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Could the JIT use such optimizations automatically, or maybe at least
remove the intermediate __add and bit.* calls operating on the integers
directly?

It already there, quoting Mike's earlier response: "The ARM port uses the dual-number mode of LuaJIT. All Lua numbers
are represented either as integers or doubles internally. This is
transparent to the user of the VM. Except for the performance
effects, of course: e.g. 'for i=1,1e7 do end' runs much faster
than 'for i=1.5,1e7 do end'."

Cheers,
-- 
Adam