rand() between 0 and 1

So the following code makes 0 < r < 1

r = ((double) rand() / (RAND_MAX))

Why does having r = ((double) rand() / (RAND_MAX + 1)) make -1 < r < 0?

Shouldn't adding one to RAND_MAX make 1 < r < 2?

Edit: I was getting a warning: integer overflow in expression

on that line, so that might be the problem. I just did cout << r << endl and it definitely gives me values between -1 and 0

This is entirely implementation specific, but it appears that in the C++ environment you're working in, RAND_MAX is equal to INT_MAX.

Because of this, RAND_MAX + 1 exhibits undefined (overflow) behavior, and becomes INT_MIN. While your initial statement was dividing (random # between 0 and INT_MAX)/(INT_MAX) and generating a value 0 <= r < 1, now it's dividing (random # between 0 and INT_MAX)/(INT_MIN), generating a value -1 < r <= 0

In order to generate a random number 1 <= r < 2, you would want

r = ((double) rand() / (RAND_MAX)) + 1

rand() / double(RAND_MAX) generates a floating-point random number between 0 (inclusive) and 1 (inclusive), but it's not a good way for the following reasons (because RAND_MAX is usually 32767):

1. The number of different random numbers that can be generated is too small: 32768. If you need more different random numbers, you need a different way (a code example is given below)
2. The generated numbers are too coarse-grained: you can get 1/32768, 2/32768, 3/32768, but never anything in between.
3. Limited states of random number generator engine: after generating RAND_MAX random numbers, implementations usually start to repeat the same sequence of random numbers.

Due to the above limitations of rand(), a better choice for generation of random numbers between 0 (inclusive) and 1 (exclusive) would be the following snippet (similar to the example at http://en.cppreference.com/w/cpp/numeric/random/uniform_real_distribution ):

#include <iostream>
#include <random>
#include <chrono>

int main()
{
    std::mt19937_64 rng;
    // initialize the random number generator with time-dependent seed
    uint64_t timeSeed = std::chrono::high_resolution_clock::now().time_since_epoch().count();
    std::seed_seq ss{uint32_t(timeSeed & 0xffffffff), uint32_t(timeSeed>>32)};
    rng.seed(ss);
    // initialize a uniform distribution between 0 and 1
    std::uniform_real_distribution<double> unif(0, 1);
    // ready to generate random numbers
    const int nSimulations = 10;
    for (int i = 0; i < nSimulations; i++)
    {
        double currentRandomNumber = unif(rng);
        std::cout << currentRandomNumber << std::endl;
    }
    return 0;
}

This is easy to modify to generate random numbers between 1 (inclusive) and 2 (exclusive) by replacing unif(0, 1) with unif(1, 2).

No, because RAND_MAX is typically expanded to MAX_INT. So adding one (apparently) puts it at MIN_INT (although it should be undefined behavior as I'm told), hence the reversal of sign.

To get what you want you will need to move the +1 outside the computation:

r = ((double) rand() / (RAND_MAX)) + 1;