Wim Couwenberg's LGMP Lua binding to GMP is excellent and widely used, but the repertoire of floating-point routines in GMP is very limited. Better arbitrary-precision floats in a GMP-compatible package are available in MPFR, but Harmut Henkel's lmpfrlib binding to it has a subtly incompatible (and sometimes awkward) API and can't interoperate with LGMP's types.
This is binding to MPFR that is API-compatible with LGMP and (if the latter is found at load time) accepts its userdata objects wherever the C library accepts GMP ones. Unfortunately, this means it must be incompatible with lmpfrlib.
No documentation is currently available, but that for LGMP can be used for the functions it provides, except that the versions here sometimes accept more combinations of arguments (either because MPFR itself more, or because arbitrary limitations of the binding have not been carried over). Most of the rest is transcendental functions with a straightforward interface.
Additionally, all functions that perform rounding accept the rounding mode as
an optional last argument (defaulting to the one set with
mpfr.set_default_rounding_mode) and return −1, 0, or 1 as the last
result when the returned value is less than, equal to, or greater than the
exact one. Accepted rounding modes are (case insensitive) U (towards
+∞), D (towards −∞), A or Y (away from 0), Z (towards
0), N (to nearest, ties to even), and F (faithful). Care should be taken
to trim the extra result when a call to such a function serves as the last
argument of a further function call.
Finally, unlike GMP floats, MPFR floats support NaNs. Lua's equality semantics make it impossible for those NaNs be unequal to themselves, as required by IEEE 754 and implemented in MPFR's comparisons, so while
print(mpfr.fr 'nan' == mpfr.fr 'nan')prints false,
local nan = mpfr.fr 'nan'; print(nan == nan)prints true. As an awkward workaround,
local nan = mpfr.fr 'nan'; print(nan <= nan and nan >= nan)does print false.