NavList:

Jaap, there seems to be some confusion here. You wrote:
"why is using the haversine formula proposed earlier the worst possible way to go when one would decide to clear the lunar distance by longhand calculation?"

Did I misunderstand you? You made it very clear in your original message that this would involve a long division and therefore it was highly impractical for longhand calculation. I added my comment only because I felt your follow-up, spelling out the calculation in detail, obscured that earlier, very important point, and, knowing some of our NavList readers as I do very well, that point was bound to get lost. You were right that this is a highly impractical way to clear lunars. These intriguing diagram solutions which you have been exploring have some applications in celestial navigation, but only when the necessary precision in the computation is rather low. There are, of course, methods for clearing lunars that work with lower precision calculating techniques: the series methods, which I have recommended multiple times. The "corner cosines" (cosines of the position angles ZMS and ZSM) are required only to a few digits precision in the series methods. 

Frank Reed