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---
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title: Rational Approximation
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pagination_prev: constellation/ssf
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pagination_next: constellation/crc32
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2024-04-24 08:40:31 +00:00
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---
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<head>
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<script src="https://cdn.sheetjs.com/frac-1.1.3/package/dist/frac.min.js"></script>
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</head>
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The SheetJS `frac` library computes rational approximations to floating point
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numbers with bounded denominator. It is a core component in number formatting,
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powering "Fraction with up to 1 digit" and related number formats.
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The library is also available for standalone use on the SheetJS CDN[^1].
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2024-10-07 21:41:19 +00:00
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Source code and project documentation are hosted on the SheetJS Git server at
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https://git.sheetjs.com/sheetjs/frac
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## Live Demo
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Formatted texts are calculated from the value and maximum denominators.
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Please [report an issue](https://git.sheetjs.com/sheetjs/frac/issues) if a
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particular result does not align with expectations.
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```jsx live
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function SheetJSFrac() {
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const [val, setVal] = React.useState(0.699450515);
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const [text, setText] = React.useState("");
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2024-05-21 17:16:30 +00:00
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if(typeof frac == "undefined") return ( <b>ERROR: Reload this page</b> );
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const fmt = arr => `${(""+arr[1]).padStart(5)} / ${(""+arr[2]).padEnd(5)}`;
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React.useEffect(() => {
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if(typeof frac == "undefined") return setText("ERROR: Reload this page!");
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let v = +val;
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if(!isFinite(v)) return setText(`ERROR: ${val} is not a valid number!`);
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try {
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fmt(frac(val, 9)); setText("");
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} catch(e) { setText("ERROR: " + (e && e.message || e)); }
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}, [val]);
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const n = { textAlign:"right" };
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const g = { backgroundColor:"#C6EFCE", color:"#006100", whiteSpace:"pre-wrap" };
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const b = { backgroundColor:"#FFC7CE", color:"#9C0006" };
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return ( <table>
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<tr><td><b>Number Value</b></td><td colspan="4">
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<input type="text" value={val} onChange={e => setVal(e.target.value)}/>
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</td></tr>
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<tr><td></td><th>Max Denom</th><th>Mediant</th><th>Continued Frac</th></tr>
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{[1,2,3,4,5].map(d => ( <tr>
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<td><b>Up to {d} Digit{d == 1 ? "" : "s"}</b></td>
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<td style={n}><code>{10**d - 1}</code></td>
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<td><code style={text?b:g}>{text||fmt(frac(val,10**d-1))}</code></td>
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<td><code style={text?b:g}>{text||fmt(frac.cont(val,10**d-1))}</code></td>
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</tr> ))}
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</table> );
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}
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```
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## API
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In the browser, the library exports the `frac` global. In NodeJS, the library
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default export is a function.
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#### Algorithms
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The "Mediant" algorithm (`frac` in the browser; the default export in NodeJS)
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calculates the exact solution.
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The "Continued Fractions" algorithm (`frac.cont` in the browser; the `cont`
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field in the NodeJS export) calculates an approximate solution but has better
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worst-case runtime performance.
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Spreadsheet software use these algorithms to render number formats including
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`?/?` and `??/??`. The algorithm choices are summarized in the following table:
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| Spreadsheet Software | Algorithm |
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|:-----------------------------------|:--------------------|
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| [SheetJS](/docs/constellation/ssf) | Continued Fractions |
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| Apple Numbers | Mediant Algorithm |
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| Google Sheets | Mediant Algorithm |
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| Lotus 1-2-3 | (unsupported) |
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| Microsoft Excel | Continued Fractions |
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| Quattro Pro | Continued Fractions |
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| WPS 电子表格 | Mediant Algorithm |
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:::danger LibreOffice bugs
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There are known rounding errors in LibreOffice[^2] which result in inaccurate
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fraction calculations.
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The LibreOffice developers believe these numerical errors are desirable:
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> "We ignore the last two bits for many stuff to improve the user experience."
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It is strongly recommended to use a different spreadsheet tool for accurate data
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processing involving fractions and numeric data.
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:::
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#### Functions
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Both functions accept three arguments:
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```js
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var frac_mediant = frac(value, denominator, mixed);
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var frac_cont = frac.cont(value, denominator, mixed);
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```
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- `value`: original value
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- `D`: maximum denominator (e.g. 99 = "Up to 2 digits")
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- `mixed`: if `true`, return a mixed fraction.
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The return value is an array with three integers:
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```js
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var [ int, num, den ] = result;
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```
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- `int` (first element) represents the integer part of the estimate.
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- `num` (second element) is the numerator of the fraction
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- `den` (second element) is the positive denominator of the fraction
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The estimate can be recovered from the array:
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```js
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var estimate = int + num / den;
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```
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2024-04-25 08:39:55 +00:00
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If `mixed` is `false`, then `int = 0` and `0` < `den` ≤ `D`
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If `mixed` is `true`, then `0` ≤ `num` < `den` ≤ `D`
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:::info Negative Values
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When `mixed` is true, `int` will be the floor of the result. For example, in
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```js
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var result = frac( -0.125 , 9, true);
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```
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the result will be `[ -1, 7, 8 ]`. This is interpreted as
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```
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-0.125 ~ (-1) + (7/8)
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```
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:::
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[^1]: See https://cdn.sheetjs.com/frac/ for more details.
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[^2]: See [issue #83511](https://bugs.documentfoundation.org/show_bug.cgi?id=83511) in the LibreOffice bug tracker.
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