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 combinationsThese 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 = 3103 +1102 + 0101 +7100

Base 2 numbers: 0,1 3107 = 1 2 4 8 16 32 64 128 256 512 1024

2048 =1211 + 1210 + 029 + 028 + 027 + 026 + 125

+ 024 + 023 + 022 + 121 + 120

=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 possibilitiesTo avoid confusion, where necessary we write 1010= 102=

Integer Representation

Integers obviously fit into this base 2 notationsRemains challenge to represent negative numbers 2s complement Excess-N

Extra choice is order of bitsChoice 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

329.25 = 0.2925 10

In binary, we can use powers of 2

29.25

Floating Point Size

In IEEE.h IEEE.h:#define IEEE_FLOAT_SIZE 4 IEEE.h:#define IEEE_DOUBLE_SIZE 8 IEEE.h:#define IEEE_QUAD_SIZE 16

Distribution

Precision

# bits

MantissaBits

Expon.Bits

SignBit

Single 32 23 8 1

Double 64 52 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 quantitiese.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 quantitiesAvoid dividing by small quantitiesAvoid sums over large loops, especially with different orders of magnitude in the sumAvoid recursive calculations, where errors will accumulate