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