C# calculations differ between const and variable locals?

I was setting up a pop quiz for my colleagues about the Banker’s Rounding approach that C# uses in the Math.Round function. But while preparing the question in repl.it I got a result that I thought was pretty weird. At first I was working with an array, but I’ve managed to boil it down to this snippet to find a small reproduction scenario:

class MainClass {
  public static void Main (string[] args) {
    double x = 10.5;
    System.Console.WriteLine(format: "Math.Round({0}) = {1}",
                             arg0: x,
                             arg1: System.Math.Round(x));


    const double y = 10.5;
    System.Console.WriteLine(format: "Math.Round({0}) = {1}",
                             arg0: y,
                             arg1: System.Math.Round(y));
  }
}

This results in the following output:

Math.Round(10.5) = 10
Math.Round(10.5) = 11

(I’ve also tried it with decimal numbers, which didn’t result in any difference between the calculation methods: both resulted in 10 which is correct according to the Banker’s Rounding rule.)

Now, I’m guessing this might have something to do with the const double version getting precompiled, but I’d expect – and I’m not sure this is reasonable – the precompiled version to use the same rules for rounding and/or (I’m not sure what the exact cause is) suffer from the exact same roundoff error – effectively, I’d expect it to perform the exact same calculation, just at a different time.

It is somewhat hard to find more information on this behavior, partially because I’m not sure whether I’ve ran into a bug in Math.Round (which is, apparently, has some issues as well) or something that is related to the precompilation that consts allow, but I’m guessing it’s the latter – and searching for things like "c# const different result" gave me nothing immediately useful.

So my question is, can anyone explain this output in terms of "this only happens if the compiler runs on platform X and the program then runs on platform Y because Z"?

(Edit: sorry, forgot to post the repl.it link. Here it is!)