Difference between revisions of "Ieee arithmetic"

(Testing for 25% NaN values and 75% non-NaN values)
(Testing for 10% NaN values and 90% non-NaN values)
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|-
 
|-
 
| '''METH1'''
 
| '''METH1'''
| 5.029s
+
|  
| 6.24s
+
|  
| 9.32s
+
|  
 
|-
 
|-
 
| '''METH2'''
 
| '''METH2'''
| 6.556s
+
|  
| '''6.59s'''
+
|  
| '''11.74s'''
+
|  
 
|-
 
|-
 
| '''METH3'''
 
| '''METH3'''
| 6.437s
+
|  
| 7.29s
+
|  
| 12.82s
+
|  
 
|-
 
|-
 
| '''METH4'''
 
| '''METH4'''
| '''6.327s'''
+
|  
| 8.80s
+
|  
| 12.06s
+
|  
 
|-
 
|-
 
| '''METH5'''
 
| '''METH5'''
| 6.800s
+
|  
| 7.73s
+
|  
| 12.22s
+
|  
 
|}
 
|}

Revision as of 15:12, 2 February 2010

We will present some of the trade-offs for computation of IEEE arithmetic for REAL_64 and REAL_32 as implemented in EiffelStudio 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.

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 value) {
	return eif_is_nan_bits(to_raw_bits (value));
}
 
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
}
 
#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:

Method used Timing VC++ 2005 x64 Timing gcc 4.4.1 x64 Timing Sun cc 5.9 x64
METH1 5.068s 6.25s 9.32s
METH2 6.495s 6.47s 11.57s
METH3 6.448s 7.35s 12.31s
METH4 6.424s 8.80s 12.31s
METH5 6.832s 7.73s 11.46s

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:

Method used Timing VC++ 2005 x64 Timing gcc 4.4.1 x64 Timing Sun cc 5.9 x64
METH1 5.242s 6.24s 9.32s
METH2 5.464s 5.17s 9.62s
METH3 5.308s 5.48s 9.26s
METH4 6.428s 8.80s 12.31s
METH5 5.384s 5.49s 9.32s

Testing for 100% NaN values

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

Method used Timing VC++ 2005 x64 Timing gcc 4.4.1 x64 Timing Sun cc 5.9 x64
METH1 4.777s 6.24s 9.25s
METH2 5.440s 5.17s 9.92s
METH3 6.266s 7.32s 16.48s
METH4 5.560s 8.80s 12.44s
METH5 7.413s 8.34s 15.74s

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:

Method used Timing VC++ 2005 x64 Timing gcc 4.4.1 x64 Timing Sun cc 5.9 x64
METH1 6.889s 6.24s 9.31s
METH2 6.914s 7.01s 13.36s
METH3 6.405s 7.01s 14.64s
METH4 6.068s 8.80s 12.07s
METH5 6.880s 7.83s 13.47s

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:

Method used Timing VC++ 2005 x64 Timing gcc 4.4.1 x64 Timing Sun cc 5.9 x64
METH1 7.553s 6.24s 9.31s
METH2 6.672s 6.73s 12.25s
METH3 8.997s 7.23s 13.47s
METH4 6.250s 8.80s 12.00s
METH5 8.063s 7.76s 12.55s

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:

Method used Timing VC++ 2005 x64 Timing gcc 4.4.1 x64 Timing Sun cc 5.9 x64
METH1 5.029s 6.24s 9.32s
METH2 6.556s 6.59s 11.74s
METH3 6.437s 7.29s 12.82s
METH4 6.327s 8.80s 12.06s
METH5 6.800s 7.73s 12.22s


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:

Method used Timing VC++ 2005 x64 Timing gcc 4.4.1 x64 Timing Sun cc 5.9 x64
METH1 4.749s
METH2 6.358s
METH3 6.109s
METH4 6.112s
METH5 6.355s

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:

Method used Timing VC++ 2005 x64 Timing gcc 4.4.1 x64 Timing Sun cc 5.9 x64
METH1
METH2
METH3
METH4
METH5

Testing for 100% NaN values

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

Method used Timing VC++ 2005 x64 Timing gcc 4.4.1 x64 Timing Sun cc 5.9 x64
METH1
METH2
METH3
METH4
METH5

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:

Method used Timing VC++ 2005 x64 Timing gcc 4.4.1 x64 Timing Sun cc 5.9 x64
METH1
METH2
METH3
METH4
METH5

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:

Method used Timing VC++ 2005 x64 Timing gcc 4.4.1 x64 Timing Sun cc 5.9 x64
METH1
METH2
METH3
METH4
METH5

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:

Method used Timing VC++ 2005 x64 Timing gcc 4.4.1 x64 Timing Sun cc 5.9 x64
METH1
METH2
METH3
METH4
METH5