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Understanding Floating Point Numbers and Computable Reals

Compute real numbers as decimal expansions with finite means; transcendental numbers like Pi; implications for precision with binary coded decimal and IEEE formats.

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Understanding Floating Point Numbers and Computable Reals

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  1. Floating point numbers

  2. 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.)

  3. 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.

  4. 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.

  5. 32 34 39 2E 37 35 First solution • Store the numbers in memory just as they are printed as a string of characters. • 249.75 Would be stored as 6 bytes as shown below Note that decimal numbers are in the range 30H to 39H as ascii codes Full stop char Char for 3

  6. 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.

  7. Example with Mathematica • 5! • Out[1]=120 • In[2]:=10! • Out[2]=3628800 • In[3]:=50! • Out[3]=30414093201713378043612608166064768844377641568960512000000000000

  8. 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”

  9. 32 34 39 2E 37 35 Representing variables • Variables are represented as pointers to character strings in this system • A=249.75 A

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

  11. 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 mantissa exponent 24 97 50 02 Each digit uses 4 bits 32 bits overall

  12. Normalise Convert N to format with one digit in front of the decimal point as follows: • If N>10 then Whilst N>10 divide by 10 and add 1 to the exponent • Else whilst N<1 multiply by 10 and decrement the exponent

  13. Add floating point • Denormalise smaller number so that exponents equal • Perform addition • Renormalise Eg 949.75 + 52.0 = 1002.75 9.49750 E02 → 9.49750 E02 5.20000 E01 →0.52000 E02 + 10.02750 E02 → 1.00275 E03

  14. 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 Mathematica 5! Out[1]=120 10! Out[2]=3628800 50! Out[3]=30414093201713378043612608166064768844377641568960512000000000000 Octave fact(5) ans = 120 fact(10) ans = 3628800 fact(50) ans = 3.0414e+64

  15. Loss of precison continued • When there is a big difference between the numbers the addition is lost with floating point Octave 325000000 + 108 ans = 3.2500D+08 Mathematica In[1]:= 325000000 + 108 Out[1]= 325000108

  16. Institution of Electrical and Electronic Engineers IEEE floating point numbers

  17. Single Precision E F

  18. Definition • N=-1s x 1.F x 2E-128 Example 1 3.25 In fixed point binary = 11.01 = 1.101 x 21 In IEEE format this is s=0 E=129, F=10100… thus in IEEE it is S E F 0|1000 0001|1010 0000 0000 0000 0000 000 Delete this bit

  19. Example 2 -0.375 = -3/8 In fixed point binary = -0.011 =-11 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|0111 1110|1000 0000 0000 0000 0000 000

  20. 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

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