Comparing code implementations, GCC, if/else->continue

I am working on horrible code.

while (x)
{
    if (check1)
    {
        if (check2)
        {
            if (check3)
            {
                function();
            }
            else
            {
                error3();
            }
        }
        else
        {
            error2();
        }
    }
    else
    {
        error1();
    }
}

I'd like it to look like this

while (x)
{
    if (!check1)
    {
        error1();
        continue;
    }
    if (!check2)
    {
        error2();
        continue;
    }
    if (!check3)
    {
        error3();
        continue;
    }
    function();
}

But... This is 'time tested code' that is in many products..... ... and horrible to work with, and will encourage errors in the future.

I need a way to prove that when I change the code to my implementation, the resulting code is equivalent.

Right now when I compile the two versions of code I get different binary files.

Questions:

Is there a way to encourage the compiler to get the same output for both implementations so that I can show management that they're the same?

Is there a different way to demonstrate that the implementations are equivalent?