Can you calculate Pi using a Solver?

[EdS2]

So, indeed, I got my trusty Limited Edition HP15C out, set flag 8 (surprised to see it not documented on the back plate) and indeed it does a very solid job of being a complex-number calculator, and is willing to have a go at solving Euler's identity for a value of pi.

It doesn't do terribly well, but it is willing.

Using the Real() method:
Solver input 1,4: result is 3.141591597
Solver input 2,3: result is 3.141583326

Using the ABS() method:
Solver input 1,4: result is Error 8, aka 3.139172931
Solver input 2,3: result is Error 8, aka 3.141225122

(Edit: I see now that using Re<>Im or the I function puts the calculator into complex mode without any flag manipulation. I wonder if the unaware user might then be stuck in complex mode forever? I must read more manual.)