The following bullet points describe the table above in more detail:
• Base 16 – uses digits 0-9 and letters A-F (to represent the values 10-15).
For example, to convert the decimal value 3000 to hexadecimal, you use the base 16
number system:
Base 16
16
7
...
16
4
16
3
16
2
16
1
16
0
268,435,456 ... 65,536 4096 256 16 1
So, the value 3000 is represented in hexadecimal as BB8
• Base 2 – uses digits 0 and 1.
For example, to convert the decimal value 184 to binary, you use the base 2 number
system:
Base 2
2
7
...
2
4
2
3
2
2
2
1
2
0
128 ... 16 8 4 2 1
So, the value 184 is represented in binary as 10111000.
• exponent bits – the number of bits reserved for storing the exponent, which
determines the magnitude of the number that you can store. The number of exponent
bits varies between operating systems. IEEE systems yield numbers of greater
magnitude because they use more bits for the exponent.
• mantissa bits – the number of bits reserved for storing the mantissa, which
determines the precision of the number. Because there are more bits reserved for the
mantissa on mainframes, you can expect greater precision on a mainframe compared
to a PC.
• round or truncate – the chosen conversion method used for handling two or more
digits. Because there is room for only two hexadecimal digits in the mantissa, a
convention must be adopted on how to handle more than two digits. One convention
is to truncate the value at the length that can be stored. This convention is used by
IBM Mainframe systems.
An alternative is to round the value based on the digits that cannot be stored, which
is done on IEEE systems. There is no right or wrong way to handle this dilemma
since neither convention results in an exact representation of the value.
Numerical Accuracy in SAS Software 65
In SAS, the LENGTH statement works by truncating the number of mantissa bits.
For more information about the effects of truncated lengths, see “Using the TRUNC
Function When Comparing Values” on page 79.
• bias – an offset used to enable both negative and positive exponents with the bias
representing 0. If a bias is not used, an additional sign bit for the exponent must be
allocated. For example, if a system uses a bias of 64, a characteristic with the value
66 represents an exponent of +2, whereas a characteristic of 61 represents an
exponent of –3.
Floating-Point Representation Using the IEEE Standard
The IEEE standard for floating-point arithmetic is a technical standard for floating-point
computation created by the Institute of Electrical and Electronic Engineers (IEEE). The
standard defines how computers store numbers in floating-point representation. The
IEEE standard for floating-point numbers is used by many operating systems, including
Windows and UNIX.
Although the IEEE platforms use the same set of specifications, you might occasionally
see varying results between the platforms due to compiler differences, and math library
differences. Also, because the IEEE standard allows for some variations in how the
standard is implemented, there might be differences in how different platforms perform
calculations even though they are following the same standard. Hosts might yield
different results because the underlying instructions that each operating system uses to
perform calculations are slightly different.
There is no standard method for performing computations. All operating systems attempt
to compute numbers as accurately as possible. It is not uncommon to get slightly
different results between operating systems whose floating-point representation
components differ. For example, there are differences between the z/OS and Windows
operating systems and between the z/OS and UNIX operating systems.
The IEEE standard for double-precision, floating-point numbers specifies an 11-bit
exponent with a base of 2 and a bias of 1023, which means that it has much greater
magnitude than the IBM mainframe representation, but sometimes at the expense of 3
bits less in the mantissa. The value of 1 represented by the IEEE standard is as follows:
3F F0 00 00 00 00 00 00
On Windows platforms, the processor performs computations in extended real precision.
This means that instead of the 64 bits that are used to store numeric values in the basic
format (52 bits for the mantissa and 11 bits for the exponent), there are 16 additional
bits: 12 additional bits for the mantissa and 4 additional bits for the exponent. Numeric
values are not stored in 80 bits (10 bytes) since the maximum width for a numeric
variable in SAS is 8 bytes. This simply means that the processor uses 80 bits to represent
a numeric value before it is passed back to its 64–bit memory slot. Intermediate
calculations might be done in 80 bits, which affects a part of the final answer.
On Windows this allows storage of numbers larger than the basic IEEE floating-point
format used by operating systems such as UNIX. This is one reason why you might see
slightly different values from operating systems that use the same IEEE standard.
Extended precision formats provide greater precision and more exponent range than the
basic floating-point formats.
66 Chapter 4 • SAS Variables
Floating-Point Representation on Windows
Storage Format
The byte layout for a 64-bit, double-precision number on Windows is as follows:
S E E E E E E E
Byte 1
E E E E M M M M
Byte 2
M M M M M M M M
Byte 3
M M M M M M M M
Byte 4
M M M M M M M M
Byte 5
M M M M M M M M
Byte 6
M M M M M M M M
Byte 7
M M M M M M M M
Byte 8
This representation corresponds to bytes of data with each character being 1 bit, as
follows:
• The S in byte 1 is the sign bit of the number. A value of 0 in the sign bit is used to
represent positive numbers.
