In the language of two binary operations, is there a single identity that is equivalent to both distributive laws?

Question

Let our language be $\{+,* \}$. $*$ is said to be left-distributive over $+$ iff $x * (y + z)=(x*y)+(x*z)$, and is said to be right-distributive over $+$ iff $(x+y)*z=(x*z)+(y*z)$. That raises the question, in the language of $\{+,*\}$, is there a single identity which is equivalent to the conjunction of the left and right distributive laws?