A missing (?) function: exponentfp(x)

The function exponent(x) gives a very coarse value of log2(x).  The
function log(x)/log(2) gives a very fine value of log2(x) (but is
slow, especially if x has high precision).  But sometimes I prefer to
know log2(x) with an intermediate precision; in fact, any sufficiently
smooth monotonic function which coincides with log2 on powers of 2
would be enough!

In particular, exponent(x) + mantissa12(x) - 1 would be OK.  (Here
mantissa12() is the (top word of) mantissa-scaled-to-[1,2].)  It
should be very cheap to calculate.  It may be beneficial to have two
flavors, returning a double, or a 1-word PARI float.

Is there something like this?

Thanks,
Ilya