Copyright © 2003-2012 Curt Hill Numerics Representation and Errors

Copyright © 2003-2012 Curt Hill

Numerics

Representation and Errors


Integers

• Typically stored as a series of bits• Two’s complement binary• Each bit represents a zero or one

– Similar to a light switch– Only two states

• Binary is base 2


Decimal

• In base 10 the 10 occurs twice• The number of digits• The power to which a place is raised• In decimal each digit may be one of

10 possibilities, 0 - 9• Each digit is multiplied by 10 raised

to some power


A Decimal Number

4,809

9 100= 90 101= 08 102= 8004 103= 4000


Binary is just the same

• In base 2 the 2 occurs twice• The number of digits• The power to which a place is raised• In binary each digit may be one of

two possibilities, 0 or 1• Each digit is multiplied by two

raised to some power


A Binary Number101101

1 23= 80 24= 0

1 20= 10 21= 01 22= 4

1 25= 32

45


Binary and other bases• We use it because it is the only

base easy to implement in hardware

• We almost never display it– Binary is quite bulky

• We often use other bases for certain displays– Octal, base 8– Hexadecimal, base 16– Converting these to and from binary

is quite simple– Conversion to decimal is much harder


Integers• Come in several sizes:• 16 bit

– -32768 to 32767

• Most common is 32 bit– Largest integer is (231)-1– -2,147,483,648 to 2,147,483,647– There is always one more negative than

positive– Zero is taken from positives

• Large integers may be 64 bit– Largest is 9,223,372,036,854,780,000


Integer Errors• Integers are only subject to one error• Overflow• Add 1 to largest positive integer and

result is smallest negative integer• Subtract 1 from largest (in absolute

value) negative and result is largest positive

• Similar to clock arithmetic, except there are positives and negatives

Another Look at Overflow



Real Numbers• Real numbers (float, double) are

actually two numbers• Similar to scientific notation:

2.543 108

• The 2.543 is called the mantissa• The 8 is called the exponent• The 10 is the base

– Not represented, since always a 2 or some other constant


Real Number Errors• Real numbers are subject to a

variety of errors– Some quite subtle

• In order to demonstrate these we will model a computation using three digit arithmetic– Two digit of mantissa– One digit of exponent– Base of 10


3 Digit Arithmetic Game• A perfect circle is found in nature• We want to know its area• The formula is a =πr2

• The radius (r) is:11.4962790000000000

• Unfortunately, we measured this radius to be:11.495

• Which becomes 11E0 on our machine


Computing…

Take the 11 E 0 and square11 E 011 E 0

121 E 0

The121E0 has too manydigits, adjust:

12 E 1

Multiply timesπ12 E 131 E -1

372 E 0

Adjust again 37 E 1


What happened?• This exercise demonstrates several

important sources of error• It is never the case that if its close

its accurate– Which is what most believe

• The three demonstrated errors are– Observational– Representational– Computational


The Results

• The real value should be 415.2060805

• We ended up with 370• About 12% error• One half of a significant digit


Observational Error

• We can never measure anything completely accurately– To infinite precision

• The introduced error in this case is 0.02%

• This is not a problem for integers• The Olympics have a problem with

this, in measuring to hundredths of a second

Olympic Observational Error• LA 1984 Women’s 100 M. Freestyle

– Carrie Steinseifer and Nancy Hogshead won in 55.92

– Both received gold – no silver awarded

• Sydney 2000 Men’s 50 M. Freestyle– Anthony Ervin and Gary Hall Jr. 21.98

• London 2012 two events, four silvers – Men’s 200 M. Freestyle

• Taehwan Park and Yang Sun

– Men’s 100 M. Butterfly• Chad le Clos and Evgeny Korotyshkin



Representational Error• The measurement cannot be represented

accurately on this machine• 11.495 to 11 causes an error amounting

to 8.4% in addition to previous

• Truncatingπwill cause an additional 1.3%• This will be minimized but not eliminated

with many more than 2 digits of precision• This is the only integer arithmetic error

– Overflow

• There is more of this than you might think


Computational Error• A digit was lost twice

– 121 to 120– 372 to 370

• We may round or truncate this but we still lose information

• This will account for another 1.3% error in addition to the previous

• Another problem that integer arithmetic does not have


Representational Error Revisited

• Recall from grade school days there are three types of decimal numbers:– Rational

• Terminating• Repeating

– Irrational

• πcould not be represented since it has infinite digits

• For computational purposes, irrationals are no worse than repeating decimals


Repeating decimals

• Many fractions convert to repeating decimals

• 1/3 repeats with one digit 0.33333…• 1/11 has two digits 0.09090909…• 1/7 repeats with 6: .142857

142857…• As does 1/13: .076923 076923…


Terminating decimals• Terminating decimals are what we want• They end in an infinite number of zeros• 1/2 is 0.50• 1/4 is 0.25• 1/5 is 0.2• 1/8 is 0.125• 1/10 is 0.1• No representational errors with these• Or is there?


Terminating and Repeating

• Why does 1/5 terminate and 1/3 repeat?

• It all has to do with the prime factors of the denominator– If the prime factors only contain 2s

and 5s the number terminates, otherwise it repeats

– These are the prime factors of 10


Terminator• A number terminates if its

denominator has only the prime factors of the base

• 1/7 terminates in base 7 and base 14 but not in base 10 or base 2

• What base does a computer use?• Binary – base 2• We have representational error for

any fraction whose denominator has something other than two for a prime factor


Example: Money

• Cents are always fractions, with a denominator of 100– 100 has prime factors of 2 and 5

• Only the cent amounts of .00, .25, .50 and .75 reduce to having a denominator of 2 or 4

• Thus 96% of all monetary amounts have representational error


Computational Error Revisited

• Certain operations produce more or less error

• The subtraction of two values that are close in value has a deleterious effect

• The most significant and most accurate digits cancel each other

• This leaves only the most error prone digits left


Multiplication and Division• Generally multiplication and division

are better at preserving significant digits than addition and subtraction– The result has the precision of the least

precise value

• The result of multiplication has the sum of digits– Most of these are thrown away

• Errors are cumulative• Once we have error digits we cannot

get rid of them


Examples

0.20348

-0.20321

0.00027

Most accurate digits are lost

0.95

0.83

0.0285

+ 0.760

0.7885

The most accurate digits contribute most

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Copyright © 2003-2012 Curt Hill Numerics Representation and Errors