Birkbeck College, U. London 1
Introduction to Computer Systems
Lecturer: Steve Maybank
Department of Computer Science and Information [email protected]
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
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Brookshear, Section 1.7 6
Binary Fraction and Powers of 2
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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?
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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
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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
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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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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
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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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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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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