Oh, you are correct, I had gamma correction in one test and not the other so they came out different - maximum is the same as the simple formula, indeed.In Reinhard's original paper he referred to it as Lwhite, because its "the smallest luminance that will be mapped to pure white". He came up with that formulation so that you could allow the image to "burn out", such that a certain subset of high values mapped >= 1.0. Using your maximum luminance means that absolutely everything will end up in the [0, 1], but that's already what you get (1 /(1 + x)) which is his original formula. Ultimately some amount of burn out is usually desirable, so that you don't compress your mids and lows too much.

So I take it I need to take the maximum luminance and multiply it by some "burn out" coefficient (between 0 and 1), and then use that as Lwhite? Is there some way to estimate a good coefficient or is it really trial and error? I understand Reinhard is a global operator and as such won't always look nice on every image (an adaptive algorithm would be a more sophisticated solution).

Ah, I missed the "key" part in the paper - I need to use the summation formula to estimate the total luminance in the image on a logarithmic scale and work with that, if I understand correctly.

EDIT: ok, I got it working - Reinhard works superbly when the luminance range isn't excessive, after which it starts to break down (everything becomes dark) - I think a local operator is required for those pathological cases, as indicated in the paper.