Wouldn't it make sense to move your first occurrence of:

$index = int(rand($symbol_size));
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... inside the while loop (right at the top of the loop), and then remove the occurrence at the end of the loop?

That is, not:

foo();
while (1) {
  bar();
  foo();
}
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But instead:

while (1) {
  foo();
  bar();
}
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It should have the same effect; be more compact and avoid repetition.

Anyway, I believe the reason you're leaking memory is that your evaluated strings can end up containing variables, and those variables are not lexicals, so they get stored in the package stash.