  Binary millisecond pulsars (BMSPs) are thought to have evolved from low-mass
X-ray binaries (LMXBs). If the mass transfer in LMXBs is driven by nuclear
evolution of the donor star, the final orbital period is predicted to be well
correlated with the mass of the white dwarf (WD), which is the degenerate He
core of the donor. Here we show that this relation can be extended to very
small WD mass ($\sim 0.14-0.17\,M_\odot$) and narrow orbital period (about a
few hours), mainly depending on the metallicities of the donor stars. There is
also discontinuity in the relation, which is due to the temporary contraction
of the donor when the H-burning shell crosses the hydrogen discontinuity. BMSPs
with low-mass He WD companions in very compact binaries can be accounted for if
the progenitor binary experienced very late Case A mass transfer. The WD
companion of PSR J1738+0333 is likely to evolve from a Pop~II star. For PSR
J0348+0432, to explain its extreme compact orbit in the Roche lobe-decoupling
phase, even lower metallicity ($Z=0.0001$) is required.

High Energy Astrophysical Phenomena;Solar and Stellar Astrophysics