Question

The discussion to this answer just got me thinking about equality and equivalence of floating point numbers. I am aware that floating point numbers can not always be represented accurately. The question is, are there mathematically equivalent expressions that will yield different results when using floating point arithmetic? Can you provide an example?

Edit: Let me be more clear. I am aware that the same code with different compilers or different machines can return different results. What I am looking for are two mathematically equivalent expressions that I can compare in my Python interpreter/C++ program/Whatever and get an unexpected result.

Answer 1

are there mathematically equivalent
expressions that will yield different
results when using floating point
arithmetic?

Absolutely. In fact, you should expect this to happen more often than not.

Even the same code can yield different results on different machines or compilers.

Can you provide an example?

Sure. This Java code should repeatably yield two different results:

public strictfp class Test {
    public static void main(String[] args) throws Exception {
        float a = 0.7f;
        float b = 0.3f;
        float c = 0.1f;

        float r1 = ((a * b) * c);
        float r2 = (a * (b * c));

        System.out.println(r1);
        System.out.println(r2);
    }
}