The most frequently seen binary companions to millisecond pulsars (MSPs) are helium white dwarfs (He WDs). The standard rejuvenation mechanism, in which a low- to intermediate-mass companion to a neutron star fills its Roche lobe between central hydrogen exhaustion and core helium ignition, is the most plausible formation mechanism. We have investigated whether the observed population can realistically be formed via this mechanism. We used the Cambridge STARS code to make models of Case B RLOF with Reimers' mass loss from the donor. We find that the range of initial orbital periods required to produce the currently observed range of orbital periods of MSPs is extremely narrow. To reduce this fine tuning, we introduce a companion enhanced wind (CEW) that strips the donor of its envelope more quickly so that systems can detach at shorter periods. Our models indicate that the fine tuning can be significantly reduced if a CEW is active. Because significant mass is lost owing to a CEW we expect some binary pulsars to accrete less than the 0.1Msun needed to spin them up to millisecond periods. This can account for mildly recycled pulsars present along the entire Mc-Porb relation. Systems with Pspin

30ms are consistent with this but too few of these mildly recycled pulsars have yet been observed to make a significant comparison.