PTX does double duty as both a virtual ISA and a intermediate compiler representation. I am not a compiler engineer, but have worked closely with compiler engineers for many years. My understanding is that use of SSA is extremely common in modern compilers as it makes various optimizations easier to apply. That is why you see a new virtual register used every time a new result is written (each virtual register is written to just once).
Since the number of registers available is architecture dependent, it is the job of the PTXAS optimizing compiler to allocate physical registers when compiling PTX into SASS (machine code). There are various conflicting goals in this process. For example, to increase latency tolerance the compiler will try to schedule loads early, but this often extends the live range of variables leading to an increase in register pressure. Likewise, common subexpression elimination reduces dynamic instruction count but can increase register pressure due to temporary variables created to hold the values of subexpressions. On the other hand, increased register use can lead to decreased occupancy and therefore lower performance.
Note that a small number of spill operations that occur infrequently may not be detrimental to performance, and the increased register use may help with other optimization that improve performance in the balance. PTXAS is reasonably smart about spills, e.g. putting them in outer rather than inner loops, trying to group and vectorize them. Not local memory stores seen in SASS disassembly are necessarily spills.
95% of the time the compiler manages to strike the right balance that delivers close to optimal performance. Note that the allocation of physical registers per thread in the hardware typically has a granularity > 1, and PTXAS may well take advantage of that to achieve other optimization goals. When you file a bug against the compiler regarding register pressure issues, it will likely be seen as non-actionable unless the resulting decrease in performance exceeds 5%. In other words, don’t sweat the small stuff.