126 lines
2.9 KiB
Markdown
126 lines
2.9 KiB
Markdown
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# Target
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In all languages, the target is a function that takes 3 parameters:
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- `x` the number we wish to approximate
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- `D` the maximum denominator
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- `mixed` if true, return a mixed fraction (default); if false, improper
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The JS implementation walks through the algorithm.
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# JS Implementation
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In this version, the return value is `[quotient, numerator, denominator]`,
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where `quotient == 0` for improper fractions. The interpretation is
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`x ~ quotient + numerator / denominator` where `0 <= numerator < denominator`
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and `quotient <= x` for negative `x`.
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```js>frac.js
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var frac = function(x, D, mixed) {
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```
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The goal is to maintain a feasible fraction (with bounded denominator) below
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the target and another fraction above the target. The lower bound is
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`floor(x) / 1` and the upper bound is `(floor(x) + 1) / 1`. We keep track of
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the numerators and denominators separately:
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```
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var n1 = Math.floor(x), d1 = 1;
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var n2 = n1+1, d2 = 1;
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```
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If `x` is not integral, we bisect using mediants until a denominator exceeds
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our target:
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```
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if(x !== n1) while(d1 <= D && d2 <= D) {
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```
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The mediant is the sum of the numerators divided by the sum of demoninators:
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```
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var m = (n1 + n2) / (d1 + d2);
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```
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If we happened to stumble upon the exact value, then we choose the closer one
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(the mediant if the denominator is within bounds, or the bound with the larger
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denominator)
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```
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if(x === m) {
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if(d1 + d2 <= D) d1+=d2, n1+=n2, d2=D+1;
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else if(d1 > d2) d2=D+1;
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else d1=D+1;
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break;
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}
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```
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Otherwise shrink the range:
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```
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else if(x < m) n2 = n1+n2, d2 = d1+d2;
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else n1 = n1+n2, d1 = d1+d2;
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}
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```
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At this point, `d1 > D` or `d2 > D` (but not both -- keep track of how `d1` and
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`d2` change). So we merely return the desired values:
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```
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if(d1 > D) d1 = d2, n1 = n2;
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if(!mixed) return [0, n1, d1];
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var q = Math.floor(n1/d1);
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return [q, n1 - q*d1, d1];
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};
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```
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Finally we put some export jazz:
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```
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if(typeof module !== undefined) module.exports = frac;
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```
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# Tests
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```js>test.js
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var frac;
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describe('source', function() { it('should load', function() { frac = require('./'); }); });
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```
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# Miscellany
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```make>Makefile
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frac.js: frac.md
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voc frac.md
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.PHONY: test
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test:
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mocha -R spec
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```
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## Node Ilk
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```json>package.json
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{
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"name": "frac",
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"version": "0.1.0",
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"author": "SheetJS",
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"description": "Rational approximation with bounded denominator",
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"keywords": [ "math", "fraction", "rational", "approximation" ],
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"main": "./frac.js",
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"dependencies": {},
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"devDependencies": {"mocha":""},
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"repository": {
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"type":"git",
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"url": "git://github.com/SheetJS/frac.git"
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},
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"scripts": {
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"test": "make test"
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},
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"bugs": { "url": "https://github.com/SheetJS/frac/issues" },
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"engines": { "node": ">=0.8" }
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}
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```
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