1. Introduction to Computer Systems
Lecturer: Steve Maybank
Department of Computer Science and Information Systems
Autumn 2013
Week 4a: Floating Point Notation for Binary Fractions
22 October 2013
Birkbeck College, U. London

2. Birkbeck College, U. London
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
22 October 2013
Birkbeck College, U. London

3. Recap: Binary Addition
column:
1 1
=====
There is a carry from column 0 to column 1 and
from column 1 to column 2.
In tests or examinations, always show the carries.
22 October 2013
Brookshear, Section 1.5

4. 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

5. Birkbeck College, U. London
Numbers in Computing
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 precision
floating point, but the following is a correct Java statement
t = 0.1;
22 October 2013
Birkbeck College, U. London

6. Spacing Between Numbers
Two’s complement:
equally spaced
numbers
Floating point:
big gaps between big numbers,
small gaps between small numbers.
22 October 2013
Birkbeck College, U. London

7. Birkbeck College, U. London
The Key: Exponents
2-4
2-3
2-2
2-1 20 21 22 23
1/ / ¼ ½
big gaps between big numbers
small gaps between small numbers
22 October 2013
Birkbeck College, U. London

8. Example of a Binary Fraction
The binary fraction has three parts:
The sign –
The position of the radix point
The bit string
22 October 2013
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
What is the binary fraction?
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
22 October 2013
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.t
where t is a bit string beginning with 1.
Examples
= +22 x
= -2-1 x
22 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
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 mantissa
1 bit bits 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.
22 October 2013
Brookshear, Section 1.7

15. Birkbeck College, U. London
Example b=
s=1
b= -2-2 x
exponent = -2, s2 =010
Floating point notation
22 October 2013
Birkbeck College, U. London

16. Birkbeck College, U. London
Second Example
Floating point notation:
s1=1, therefore negative.
s2 = 011, exponent=-1
s3 = 1100
Binary fraction = -3/8
22 October 2013
Birkbeck College, U. London

17. Birkbeck College, U. London
Class Examples
Find the floating point representation of the decimal number -1 1/8
Find the decimal number which has the floating point representation
22 October 2013
Birkbeck College, U. London

18. Round-Off Error
2+5/8=
2 ½ =
The 8 bit floating point notations for 2 5/8 and 2 ½ are the same:
The error in approximating 2+5/8 with is round-off error or truncation error.
22 October 2013
Brookshear, Section 1.7

19. Floating Point Addition
Let [x] be the floating point number closest to the number x.
Floating point addition, is defined by
Each operation w |-> [w] may introduce round off.
22 October 2013
Birkbeck College, U. London

20. Examples of Floating Point Addition
2 ½:
1/8: ¼:
2 ¾:
1/8)=2 1/4=2 ¾
(2 1/8=2 1/8=2 ½
22 October 2013
Birkbeck College, U. London

21. Round-Off in Decimal and Binary
1/5=0.2 exactly in decimal notation
1/5= ….. 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.
22 October 2013
Birkbeck College, U. London

22. 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.
22 October 2013
Birkbeck College, U. London

23. Birkbeck College, U. London
Examples
4 1/4: , fpoint =
4: , fpoint =
4 ¼-4 -> fpoint =
½: 0.1, fpoint =
¼: 0.01, fpoint =
½-¼ -> fpoint =
22 October 2013
Birkbeck College, U. London

24. Birkbeck College, U. London
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
22 October 2013
Birkbeck College, U. London

25. IEEE Standard for Floating Point Arithmetic
Single precision, 32 bits. 1 … 8 9 31
Mantissa m
bits 9-31
Sign s
bit 0
Exponent e
bits 1-8
If 0<e<255, then value = (-1)s x 2e-127 x 1.m
If e=0, s=0, m=0, then value = 0
If e=0, s=1, m=0, then value = -0
For a general discussion of fp arithmetic see
22 October 2013
Birkbeck College, U. London