I did the work that Lemire links there. It started with realizing that Lemire's rejection method becomes more likely to need a division/modulo as the range max

I did the work that Lemire links there. It started with realizing that Lemire's rejection method becomes more likely to need a division/modulo as the range maxes out (the gate on division is fraction < range, so as range -> maxint the division case approaches 100%, if the fraction and range have the same precision). If the range is 32 bit but the fraction is 64, the division case is 2^32 times less likely, so I worked out Lemire's method for 32.64 fixed point. That turned out to be similar to the infinite-precision case described here, as we only need 32 bits of fraction if it's far enough from the edges, ie nothing can carry into the integer part if we extend to 64. Once I worked out when to extend to 64 bits in the rejection method, I realized that the infinite-precision case is similar, using a loop instead of a conditional. It's simpler since there's no rejection, just a decision to continue as long as it's possible to carry.