[joshAg]

Attmepting to just use simple equivalence with any floating point type a horrible idea to begin with, even without NaNs. You should instead declare equivalence if the absolute difference between the two numbers is less than some bound based on what you're doing and not look for bit equivalency.

However, you should be able to examine two NaNs and declare them "equivalent" (for certain definitions of equivalence) by intelligently examining the bits based on the hardware that you're running the program on. In the case of a binary Nan [1] that would entail checking that the exponential fields are both entirely high (eg 0x8 == (a.exponent & b.exponent), assuming a standard 8 bit exponent) and that the mantissas are nonzero (eg a.mantissa && b.mantissa).

[1]: "Binary format NaNs are represented with the exponential field filled with ones (like infinity values), and some non-zero number in the significand (to make them distinct from infinity values)." --http://en.wikipedia.org/wiki/NaN

[shrughes]

That's not true, there are plenty of cases where using equivalence is just fine. Integer arithmetic, and algorithms that are more reliably written not to contain any empty intervals are two examples.

[noblethrasher]

He was specifically talking about equivalence on floating point types. Integers don't have or need NaN.

[shrughes]

I was talking about integer values (with floating point representation) being multiplied and added (and divided and floored, I suppose).

[joshAg]

even just addition and multiplication with floats make simple equivalence a horrible idea, due to uncertainty.

For example:

    float a = 1.0;
    float b = 1000.0;
    for (int i = 0; i < 1000000; ++i)
        a+=1.0;
    b *= b;

There is no guarantee that a == b. Floats make everything more complicated, even simple addition: http://en.wikipedia.org/wiki/Kahan_summation_algorithm

[shrughes]

There is no uncertainty. There is a guarantee that a == b (if we ignore the off-by-one error in your post), because IEEE operations are guaranteed to be accurate within half an ulp. You can safely perform addition, subtraction, and multiplication, and truncated or floored division, within the 24-bit integer range for single-precision floats and the 53-bit integer range for doubles. This is why people can safely use integers in Javascript.

[joshAg]

i guess that's what a i get for not double checking my math. here's a revised version that (as long as i haven't made any other math mistakes) still fits within a 32 bit signed int but doesn't guarantee simple equality:

    float a = 0.0;
    float b = 10000.0;
    for (int i = 0; i < 100000000; ++i)
        a+=1.0;
    b *= b;

why in the world would you use a float instead of an int for addition, subtraction, and multiplication, and truncated or floored division, within the 24-bit integer range? it seems like there's no benefit to offset the facts that floating point operations are slower than integer operations and that ints can store integers 7 or 8 bits larger.

and what happens when you go beyond 24 bits? since it's a float no error or warning will be thrown, but now equivalence won't work for numbers that are easily stored by an int.

[joshAg]

If you mean arithmetic on floating point numbers that only store integers, then no equivalence is not ok, because there are examples where bit equivalence fails. One example is repeatedly adding 1.0 to a float something like 1 trillion times versus multiplying 1000000.0 by 1000000.0

What do you mean by

    algorithms that are more reliably written not to contain any empty intervals are two examples.

[shrughes]

Obviously I'm referring to integer operations within a 52-bit or 23-bit range, not outside the ability of floating point representations to represent integers!

> What do you mean by

I mean exactly that sort of thing. Algorithms that, for example, divide a line into intervals, where empty intervals are not needed, or desired, and are generally risky with respect to the probability of having a correct implementation. An example of this would be computations of the area of a union of rectangles. You might make a segment tree -- and avoid having degenerate rectangles in your segment tree.