Birkbeck College, U. London1 Introduction to Computer Systems Lecturer: Steve Maybank Department of Computer Science and Information Systems [email protected].

Birkbeck College, U. London 1

Introduction to Computer Systems

Lecturer: Steve Maybank

Department of Computer Science and Information [email protected]

Autumn 2013

Week 4a: Floating Point Notation for Binary Fractions

22 October 2013

Recap: Binary Numbers In the standard notation for binary

numbers a string of binary digits such as 1001 stands for a sum of powers of 2: 1x23+0x22+0x21+1x20

Binary[1001] and Decimal[9] are different names for the same number

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Recap: Binary Addition

22 October 2013 Brookshear, Section 1.5 3

column: 3 2 1 0

1 0 0 1 1 1 ===== 1 1 0 0

There is a carry from column 0 to column 1 andfrom column 1 to column 2.

In tests or examinations, always show the carries.

Recap: Two’s Complement Two’s complement representations

can be added as if they were standard binary numbers.

22 October 2013 Brookshear, Section 1.6 4

-3 1101 2 0010== ==== -1 1111

Numbers in Computing

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cells in table: all numbers*: numbers that can be stored in memory#: numbers that can be referred to in a program

*#

#

#

* * *#

#

* * #

*#

*

Example: 0.1 cannot be stored in memory in IEEE double precisionfloating point, but the following is a correct Java statementt = 0.1;

Spacing Between Numbers

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Two’s complement:equally spacednumbers

0

Floating point:big gaps between big numbers,small gaps between small numbers.

0

The Key: Exponents

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2-4 232-22-3 2-1 20 2221

1/16 1/8 ¼ ½ 1 2 4 8

-4 -3 -2 -1 0 1 2 3

big gaps between big numberssmall gaps between small numbers

Brookshear, Section 1.7 8

Example of a Binary Fraction -101.11001 The binary fraction has three parts:

The sign –The position of the radix pointThe bit string 10111001

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Brookshear, Section 1.7 9

Reconstruction of a Binary Fraction

The sign is + The position of the radix point is

just to the right of the second bit from the left

The bit string is 101101 What is the binary fraction?

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22 October 2013 Brookshear, Section 1.7 10

Summary To represent a binary fraction three

pieces of information are needed:

Sign Position of the radix point Bit string

Brookshear, Section 1.7 11

Standard Form for a Binary Fraction

Any non-zero binary fraction can be written in the form

±2r x 0.twhere t is a bit string beginning with 1.

Examples11.001 = +22 x 0.11001

-0.011011 = -2-1 x 0.1101122 October 2013

Brookshear, Section 1.7 12

Floating Point Representation Write a non-zero binary fraction in

the form ± 2r x 0.t Record the sign – bit string s1 Record r – bit string s2 Record t – bit string s3 Output s1||s2||s3

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22 October 2013 Brookshear, Section 1.7 13

Floating Point Notation 8 bit floating point:

s e1 e2 e3 m1 m2 m3 m4

sign exponent mantissa1 bit 3 bits 4 bits radix r bit string t

The exponent is in 3 bit excess notation

22 October 2013 Brookshear, Section 1.7 14

To Find the Floating Point Notation

Write the non-zero number as ± 2r x 0.t

If sign = -1, then s1=1, else s1=0.

s2 = 3 bit excess notation for r.

s3= rightmost four bits of t.

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Example b= - 0.00101011101 s=1 b= -2-2 x 0.101011101

exponent = -2, s2 =010 Floating point notation

10101010

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Second Example

Floating point notation: 10111100 s1=1, therefore negative. s2 = 011, exponent=-1 s3 = 1100 Binary fraction -0.011 = -3/8

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Class Examples Find the floating point

representation of the decimal number -1 1/8

Find the decimal number which has the floating point representation

01101101

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22 October 2013 Brookshear, Section 1.7 18

Round-Off Error 2+5/8= 10.101 2 ½ = 10.100 The 8 bit floating point notations for

2 5/8 and 2 ½ are the same: 01101010

The error in approximating 2+5/8 with 10.100 is round-off error or truncation error.

Floating Point Addition Let [x] be the floating point

number closest to the number x. Floating point addition, x@y is

defined byx@y=[[x]+[y]]

Each operation w |-> [w] may introduce round off.

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Examples of Floating Point Addition

2 ½: 01101010 1/8: 00101000 ¼: 00111000 2 ¾: 01101011 2 ½ @(1/8 @ 1/8)=2 ½ @ 1/4=2 ¾ (2 ½ @ 1/8) @ 1/8=2 ½ @ 1/8=2 ½

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Round-Off in Decimal and Binary

1/5=0.2 exactly in decimal notation 1/5=0.0011001100110011….. in

binary notation 1/5 cannot be represented exactly

in binary floating point no matter how many bits are used.

Round-off is unavoidable but it is reduced by using more bits.

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Size of Round-Off Error E(x)

E(x)/x≈α where α is constant. If x > 0, y > 0 and

|x-y|<α x, then x-y cannot be found accurately using

floating point arithmetic.

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Examples 4 1/4: 100.01, fpoint = 01111000 4: 100.00, fpoint = 01111000 4 ¼-4 -> fpoint = 00000000

½: 0.1, fpoint = 01001000 ¼: 0.01, fpoint = 00111000 ½-¼ -> fpoint = 00111000

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Floating Point Errors

Overflow: number too large to be represented.

Underflow: number <>0 and too small to be represented.

Invalid operation: e.g. SquareRoot[-1].

See http://en.wikipedia.org/wiki/Floating_point

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IEEE Standard for Floating Point Arithmetic

For a general discussion of fp arithmetic seehttp://www.ee.columbia.edu/~marios/matlab/Fall96Cleve.pdf

0 1 … 8 9 … 31

Sign sbit 0

Exponent ebits 1-8

Mantissa mbits 9-31

If 0<e<255, then value = (-1)s x 2e-127 x 1.mIf e=0, s=0, m=0, then value = 0If e=0, s=1, m=0, then value = -0

Single precision, 32 bits.

22 October 2013