DECIMAL

ARITHMETICPeeyush Pashine

Contents

• Why increasing emphasis on Decimal

Arithmetic?

• Problems caused by Floating Point?

• Disaster ( Examples)

• How Floating to Decimal?

• Is Hardware supports decimal arithmetic

directly?

Why Decimal Arithmetic?

Most computers today support binary floating-

point in hardware.

But for financial, commercial, and human-

centric (such as Web) applications – NOT

suitable.

In many commercial applications, including

financial analysis, banking, tax calculation,

currency conversions, insurance, accounting,

and e-commerce, the errors introduced by

converting between decimal and binary

numbers are unacceptable and may violate

legal accuracy requirements.

What problems are caused by

using binary floating-point?

Binary floating-point cannot exactly represent

decimal fractions.

DECIMAL BINARY

0.9

0.09

0.009

0.0009

0.00009

0.000009

9E-7

9E-8

9E-9

9E-10

0.9

0.089999996

0.0090

9.0E-4

9.0E-5

9.0E-6

9.0000003E-7

9.0E-8

9.0E-9

8.9999996E-10

Example

For example, using the Java or C double datatype, 0.1 × 8 (a binary multiple) gives the result 0.8000000000000000444089209850062616169452667236328125

but 0.1 added to itself 8 times gives the different answer 0.79999999999999993338661852249060757458209991455078125.

The two results would not compare equal, and further, if these values are multiplied by ten and rounded to the nearest integer below (the „floor‟ function), the result will be 8 in one case and 7 in the other.

Similarly, the result of 1.30 x 1.05 using binary is

1.3650000000000002131628207280300557613

372802734375; this would be rounded up to

$1.37. However, the result should have been

1.365 – which would be rounded down to $1.36

(using “Banker‟s rounding”).

Taken over a million transactions of this kind, as

in the „telco‟ benchmark, these systematic errors

add up to an overcharge of more than $20. For a

large company, the million calls might be two-

minutes-worth; over a whole year the error then

exceeds $5 million.

for (double d = 0.1; d <= 0.5; d += 0.1)

System.out.println(d);

for (double d = 1.1; d <= 1.5; d += 0.1)

System.out.println(d);

FAILURES

The Patriot Missile Failure

On February 25, 1991, during the Gulf War, an American Patriot Missile

battery in Dharan, Saudi Arabia, failed to track and intercept an incoming

Iraqi Scud missile.

A report of the General Accounting office, GAO/IMTEC-92-26,

entitled Patriot Missile Defense: Software Problem Led to System Failure

at Dhahran, Saudi Arabia reported on the cause of the failure.

The cause was an inaccurate calculation of the time since boot due to

computer arithmetic errors.

Specifically, the time in tenths of second as

measured by the system's internal clock was

multiplied by 1/10 to produce the time in

seconds.

Explosion of the Ariane 5

On June 4, 1996 an unmanned Ariane 5 rocket launched by the European

Space Agency exploded just forty seconds after lift-off (918K QuickTime

movie).

The cause of the failure was a software error in the inertial reference

system.

Specifically a 64 bit floating point number relating to

the horizontal velocity of the rocket with respect to the

platform was converted to a 16 bit signed integer. The

number was larger than 32,768, the largest integer

storeable in a 16 bit signed integer, and thus the conversion

failed.

The applications actually use

decimal data

Airline systems, Banking, Financial Analysis,

Insurance, Inventory control, Management

reporting, Marketing services, Order entry,

Order processing, Pharmaceutical

applications, and Retail sales.

How much precision and range is

needed for decimal arithmetic?

This depends, of course, on the application.

What rounding modes are needed

for decimal arithmetic?

The three most important are

Round-half-even

Round-half-up

Round-half-up

What disadvantages are there in

using decimal arithmetic?

Decimal numbers are traditionally held in a

binary coded decimal form which uses about

20% more storage than a purely binary

representation.

Some properties that hold for binary do not hold

for any other base. For example, (d ÷ 2) × 2

gives d in binary (unless there is an underflow),

but with base 10 it might not if d is full precision

and d ÷ 2 is Inexact.

The first division illustrates dividing 2.1 into 123.4 . Since we are working

with integers, this is equivalent to dividing 21 into 1234. The result is 58

and contains no decimal point. This will not give us the precision we

demand in the quotient. In the second division, by shifting the dividend to

the left by one digit (bringing in a 0 on the right), we are effectively dividing

21 into 12340, and producing a quotient of 587 which could be edited to

58.7 for printing.

for (i=0.1; i<0.5; i=i+0.1) printf

("%f\n",100000000*i);

will print out:

100000001.490116

200000002.980232

300000011.920929

400000005.960464

for (num=1.1; num<=1.5;

num=num+0.1)

printf ("%f\n",num);

prints:

1.100000

1.200000

1.300000

1.400000

for (num=0.1; num<=0.5;

num=num+0.1)

printf ("%f\n",num);

prints:

0.100000

0.200000

0.300000

0.400000

0.500000

Why doesn’t hardware support

decimal arithmetic directly?

Most computer architectures other than „pure

RISC‟ machines do, in fact, provide some form

of decimal arithmetic instructions or support.

The native (hardware) decimal floating-point

arithmetic now available in the IBM Power6

processor and expected in the z6

microprocessor makes programming far

simpler and more robust, and with much better

performance than software

DIFFERENT WAYS

INVOLVED

Why is decimal arithmetic

unnormalized?

Unnormalized arithmetic is compatible with

existing languages and applications.

The arithmetic of all existing decimal datatypes

can be derived by constraining the

unnormalized arithmetic.

Unnormalized arithmetic often permits

performance improvements.

Gradual underflow is „free‟.

Why is decimal arithmetic

unnormalized?

Zeros are not special cases.

Conversions to and from existing decimal

datatypes are faster.

Unnormalized arithmetic results match human

expectations.

Application design is simpler.

How are decimal numbers

encoded?

Plimit, the maximum precision of the coefficient. This is the

maximum length of the coefficient, in digits. Any result from an

operation which needs more digits than this will be rounded to fit.

If this rounding caused non-zero digits to be removed, the result

is Inexact.

Elimit, the maximum encoded exponent. The encoded exponent is

a non-negative number, in the range 0 through Elimit, from which

the exponent parameter is calculated by subtracting a bias. (This

use of a bias makes it easier to compare exponents in a hardware

implementation.)

How is the exponent bias

chosen?

A balanced range of exponents is defined by the parameters,

Emax and Emin, which determine the overflow threshold

(10×10Emax) and the underflow threshold (1×10E

min) respectively.

In addition to the normal numbers, it must also be possible to

encode a further range of numbers of lower precision which are

smaller than the underflow precision. These numbers are

called subnormal numbers. The smallest subnormal number

must be 1×10Etiny, where Etiny is given by Emin - (Plimit - 1). In our

example format, the range of subnormal numbers is

1×10Etiny through 999999×10E

tiny (which is

0.000001×10E through 0.999999×10E )

How is the exponent bias

chosen?(cont..)

In a redundant encoding, more than one coefficient

(with an appropriate exponent) can be used to

represent a given numerical value (for example, the

underflow threshold could be represented as either

1000000×10Etiny or 1×10E

min).

In a non-redundant encoding, only one coefficient is

used for a given numerical value. (The coefficient

chosen usually depends on the scale – when the scale

is 0 the smallest coefficient is preferred).

Decimal Format Parameters

Conversion

How many decimal digits are needed to

represent a binary floating-point number as an

exact fraction?

How many decimal digits are needed to

represent a binary floating-point number

reversibly?