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Floating point numbers

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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 , )

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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.

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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.

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First solution Store the numbers in memory just as they are printed as a string of characters Would be stored as 6 bytes as shown below Note that decimal numbers are in the range 30H to 39H as ascii codes E3735 Full stop char Char for 3

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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.

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Example with Mathematica 5! Out[1]=120 In[2]:=10! Out[2]= In[3]:=50! Out[3]=

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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

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Representing variables Variables are represented as pointers to character strings in this system A= A E3735

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Advantages Arbitrarily precise Needs no special hardware Disadvantages Slow Needs complex memory management

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Binary Coded Decimal (BCD) or Calculator style floating point Note that can be represented as x 10 2 Store this 2 digits to a byte to fixed precision as follows bits overall Each digit uses 4 bits exponentmantissa

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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

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Add floating point 1.Denormalise smaller number so that exponents equal 2.Perform addition 3.Renormalise Eg = E E E E E03

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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 = 120 fact(10) ans = fact(50) ans = e+64 Mathematica 5! Out[1]=120 10! Out[2]= ! Out[3]=

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Loss of precison continued When there is a big difference between the numbers the addition is lost with floating point Octave ans = D+08 Mathematica In[1]:= Out[1]=

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IEEE floating point numbers Institution of Electrical and Electronic Engineers

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Single Precision EF

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Definition N=-1 s x 1.F x 2 E-128 Example In fixed point binary = = x 2 1 In IEEE format this is s=0 E=129, F=10100… thus in IEEE it is S E F 0| | Delete this bit

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Example = -3/8 In fixed point binary = =-1 1 x 1.1 x 2 -2 In IEEE format this is s=1 E=126, F=1000 … thus in IEEE it is S E F 1| |

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Range IEEE * 10 –38 to * IEEE * 10 –308 to * bit 3.37 * 10 –4932 to *

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