describes "RLibm" that correctly rounds 32-bit results for many common math/trig functions, and compares their library to glibc and Intel.
It's interesting that the 32-bit gnu library has significant numbers of rounding errors; and, when the same were tried with gnu 64-bit doubles, there were *still* errors when considered as 32-bit floats (a few, but still...).
How well do we do on this metric?
I do have one sort-of datapoint: I converted an orbital prediction program that used doubles to using long doubles. Ran long predictions for several satellites, and found a difference of e.g. 1 microsecond when a body reaches apogee for long double vs double; and only once. I conclude for this application, doubles are totally adequate. And, although floating point numerics can be complicated, it would seem we do 'pretty good' when using doubles.
-Mike