it is interesting how this is a dichotomy made by two complementary fields in the battle for authorship. I agree with the previous comments that it should be decided prior writing it up but it also counts how much leverage the results have on the paper's message.
author's contribution is also key. now that lots of journals added this section in their papers we can know exactly what a person did. this helps laying out the fairness of the authorship order, which can then be interpreted subjectively, talking to lots of PIs at conferences/seminars etc I had the impression that, beside looking at first authorship, most also focus on the contribution (when they have the time to get to the paper are read it) and then draw their own conclusions. so, where it can be unfair that a person is put as second author, hopefully he/she'll be fairly recognized for his/her contribution in the apposite section.
a simple solution that i.e. I went through with a colleague in the lab is to look at the number of figures generated, the person who made the most becomes the first author, then second and so on. he generated one result figure more than I did, therefore he was first and I was second...this is pretty straightforward and pragmatic.
otherwise...it seems that tossing a coin is a valuable solution ;-P (cropped from web-page snapshot):

One of my friend's condition is reverse. He would not be given even authorship stating that he is just doing his job as a service. Seems like unfairness is existing in both ways.
Based on my personal experience as a computational biologist, I was not a co-first author even if I prioritized the candidates from the public dataset and other molecular biologists did the validation part, and when it comes to credits contribution, the supervisor said the computation is not important at all. Seems that many of the reversed situations happen to me.