Page 48 of Smuel H.Preston book ( Demograpy measuring and modeling population processes) gave the Coale and Demeny table for nax . They gave different formula for computing nax given different condition of nmx eg

values of 1a0

if 1m0 >= 0.107 .330 for male and .350 for female

if 1m0 < 0.107 .045 + 2.684*1m0 for male and .053 +2.800*1m0 for female etc

I also noticed elsewhere ( not in the book mentioned above) that the same conditions were used when calculating nax when converting from nqx to nmx. In this case  presented the following for computing nax.

value of 1a0

if 1q0 >= 0.107  0.330 for male and 0.35 for female

if 1q0 < 0.107  a10 = 0.045 + 2.684*1q0 for male and a10 =0.053 + 2.800*1q0 for female

Does anyone know why the formula for nax is same when converting from nqx to nmx and when converting from nmx to nqx? 

Thanks