uniform_real_distribution

I am currently working on importance sampling, and for testing purposes I need to be able to generate all possible values that uniform_real_distribution<float> may generate. My idea was to generate integer numbers which I can then convert to floating point numbers. From the tests I made it seems that there is a perfect bijection between uniform single-precision floats in [0,1] and integers in [0,2^24] (I am a bit bothered by the fact that it is not [0,2^24-1] and I am still trying to figure out why, my best guess is that 0 is simply special for floats and 1 to 2^24 all result in floats that have the same exponent). My question is whether the floats generated this way are exactly the floats that can be generated from uniform_real_distribution<float>. You can find my integer <-> float tests below:

void floatIntegerBitsBijectionTest()
{
    uint32 two24 = 1 << 24;
    bool bij24Bits = true;
    float delta = float(1.0) / float(two24);
    float prev = float(0) / float(two24);
    for (uint32 i = 1; i <= two24; ++i)
    {
        float uintMap = float(i) / float(two24);
        if (uintMap - prev != delta || uint32(uintMap*float(two24)) != i)
        {
            std::cout << "No bijection exists between uniform floats in [0,1] and integers in [0,2^24].\n";
            bij24Bits = false;
            break;
        }
        prev = uintMap;
    }
    if(bij24Bits) std::cout << "A bijection exists between uniform floats in [0,1] and integers in [0,2^24].\n";
    std::cout << "\n";

    uint32 two25 = 1 << 25;
    bool bij25Bits = true;
    delta = float(1.0) / float(two25);
    prev = float(0) / float(two25);
    for (uint32 i = 1; i <= two25; ++i)
    {
        float uintMap = float(i) / float(two25);
        if (uintMap - prev != delta || uint32(uintMap*float(two25)) != i)
        {
            std::cout << "No bijection exists between uniform floats in [0,1] and integers in [0,2^25].\n";
            if (i == ((1 << 24) + 1)) std::cout << "The first non-uniformly distributed float corresponds to the integer 2^24+1.\n";

            bij25Bits = false;
            break;
        }
        prev = uintMap;
    }
    if (bij25Bits) std::cout << "A bijection exists between uniform floats in [0,1] and integers in [0,2^25].\n";
    std::cout << "\n";


    bool bij25BitsS = true;
    delta = 1.0f / float(two24);
    prev = float(-two24) / float(two24);
    for (int i = -two24+1; i <= two24; ++i)
    {
        float uintMap = float(i) / float(two24);
        if (uintMap - prev != delta || int(uintMap*float(two24)) != i)
        {
            std::cout << i << " " << uintMap - prev << " " << delta << "\n";
            std::cout << "No bijection exists between uniform floats in [-1,1] and integers in [-2^24,2^24].\n";
            bij25BitsS = false;
            break;
        }
        prev = uintMap;
    }
    if (bij25BitsS) std::cout << "A bijection exists between uniform floats in [-1,1] and integers in [-2^24,2^24].\n";
}