• The remaining M characters in bytes 2 through 8 represent the bits of the mantissa.
There is an implied radix point before the left-most bit of the mantissa. Therefore,
the mantissa is always less than 1. The term radix point is used instead of decimal
point because decimal point implies that you are working with decimal (base 10)
numbers, which might not be the case. The radix point can be thought of as the
generic form of decimal point.
The exponent has a base associated with it. Do not confuse this with the base in which
the exponent is represented; the exponent is always represented in binary format, but the
exponent is used to determine how many times the base should be multiplied by the
mantissa.
Conversion Example
This example shows the conversion process for the decimal value 255.75 to floating-
point representation.
1. Use the base 2 number system to write out the value 255.75 in binary.
Note: Each bit in the mantissa represents a fraction whose numerator is 1 and whose
denominator is a power of 2; that is, the mantissa is the sum of a series of
fractions such as 1 half , 1 fourth , 1 eighth , and so on. Therefore, for any
floating-point number to be represented exactly, you must express it as the
previously mentioned sum.
Base 2
2
7
2
6
2
5
2
4
2
3
2
2
2
1
2
0
.2
-1
2
-2
128 64 32 16 8 4 2 1 1/2 1/4
255.75 =
1 x 2
7
1 x 2
6
1 x 2
5
1 x 2
4
1 x 2
3
1 x 2
2
1 x 2
1
1 x 2
0
1 x 2
-1
1 x 2
-2
Numerical Accuracy in SAS Software 67
So, the value 255.75 is represented in binary format as 1111 1111.11
2. Move the decimal over until there is only one digit to the left of it. This process is
called normalizing the value. Normalizing a value in scientific notation is the process
by which the exponent is chosen so that the absolute value of the mantissa is at least
one but less than ten. For this number, you move the decimal point 7 places:
1.111 1111 11
Because the decimal point was moved 7 places, the exponent is now 7.
3. The bias is 1023, so add 7 to 1023 to get
1030
4. Convert the decimal value, 1030, to hexadecimal using the base 16 number system:
Base 16
16
7
...
16
4
16
3
16
2
16
1
16
0
268,435,456 ... 65,536 4096 256 16 1
The converted hexadecimal value for 1030 will be placed in the exponent portion of
the final result.
5. Convert 406 to binary format:
0100 0000 0110
4 0 6
If the value that you are converting is negative, change the first bit to 1:
1100 0000 0110
This translates in hexadecimal to
C 0 6
6. In Step 2 above, delete the first digit and decimal (the implied one-bit):
11111111
7. Break these up into nibbles (half bytes) so that you have
1111 1111 1
8. To have a complete nibble at the end, add enough zeros to complete 4 bits:
1111 1111 1000
9. Convert
1111 1111 1000
to its hexadecimal equivalent to get the mantissa portion:
1111 1111 1000
F F 8
68 Chapter 4 • SAS Variables
The final floating-point representation for 255.75 is
406F F800 0000 0000
The final floating-point representation for –255.75 is
C06F F800 0000 0000
In this example, the starting decimal value, 255.75, conveniently converts to a finite
binary value that can be represented without rounding in both binary and hexadecimal.
The following section shows the conversion process for a decimal number that cannot be
represented precisely in floating-point representation.
Accuracy on x64 Windows Processors
Consider this example:
data _null_;
x=.500000000000000000000000;
y=.500000000000000000000000000;
if x=y then put 'equal';
else put 'not equal';
run;
Log Output
not equal
Although these values appear to be alike, the internal representations differ slightly,
because the IEEE floating-point representation can only represent 15 digits. Here is the
floating-point representation of both variables using the HEX16. format.
x=3FE0000000000000
y=3FDFFFFFFFFFFFFF
When the number of significant digits is reduced to 15 or less, the floating-point
representation is the same and the values are equal.
data _null_;
x=.5000000000000000;
y=.500000000000000;
if x=y then put 'equal';
else put 'not equal';
put x=hex16./
y=hex16.;
run;
Log Output
equal
x=3FE0000000000000
y=3FE0000000000000
This issue pertains to floating-point representation on the x64 processors. The routine
used to compute the result is slightly different on Windows than on any other host
(Linux, UNIX, AIX, and so on). The routine is written in x64 assembly to maximize
performance, and any changes to the routine for Windows x64 could not only lead to
poorer performance, but they could also potentially introduce other unintended side
effects or errors in other computations.
Numerical Accuracy in SAS Software 69
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