# Arithmetic coding issue

Since a few days I am fighting my way through implementing arithmetic coding. I found a really great source of information which made me understand how it should work. Long story short, it implements arithmetic coding on integers using two registers: HIGH and LOW. Those register store a fraction, ex.

``````HIGH = 98765 //means 0.98765(9)
LOW = 91234 // means 0.91234(0)
``````

Then comes magic, when the most significant numbers match (in this case it would be 9s) they are printed out. It's all clear to me.

However, when we take the following example

``````                         HIGH    LOW    RANGE   CUMULATIVE OUTPUT

Initial state           99999  00000   100000
Encode B (0.2-0.3)      29999  20000
Shift out 2             99999  00000   100000    .2
Encode I (0.5-0.6)      59999  50000             .2
Shift out 5             99999  00000   100000    .25
Encode L (0.6-0.8)      79999  60000   20000     .25 //here starts problems
Encode L (0.6-0.8)      75999  72000             .25 //how possible?
Shift out 7             59999  20000   40000     .257
Encode SPACE (0.0-0.1)  23999  20000             .257
Shift out 2             39999  00000   40000     .2572
Encode G (0.4-0.5)      19999  16000             .2572
Shift out 1             99999  60000   40000     .25721
Encode A (0.1-0.2)      67999  64000             .25721
Shift out 6             79999  40000   40000     .257216
Encode T (0.9-1.0)      79999  76000             .257216
Shift out 7             99999  60000   40000     .2572167
Encode E (0.3-0.4)      75999  72000             .2572167
Shift out 7             59999  20000   40000     .25721677
Encode S (0.8-0.9)      55999  52000             .25721677
Shift out 5             59999  20000             .257216775
Shift out 2                                      .2572167752
Shift out 0                                      .25721677520
``````

I really don't know how the marked lines are achieved on computers, on paper it's easy, since:

``````/*
range, HIGH, LOw - integer
symbol->high, symbol->low - real
*/
HIGH = HIGH - (range - symbol->high*range)
LOW = LOW + range*symbol->low
``````

But on computers? The real-number inaccuracy comes in and my intervals are much different.

``````[0; 99999)  100000  b   [0.2; 0.3)
[20000; 29999)  100000  ===> 2
[0; 99999)  100000  i   [0.5; 0.6)
[50000; 59998)  100000  ===> 5 //inaccuracy
[0; 99989)  99990   l   [0.6; 0.8)
[59993; 79990)  19998   l   [0.6; 0.8)
[71991; 75990)  19998   ===> 7
[19910; 59909)  40000       [0; 0.1)
[19910; 23908)  3999    g   [0.4; 0.5)
[21509; 21908)  3999    ===> 2
[15090; 19089)  3999    ===> 1
[50900; 90899)  40000   a   [0.1; 0.2)
[54900; 58898)  40000   ===> 5
[49000; 88989)  39990   t   [0.9; 1)
[84991; 88988)  39990   ===> 8
[49910; 89889)  39980   e   [0.3; 0.4)
[61904; 65900)  39980   ===> 6
[19040; 59009)  39970   s   [0.8; 0.9)
[51016; 55011)  39970   ===> 5
zakodowano: 257215865
``````

I would appreciate any help.

-

Probabilities can be represented as rational numbers with denominator `10^something`, this allows you to divide current interval exactly without rounding errors.
So in your case 0.6 becomes 6/10, 0.8 - 8/10. How you will store such things is your choice. Usually some kind of fixed point format is used. For example you store `60` and know that this is actually `probability*100`, so whenever you multiply by it, you also divide by 100.