Is there a symbol for ‘equal if defined’

As far as functions go, note that we already have a symbol for this (coming from set theory): "$\subseteq$"! If $f$ and $g$ are partial functions, identifying them with their graphs means that we can write "$f\subseteq g$" to mean "$f(x)=g(x)$ whenever $f(x)$ is defined."

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More generally (or in contexts where imposing set theory might be undesirable), here's a possible recommendation:

First, the symmetric version. I don't know how common this is elsewhere, but in computability theory, we often write $$x\simeq y$$ if $x$ and $y$ are expressions which are either both undefined, or both defined and equal. (Occasionally "$\cong$" is used instead of "$\simeq$," but I prefer "$\simeq$" since (at least within computability theory) it's less often used for isomorphism.)

For the asymmetric version, I would recommend "$\gtrsim$" in analogy with "$\simeq$." But I haven't seen that before, so I don't actually know if it's commonly used; it's just what seems to me the natural choice.