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Floating Point Numbers
It's all just 1s and 0s
Computers are fundamentally driven by logic
and thus bits of data

Manipulation of bits can be done incredibly quickly
Given n bits of information, there are 2n
possible combinations
These 2n representations can encode pretty
much anything you want, letters, numbers,
instructions….
Bases of number systems
Base 10 numbers: 0,1,2,3,4,5,6,7,8,9

3107 = 3103 +1102 + 0101 +7100
Base 2 numbers: 0,1
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3107 = 1 2 4 8 16 32 64 128 256 512 1024 2048
=1211 + 1210 + 029 + 028 + 027 + 026
+ 125 + 024 + 023 + 022 + 121 + 120
=110000100011
Addition, multiplication etc, all proceed same
way
Base Notation
What does 10 mean?



10 in binary = 2 decimal
10 in octal (base 8) = 8 decimal
10 in decimal = 10 decimal
Need some method of differentiating between
these possibilities
To avoid confusion, where necessary we write


1010=
102=
Integer Representation
Integers obviously fit into this base 2
notations
Remains challenge to represent negative
numbers


2s complement
Excess-N
Extra choice is order of bits
Choice is made chip-by-chip

portability
Floating Point Representation
Computers represent oating point
numbers in binary form
For generality, they use a binary form of
scientic notation
29.25 = 0.2925  10
3
In binary, we can use powers of 2
29.25 
Floating Point Size
In IEEE.h


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IEEE.h:#define IEEE_FLOAT_SIZE 4
IEEE.h:#define IEEE_DOUBLE_SIZE 8
IEEE.h:#define IEEE_QUAD_SIZE 16
Distribution
Precision # bits
Single
32
Mantissa
Bits
23
Double
64
52
Expon.
Bits
8
Sign
Bit
1
11
1
In Decimal Terms
Each binary floating point double holds
roughly 16 decimal digits

technically, 2^(-52)
MATLAB example
Advantages
Scientific notation can work on any
scale (all handled by exponent)
So long as errors are small relative to
scale of data values, calculations are
accurate

right?
Example 1
1e12 + 0.2 – 1e12
Problem
Nice decimal numbers (0.2) have
continuing binary representations

like 1/3 = 0.3333333, 0.2 has binary
0.0011 0011 0011 0011…
Analogy with adding, subtracting large
number
Roundoff Error
Round-off error will always be present
e.g.
Roundoff error is more significant when
you are subtracting two almost equal
quantities
e.g in decimal, 255.67 – 255.69
Example 2
A = 112000000
B = 100000
C = 0.0009
X=A-B/C
Common occurrence
Delta x in


finite element methods
numerical differentiation
Places where more closely packed data
gives
Example 3: Numerical Diff.
Example 4: Recursion
Comparing sum of delta x and real sum



t = 0;
N = 10000; dx = 1/N;
for (I = 1:N)
 t = t + dx;

end
Avoiding (Large) Roundoff
Error
Avoid substracting almost-equal
quantities
Avoid dividing by small quantities
Avoid sums over large loops, especially
with different orders of magnitude in
the sum
Avoid recursive calculations, where
errors will accumulate