We will present some of the trade-offs for computation of IEEE arithmetic for REAL_64 and REAL_32 as implemented in EiffelStudio when enabling the Total Order on REALs option where NaN is not an unordered value but a value less than all the other values (note that in some other frameworks, we have seen it defined as the largest value). In other words:

• NaN = NaN yields True
• NaN < x for all x but NaN

To best show the trade-offs we will start by showing some benchmark results.

Note: This is an EiffelStudio specific extension to potentially help the ECMA standard committee in making a reasonable decision on the semantic of NaN.

Benchmarks

The code

The code below defines an equality function as well as a comparison function. The test is divided in two parts, first the initialization and then the computation.

static EIF_NATURAL_64 to_raw_bits (EIF_REAL_64 d) {
	return *((EIF_NATURAL_64 *)&d);
}
 
static int eif_is_nan_bits (EIF_NATURAL_64 value) {
		/* Clear the sign mark. */
    EIF_NATURAL_64 jvalue = (value & ~RTU64C(0x8000000000000000));
		/* Ensure that it starts with 0x7ff and that the mantissa is not 0. */
    return (jvalue > RTU64C(0x7ff0000000000000));
}
 
static int eif_is_nan (EIF_REAL_64 v) {
	EIF_NATURAL_64 value = *((EIF_NATURAL_64 *)&v);
	value &= ~RTU64C(0x8000000000000000);
	return (value > RTU64C(0x7ff0000000000000));
}
 
static int eif_is_nan_real_64 (EIF_REAL_64 v) {
#ifdef NAN1
	return v != v;
#elif defined(NAN2)
	EIF_NATURAL_64 value = *((EIF_NATURAL_64 *)&v);
	value &= ~RTU64C(0x8000000000000000);
	return (value > RTU64C(0x7ff0000000000000));
#elif defined(NAN3)
	EIF_REAL_64 *l_v = &v;
	return ((*((EIF_NATURAL_64 *)(l_v)) & RTU64C(0x7FF0000000000000))==RTU64C(0x7FF0000000000000)) &&
	          (*((EIF_NATURAL_64 *)(l_v)) & RTU64C(0x000FFFFFFFFFFFFF));
#elif defined(NAN4)
#ifdef _WIN32
	return _isnan(v);
#else
	return isnan(v);
#endif
#endif
 
static int eif_equal_real_64 (EIF_REAL_64 d1, EIF_REAL_64 d2) {
#ifdef METH1
		/* Here the base comparison is IEEE arithmetic. */
	return (d1 == d2);
#elif defined(METH2)
		/* Conversion to perform comparison on the binary representation. */
	EIF_NATURAL_64 f1 = to_raw_bits(d1);
	EIF_NATURAL_64 f2 = to_raw_bits(d2);
	return (f1 == f2 ? 1 : (eif_is_nan_bits (f1) && eif_is_nan_bits(f2)));
#elif defined(METH3)
		/* Use IEEE arithmetic to compare and find out if we have NaNs. */
	return (d1 == d2 ? 1 : ((d1 != d1) && (d2 != d2)));
#elif defined (METH4)
		/* Pessimist case, we assume that we compare mostly NaNs. */
	return (d1 == d1 ? d1 == d2 : d2 != d2);
#elif defined(METH5)
		/* Use IEEE arithmetic to compare but use binary representation to
		 * find out if we have NaNs. */
	return (d1 == d2 ? 1 : (eif_is_nan (d1) && eif_is_nan(d2)));
#endif
}
 
static int eif_is_less_real_64 (EIF_REAL_64 d1, EIF_REAL_64 d2) {
#ifdef METH1
		/* Here the base comparison is IEEE arithmetic. */
	return d1 < d2;
#elif defined(METH2)
		/* Use IEEE arithmetic to compare but use binary representation to
		 * find out if we have NaNs. */
	return (d1 < d2 ? 1 : eif_is_nan(d1) && !eif_is_nan(d2));
#elif defined(METH3)
		/* Use IEEE arithmetic to compare and find out if we have NaNs. */
	return (d1 < d2 ? 1 : (d1 != d1) && (d2 == d2));
#elif defined(METH4)
		/* Pessimist case, we assume that we compare mostly NaNs. */
	return (d1 == d1 ? d1 < d2 : d2 == d2);
#elif defined(METH5)
		/* Variation on METH3 using a different order for comparison. */
	return (eif_is_nan(d1) ? !eif_is_nan(d2) : d1 < d2);
#endif
}
 
