How to combine float representation with discontinous function?

Question

I have read tons of things about floating error, and floating approximation, and all that.
The thing is : I never read an answer to a real world problem. And today, I came across a real world problem. And this is really bad, and I really don't know how to escape.

Take a look at this example :

    [TestMethod]
    public void TestMethod1()
    {
        float t1 = 8460.32F;
        float t2 = 5990;
        var x = t1 - t2;
        var y = F(x);

        Assert.AreEqual(x, y);
    }

    float F(float x)
    {
        if (x <= 2470.32F) { return x; }
        else { return -x; }
    }

x is supposed to be 2470.32. But in fact, due to rounding error, its value is 2470.32031.
Most of the time, this is not a problem. Functions are continuous, and all is good, the result is off by a little value.
But here, we have a discontinous function, and the error is really, really big. The test failed exactly on the discontinuous point.

How could I handle the rounding error with discontinuous functions?

Answer 1

The key problem here is:

• The function has a large (and significant) change in output value in certain cases when there is a small change in input value.
• You are passing an incorrect input value to the function.

As you write, “due to rounding error, [x’s value] is 2470.32031”. Suppose you could write any code you desire—simply describe the function to be performed, and a team of expert programmers will provide complete, bug-free source code within seconds. What would you tell them?

The problem you are posing is, “I am going to pass a wrong value, 2470.32031, to this function. I want it to know that the correct value is something else and to provide the result for the correct value, which I did not pass, instead of the incorrect value, which I did pass.”

In general, that problem is impossible to solve, because it is impossible to distinguish when 2470.32031 is passed to the function but 2470.32 is intended from when 2470.32031 is passed to the function and 2470.32031 is intended. You cannot expect a computer to read your mind. When you pass incorrect input, you cannot expect correct output.

What this tells us is that no solution inside of the function F is possible. Therefore, we must zoom out and look at the larger problem. You must examine whether the value passed to F can be improved (calculated in a better way or with higher precision or with supplementary information) or whether the nature of the problem is such that, when 2470.32031 is passed, 2470.32 is always intended, so that this knowledge can be incorporated into F.