Floating point numbers

Computable reals

“computable numbers may be described briefly as the real numbers whose expressions as a decimal are calculable by finite means.”(A. M. Turing, On Computable Numbers with an Application to the Entschiedungsproblem, Proc. London Mathematical

Soc., Ser. 2 , Vol 42, pages 230-265, 1936-7.)

Look first at decimal reals A real number may be approximated by a

decimal expansion with a determinate decimal point.

As more digits are added to the decimal expansion the precision rises.

Any effective calculation is always finite – if it were not then the calculation would go on for ever.

There is thus a limit to the precision that the reals can be represented as.

Transcendental numbers In principle, transcendental numbers

such as Pi or root 2 have no finite representation

We are always dealing with approximations to them.

We can still treat Pi as a real rather than a rational because there is always an algorithmic step by which we can add another digit to its expansion.

First solution Store the numbers in memory just as they

are printed as a string of characters. 249.75Would be stored as 6 bytes as shown belowNote that decimal numbers are in the range 30H

to 39H as ascii codes

32 34 39 2E 37 35

Full stop charChar for 3

Implications

The number strings can be of variable length.

This allows arbitrary precision. This representation is used in systems

like Mathematica which requires very high accuracy.

Example with Mathematica

5! Out[1]=120 In[2]:=10! Out[2]=3628800 In[3]:=50! Out[3]=3041409320171337804361260816

6064768844377641568960512000000000000

Decimal byte arithmetic

“9”+ “8”= “17” decimal 39H+38H=71H hexadecimal ascii 57+56=113 decimal ascii Adjust by taking 30H=48 away ->

41H=65 If greater than “9”=39H=57 take away

10=0AH and carry 1 Thus 41H-0Ah = 65-10=55=37H so the

answer would be 31H,37H = “17”

Representing variables

Variables are represented as pointers to character strings in this system

A=249.75

A 32 34 39 2E 37 35

Advantages

Arbitrarily precise Needs no special hardwareDisadvantages Slow Needs complex memory management

Binary Coded Decimal (BCD) or Calculator style floating point

Note that 249.75 can be represented as 2.4975 x 102

Store this 2 digits to a byte to fixed precision as follows

24 97 50 02

32 bits overallEach digit uses 4 bits

exponentmantissa

Normalise

Convert N to format with one digit in front of the decimal point as follows:

1. If N>10 then Whilst N>10 divide by 10 and add 1 to the exponent

2. Else whilst N<1 multiply by 10 and decrement the exponent

Add floating point

1. Denormalise smaller number so that exponents equal

2. Perform addition3. RenormaliseEg 949.75 + 52.0 = 1002.759.49750 E02 → 9.49750 E02

5.20000 E01 → 0.52000 E02 +

10.02750 E02 → 1.00275 E03

Note loss of accuracy Compare Octave which uses floating point

numbers with Mathematica which uses full precision arithmetic

Octave floating point gives only 5 figure accuracy

Octave fact(5)ans = 120fact(10)ans = 3628800fact(50)ans = 3.0414e+64

Mathematica5!Out[1]=12010!Out[2]=362880050!Out[3]=30414093201713378043612608166064768844377641568960512000000000000

Loss of precison continued

When there is a big difference between the numbers the addition is lost with floating point

Octave325000000 + 108 ans =

3.2500D+08

Mathematica

In[1]:=

325000000 + 108

Out[1]=

325000108

IEEE floating point numbers

Institution of Electrical and Electronic Engineers

Single Precision

E F

Definition N=-1s x 1.F x 2E-128

Example 13.25In fixed point binary = 11.01 = 1.101 x 21

In IEEE format this iss=0 E=129, F=10100… thus in IEEE it isS E F

0|1000 0001|1010 0000 0000 0000 0000 000

Delete this bit

Example 2

-0.375 = -3/8In fixed point binary = -0.011 =-11 x 1.1 x 2-2

In IEEE format this iss=1 E=126, F=1000 … thus in IEEE it isS E F

1|0111 1110|1000 0000 0000 0000 0000 000

Range

IEEE32 1.17 * 10–38 to +3.40 * 1038

IEEE64 2.23 * 10–308 to +1.79 * 10308

80bit 3.37 * 10–4932 to +1.18 * 104932