Birkbeck College, U. London 1

Introduction to Computer Systems

Lecturer: Steve Maybank

Department of Computer Science and Information Systemssjmaybank@dcs.bbk.ac.uk

Autumn 2015

Week 3b: Floating Point Notation for Binary Fractions

13 October 2015

Binary Fractions A binary fraction has the form

sign||bit string 1||radix point||bit string 2

E.g. +1.01, -10011.11 The + is usually omitted Digits to the right of the radix point

specify powers of 2 with negative exponents

E.g. 1.01 is13 October 2015 Birkbeck College, U. London 2

Properties of Binary Fractions 1

Multiply by 2: move the radix point one place to the right, e.g.

1.01x2 = 10.1

Divide by 2: move the radix point one place to the left, e.g.

1.02÷2 = 0.101

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Properties of Binary Fractions 2

A number can be specified exactly by a binary fraction if and only if it has the form

integer/power of 2 E.g.

1.01 specifies the decimal fraction 5/4

0.0101010101010101…. specifies 1/3

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Brookshear, Section 1.7 5

Specification of a Binary Fraction

-101.11001 The binary fraction has three parts:

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

13 October 2015

Brookshear, Section 1.7 6

Binary Fraction and Powers of 2

13 October 2015

 1  0 1  .  1  1  0  0  1 

 22 21  20 2-1 2-2 2-3 2-4

 2-5 

22+20+2-1+2-2+2-5 = 5+(25/32)

Brookshear, Section 1.7 7

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?

13 October 2015

13 October 2015 Brookshear, Section 1.7 8

Summary To represent a binary fraction three

pieces of information are needed:

Sign Position of the radix

point Bit string

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 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 0.1101113 October 2015

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

13 October 2015

13 October 2015 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

13 October 2015 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= leftmost 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

Birkbeck College, U. London 17

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

13 October 2015

13 October 2015 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 of Numbers x, y

a = floating point number nearest to x

b = floating point number nearest to y

c=a+b z=floating point number nearest to

c z=x@y13 October 2015 Birkbeck College, U. London 19

13 October 2015 Birkbeck College, U. London 20

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

13 October 2015 Birkbeck College, U. London 21

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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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

13 October 2015

Birkbeck College, U. London 23

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.

13 October 2015

Numbers in Computing

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q = 0.1

The value stored in the memory location q is not 0.1!

E.g. in Python the value stored is

0.1000000000000000055511151231257827021181583404541015625

See https://docs.python.org/2/tutorial/floatingpoint.html