Complicating this further, doing this in a loop requires that the stack state match between the start and end of the loop. This can be challenging to do with minimal FXCH instructions. I've seen compilers emit 3+ FXCH instructions in a row at the end of a loop to match the stack state, where with some hairy rearrangement it was possible to get it down to 2 or 1.
Finally, the performance characteristics of different x87 implementations varied in annoying ways. The Intel Pentium, for instance, required very heavy use of FXCH to keep the add and multiply pipelines busy. Other x87 FPUs at the time, however, were non-pipelined, some taking 4 cycles for an FADD and another 4 cycles for FXCH. This meant that rearranging x87 code for Pentium could _halve_ the speed on other CPUs.
Quake wouldnt happen until Pentium 2 if Intel didnt pipeline FPU.
K6 did have the advantage of being OOO, which reduced the importance of instruction scheduling a lot, and having good integer performance. It also had some advantage with 3DNow! starting with K6-2, for the limited software that could use it.
Once you have a parse tree, visiting it in post order (left tree, right tree, operation) produces the RPN.