end;

j=1;

do k = 1 to 5;

if indexc(old(k),from(j)) > 0 then do;

new(k)=translate(old(k),to(j),from(j));

j+1;

if j=4 then j=1;

end;

end;

decrypt_num=input(cats(of new1-new5),5.);

keep num encrypt_num decrypt_num;

run;

proc print;

run;

The following output shows the results of the PROC PRINT for Example 3:

Numerical Accuracy in SAS Software

Overview

In any number system, whether it is binary or decimal, there are limitations to how

precise numbers can be represented. As a result, approximations have to be made. For

example, in the decimal number system, the fraction 1/3 cannot be perfectly represented

as a finite decimal value because it contains infinitely repeating digits

(.333...). On

computers, because of finite precision, this number must be approximated. Numerical

precision is the accuracy with which numbers are approximated or represented.

In computing, software applications are particularly susceptible to numerical precision

errors due to finite precision and machine hardware limitations. Computers are finite

machines with finite storage capacity, so they cannot represent an infinite set of numbers

with perfect precision.

The problem is further compounded by the fact that computers use a different number

system than people do. Decimal infinite-precision arithmetic is the norm for human

calculations but computers use finite binary representations of values and finite-

precision arithmetic. This representation has been proven adequate for many

calculations. Yet, depending on the problem, you may need an extended precision that is

wider than what the hardware offers. In that case, representation and arithmetic are done

mostly in software and are relatively much slower than hardware arithmetic.

60 Chapter 4 • SAS Variables

Furthermore, although computers do allow the use of decimal numbers and decimal

arithmetic via human-centric software interfaces, all numbers and data are eventually

converted to binary format to be stored and processed by the computer internally. It is in

the conversion between these 2 number systems – decimal to binary – that precision is

affected and rounding errors are introduced.

Truncation in Binary Numbers

Just like there are decimal values with infinitely repeating representations, there are also

binary values that have infinitely repeating representations. However, the numbers that

are imprecise in decimal are not always the same ones that are imprecise in binary.

For example, the decimal value 1/10 has a finite decimal representation (0.1), but in

binary it has an infinitely repeating representation. In binary, the value converts to

0.000110011001100110011 ...

where the pattern 0011 is repeated indefinitely. As a result, the value will be rounded

when stored on a computer.

Performing calculations and comparisons on imprecise numbers in SAS can lead to

unexpected results. Even the simplest calculations can lead to a wrong conclusion.

Hardware cannot always match what might seem obvious and expected in the decimal

system.

For example, in decimal arithmetic, the expression (3 x 0.1) is expected to be equal

to

0.3, so the difference between (3 x 0.1) and (0.3), must be 0. Because the

decimal values 0.1 and 0.3 do not have exact binary representations, this equality does

not hold true in binary arithmetic. If you compute the difference between the two values

in a SAS program, the result is not 0, as Example Code 4.6 on page 61 illustrates.

In the example, SAS sets the variables point_three and

three_times_point_three to 0.3 and (3 x 0.1), respectively. It then compares

the two values by subtracting one from the other and writing the result to the SAS log:

Example Code 4.6 Comparing Imprecise Values in SAS

data a;

point_three=0.3;

three_times_point_one= 3 * 0.1;

difference= point_three - three_times_point_one;

put 'The difference is ' difference;

run;

Output 4.5 Log Output for Comparing Imprecise Values in SAS

The log output shows that (3 x 0.1) — 0.3 does not equal 0, as it does in decimal

arithmetic. This is because the variable "difference" is the result of calculations that are

performed on rounded values, or, infinitely repeating binary values.

Numerical Accuracy in SAS Software 61

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