & x = 0 \\[2pt] x \log(1 + y), & \text{otherwise}. \end{cases}$$

The convention covers only x == 0: with x != 0 the plain definition applies, so xlog1py(2, -1) is -inf. Using $\log(1 + y)$ avoids precision loss for small y where 1 + y would round to 1.0.

Gradients are $\partial/\partial x = \log(1 + y)$ and $\partial/\partial y = x / (1 + y)$, except where x == 0 and $\log(1 + y)$ is not finite (y <= -1 or y infinite): there both are 0, so the backward never forms 0 \cdot \infty. This is the convention of lucid.xlogy, and the two functions share one implementation. Two cells differ from the reference framework on purpose: at (0, -1) its y gradient is 0 / 0 = NaN and Lucid's is 0, and at (0, +\infty) its x gradient is $+\infty$ and Lucid's is 0.