The issues that the GP is grappling with are largely due to the fact that they are trying to "construct" real numbers from a stream of bits. That is always going to lead to bias issues. On the other hand, with this particular algorithm (assuming a truly random source) the resulting number should be more or less completely uniform. It works because we are partitioning the search space itself in such a way that all numbers are as likely as any other. In fact, that the algorithm terminates rather predictably essentially proves just that. After one million invocations, for example, the average number of iterations was something like 57 (with the minimum being 55 and the maximum outlier 74). Which is to say you could pick any number whatsoever and expect to see it no more than once per ~2^57 invocations.
I was curious about this. On the one hand, comparing doubles with == is rarely a good idea but, on the other hand, your explanation seems valid.
After some testing I discovered a problem but not with the comparison. The problem is with calculating the halfway value. There are some doubles where their difference cannot be represented as a double:
#include <stdio.h>
#include <stdlib.h>
#include <time.h>
#include <float.h>
double random_real(double low, double high, int (*random_bit)(void)) {
if (high < low)
return random_real(high, low, random_bit);
double halfway, previous = low;
while (1) {
halfway = low + (high - low) / 2;
if (halfway == previous)
break;
if (random_bit() & 1)
low = halfway;
else
high = halfway;
previous = halfway;
}
return halfway;
}
int main(int argc, char *argv[]) {
srand(time(NULL));
for (int i = 0; i < 1000000; i++) {
double r = random_real(-DBL_MAX, DBL_MAX, rand);
printf("%f\n", r);
}
}
Actually, the problem is that the algorithm is having to calculate DBL_MAX+DBL_MAX, which of course is going to exceed the maximum value for a double-precision number (by definition). That isn't a very realistic use-case either, but in any case you could just clamp the inputs like so:
double random_real(double low, double high, int (*random_bit)(void)) {
if (high < low)
return random_real(high, low, random_bit);
const double max = DBL_MAX / 2;
if (high > max)
high = max;
const double min = -max;
if (low < min)
low = min;
double halfway, previous = low;
while (1) {
halfway = low + (high - low) / 2;
if (halfway == previous)
break;
if (random_bit() & 1)
low = halfway;
else
high = halfway;
previous = halfway;
}
return halfway;
}
