Calculator
Example Data Table
Example message: BANANA
Auto frequencies give B = 1, A = 3, and N = 2. Those counts become normalized probabilities before interval updates begin.
| Symbol | Count | Probability | Cumulative Low | Cumulative High |
|---|---|---|---|---|
| B | 1 | 0.1667 | 0.0000 | 0.1667 |
| A | 3 | 0.5000 | 0.1667 | 0.6667 |
| N | 2 | 0.3333 | 0.6667 | 1.0000 |
Formula Used
Arithmetic coding repeatedly shrinks one interval inside another. Start with low = 0 and high = 1.
Range: range = high - low
Updated lower bound: newLow = low + range × cumulativeLow(symbol)
Updated upper bound: newHigh = low + range × cumulativeHigh(symbol)
Tag value: any number inside the final interval, commonly (low + high) / 2
Estimated bit length: ceil(-log2(high - low))
Entropy: H = -Σ p(x) log2 p(x)
How to Use This Calculator
- Choose Encode to convert a message into a tag, or Decode to reconstruct a message from a known tag.
- For encoding, enter a message. Leave symbols and values blank for automatic frequencies, or provide your own manual model.
- For decoding, enter the tag, decoded length, symbols, and probabilities or counts.
- Use placeholders like
[space]or[comma]inside the symbol list when needed. - Press Calculate. The result appears below the header and above the form.
- Use the CSV and PDF buttons to export the summary, distribution table, and detailed interval steps.
FAQs
1. What does arithmetic coding do?
Arithmetic coding maps a whole message into one fractional interval. The narrower that interval becomes, the more information it represents. A single number inside the final interval can reproduce the original message when the same probability model is used.
2. Why are probabilities important?
The symbol probabilities determine interval sizes. Frequent symbols receive larger subranges, while rare symbols receive smaller ones. A better probability model usually leads to tighter compression because it matches the message statistics more closely.
3. Can I use raw counts instead of probabilities?
Yes. This calculator automatically normalizes positive values. That means counts like 3,2,1 work the same way as probabilities that sum to one. It is useful when you know symbol frequencies but do not want to convert them manually.
4. Why does decoding require message length?
A tag alone may fit many valid prefixes inside the same interval. The decoded length tells the algorithm how many symbol selections to perform. Without that limit, decoding would not know when to stop.
5. Why can floating-point precision matter?
Arithmetic coding repeatedly multiplies small interval widths. Floating-point rounding can slightly shift bounds, especially for long messages. This calculator is excellent for learning, analysis, and moderate inputs, but production compressors often use integer range coding for stronger numerical stability.
6. What does estimated arithmetic bits mean?
It estimates how many binary digits are needed to identify one value inside the final interval. The formula uses the interval width. Smaller widths imply more precision and therefore more code bits.
7. Does symbol order matter in the manual model?
Yes. The cumulative ranges are built in the exact order you enter symbols. Different orders change subinterval boundaries, though decoding still works correctly as long as encoder and decoder use the same ordered model.
8. What does the Plotly graph show?
The graph shows the active symbol probability distribution used during the current calculation. It makes dominant and rare symbols easy to compare visually, which helps explain why some messages compress more efficiently than others.