Bruce
What an interesting question! As many of you know, I'm having fun porting the G3RUH's Plan13 algorithm to the Atmel platform. One of the first things you want to be able to do in programs like this is to convert degrees to radians; and that means pi.
FWIW I used 32 bit accuracy (equivalent to IEEE754 single precision) in my PIC LVB Tracker 2 code (based on G3RUH Plan13 too) without any significant difference in results over 64 bit. Although the accuracy did cause some anomalies, they were small and inconsequential for amateur purposes, typically a second or two in predictions. I tested this by running the same code on a PC and carefully coding constants, stored values and datatypes with #defines so I could recompile on demand and see the differences in the results.
I was lucky, ISTR I had 3.5KB, but even that was a struggle, although I did have the additional dimension of having to deal with manual bank switching too of the PIC18 - you might not have this on your device. And the compiler I used, although generating very compact code, wouldn't compile anything slightly complex, so I had to manually split the statements into simpler steps.
If I did it again, I'd head for at least a 16 bit and probably a 32 bit device, and a different compiler. But five plus years ago we didn't have such devices or capabilities!
73, Howard G6LVB