mpn_sqrtrem1

[paul zimmermann]

       Hi,

in my current effort of optimizing MPFR for small precisions (1-2 limbs),
I am currently reviewing mpn_sqrtrem1() and mpn_sqrtrem2(), since we need
such functions to speed up mpfr_sqrt.

In mpn_sqrtrem1, I wonder what the 0x30000 constant is used for. By exhaustive
search on all possible values of a1 (there are only 3/4*2^33) on a 64-bit computer,
the maximal error on the 16-bit approximation x0 is the same without this constant,
more precisely -0.000023 <= x0/2^32 - (a0/2^64)^(-1/2) <= 0.

The worst error is even smaller (in absolute value) without the 0x30000 constant,
since with - 0x30000 it is attained for a1=2248147012 (error -0.0000231867), and
without 0x30000 it is attained for a1=2248147013 (error -0.0000231863).

Also "make check" passes without the 0x30000 constant. If the 0x30000 constant is
really needed, it would be nice to add an example that breaks make check if we
remove it.

Paul