How can I simulate uncertainty propagation

Question

I am learning about uncertainty propagation, but it is all very obscure and I am not sure I understand it very well.

I have a couple of exercises, but I want to run some simulations just to make sure I understand what is going on (I want to make sure uncertainty propagation means what I think it does).

The thing I want to simulate is:

There is a measurement of $A$ that quotes: $$ A = a_0 \pm \sigma_A $$ There is a measurement of $B$ that quotes: $$ B = b_0 \pm \sigma_B$$ There are these constraints: $$ AB^3 = c_0 \pm \sigma_C$$ $$ A+D=1$$ Finally I want to estimate: $$E = f(A,B,D)$$ and its uncertainty.

My idea (what I expect the uncertainty propagation to represent) is to run the following simulation a million times and get a distribution for $E$:

1. Draw a random number for $A$ (from Gaussian($\mu=a_0$, $\sigma=\sigma_A$)).
2. Draw a random number for $B$ (from Gaussian($\mu=b_0$, $\sigma=\sigma_B$)).
3. Set $D = 1-A$.
4. Get $E = f(A,B,D)$.
5. Repeat.

I probably need to do something fancier for steps 1 and 2 (in order to satisfy the first constraint), but I don't know how.

Does a simulation like this make sense in order to estimate $E$ and its uncertainty? How can I go about satisfying the first constraint?

Cross-posting here. If this is not allowed/off-topic please close the question.