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Here is the Nth patch for a globals/builtins cache. As other caches at the same kind, it shows very small to no gain on non-micro benchmarks, showing that contrary to popular belief, globals/builtins lookup are not a major roadblock in today's Python performance.
However, this patch could be useful in combination with other optimizations such as issue10399. Indeed, using the globals/builtins version id, it is easy and very cheap to detect whether the function pointed to by a global name has changed or not.
As for micro-benchmarks, they show that there is indeed a good improvement on builtins lookups:
$ ./python -m timeit "x=len;x=len;x=len;x=len;x=len;x=len;x=len;x=len;x=len;x=len;"
-> without patch:
1000000 loops, best of 3: 0.282 usec per loop
-> with patch:
10000000 loops, best of 3: 0.183 usec per loop

Might such a cache lay the groundwork for more aggressive optimizations
by JITs like Unladen Swallow?

Unladen actually has something like this in place for performance optimizations. Not sure how Antoine's approach differs, though.

There aren't many possible approaches. The more complex variants of globals caches try to also speedup writes, which is IMO a waste of time since rebinding globals is not a good coding practice, and especially not in the middle of time-critical loops.
(by the way, the patch only addresses normal functions, but generators would easily benefit from a similar treatment)
And Skip is right that this would be most useful when paired with a JIT (allowing for aggressive specialization, and possibly inlining).

dict has ma_version for now.
@haypo, how do you think about this patch?
Would you reimplement global cache? Or may I update this patch?

This is implemented in #26219.