Birkbeck College, U. London 1
Introduction to Computer Systems
Department of Computer Science and Information Systems
Lecturer: Steve Maybank
Spring 2020
Week 3b: Floating Point Notation for Binary Fractions
28 January 2020
Binary Fractions
A binary fraction has the form
sign||bitString1||radix point||bitString2
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 is20 + 0 × 2−1 + 2−2 = 1 + 1/4
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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.
1.01÷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
Examples
1.01 specifies the decimal fraction 5/4
An infinite number of bits is required to specify 1/3: 0.0101010101010101….
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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 point
The bit string 10111001
Brookshear, Section 1.7 6
Binary Fraction and Powers of 2
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 radix point is between the second bit from the left and the third bit from the left
The bit string is 101101
What is the binary fraction?
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 numberssmall gaps between numbers near to 0
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 numbers close to 0
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
11.001 = +22 x 0.11001
-0.011011 = -2-1 x 0.11011
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
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
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
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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
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
result = floating point number nearest to c
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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 + ¼ = 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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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]
https://en.wikipedia.org/wiki/Floating-point_arithmetic
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IEEE Standard for Floating Point Arithmetic
See http://steve.hollasch.net/cgindex/coding/ieeefloat.html
sign
1 bit
exponent
8 bits
mantissa
23 bits
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:
Numbers in Computing
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q = 0.1
The value stored in the memory location q may not be 0.1!
E.g. in Python the value stored is
0.1000000000000000055511151231257827021181583404541015625
See https://docs.python.org/2/tutorial/floatingpoint.html