 
#define ARR_SIZE 100000
 
int main(void) {
	EIF_NATURAL_64 res, i;
	EIF_REAL_64 *d = (EIF_REAL_64 *) malloc (sizeof(EIF_REAL_64) * ARR_SIZE + 1);
 
		/* Initialization of `d'. */
	...
 
	for (i = 0; i <= 0x3FFFFFFF; i++) {
			/* Substitute comparison_function with what needs to be tested. */
		res = res + comparison_function (d [i % ARR_SIZE], d[(i - 1) % ARR_SIZE]);
	}
	printf ("%d\n", res);
}

The configuration

• On Windows XP 64-bit with a Intel Q9450 @ 3 GHz with VC++ 2005 using the following command line:

/O2 /GL /FD /MT /GS-

• On Linux Ubuntu 9.04 x64 with Intel Xeon E5420 @ 2.5 GHz with gcc 4.4.1 using the following command line:

-O3 -funroll-loops -lm

• On Solaris 10 x64 with AMD Opteron 248 @ 2.2 GHz with Sun C 5.9 using the following command line:

-xO5 -m64 -lm

Equality Testing

Testing for non-NaN values which are always different

The array is filled with non-NaN values which are always different, which gives the following results:

Testing for non-NaN values which are always the same

The array is filled with non-NaN values which are always the same, which gives the following results:

Testing for 100% NaN values

The array is filled with NaN values, which gives the following results:

Testing for 50% NaN values and 50% non-NaN values

The array is filled at 50% with NaN values and the rest with different values which gives the following results:

Testing for 25% NaN values and 75% non-NaN values

The array is filled at 25% with NaN values and the rest with different values which gives the following results:

Testing for 10% NaN values and 90% non-NaN values

The array is filled at 10% with NaN values and the rest with different values which gives the following results:

Less than Testing

Testing for non-NaN values which are always different

The array is filled with non-NaN values which are always different, which gives the following results:

Testing for non-NaN values which are always the same

The array is filled with non-NaN values which are always the same, which gives the following results:

Testing for 100% NaN values

The array is filled with NaN values, which gives the following results:

Testing for 50% NaN values and 50% non-NaN values

The array is filled at 50% with NaN values and the rest with different values which gives the following results:

Testing for 25% NaN values and 75% non-NaN values

The array is filled at 25% with NaN values and the rest with different values which gives the following results:

Testing for 10% NaN values and 90% non-NaN values

The array is filled at 10% with NaN values and the rest with different values which gives the following results:

IsNaN Testing

We are testing eif_is_nan_real_64 which has 4 different implementations: NAN1, NAN2, NAN3 and NAN4. As we can see in the results below, it is always best to check for NaN using the `x != x' yielding True pattern.

Testing for non-NaN values which are always different

The array is filled with non-NaN values which are always different, which gives the following results:

Testing for non-NaN values which are always the same

The array is filled with non-NaN values which are always the same, which gives the following results:

Testing for 100% NaN values

The array is filled with NaN values, which gives the following results:

Testing for 50% NaN values and 50% non-NaN values

The array is filled at 50% with NaN values and the rest with different values which gives the following results:

Testing for 25% NaN values and 75% non-NaN values

The array is filled at 25% with NaN values and the rest with different values which gives the following results:

Testing for 10% NaN values and 90% non-NaN values

The array is filled at 10% with NaN values and the rest with different values which gives the following results: