Avoid division by zero in C when taking log with respect to a random number

I am currently using C to generate a Gaussian noise. In one step, I need to take the log of a uniformly distributed number u1 = (double) rand() / RAND_MAX. Because u1 may be zero, there is a risk in doing log(u1). So, I need to check. Should I use

do {
    u1 = ((double) rand() / RAND_MAX);
} while (u1 == 0.);

Or, should I use

do {
    u1 = ((double) rand() / RAND_MAX);
} while (u1 < epsilon);

where epsilon is a small number? If the latter is preferred, how should I choose the value of epsilon? (In Fortran there is TINY, but I do not know what to do in C).

Thanks!

Attached is the complete code:

#include <stdio.h>
#define _USE_MATH_DEFINES
#include <math.h>
#include <stdlib.h>

double gaussian_noise(double mean, double std)
{
    static int have_spare = 0;
    static double u1, u2, z1, z2;
    if(have_spare)
    {
        have_spare = 0;
        z2 = sqrt(-2. * log(u1)) * sin(2. * M_PI * u2);
        return mean + std * z2;
    }
    have_spare = 1;
    do {
    u1 = ((double) rand() / RAND_MAX);
    } while (u1 == 0.);
    u2 = ((double) rand() / RAND_MAX);
    z1 = sqrt(-2. * log(u1)) * cos(2. * M_PI * u2);
    return mean + std * z1;
}

void main()
{
    const double mean = 0., std = 1.;
    double noise;
    int i;
    for(i=0; i<100000; i++)
    {
        noise = gaussian_noise(mean, std);
        printf("%lf\t", noise);
    }
}