frac/frac.md

Target

In all languages, the target is a function that takes 3 parameters:

• x the number we wish to approximate
• D the maximum denominator
• mixed if true, return a mixed fraction (default); if false, improper

The JS implementation walks through the algorithm.

JS Implementation

In this version, the return value is [quotient, numerator, denominator], where quotient == 0 for improper fractions. The interpretation is x ~ quotient + numerator / denominator where 0 <= numerator < denominator and quotient <= x for negative x.

var frac = function(x, D, mixed) {

The goal is to maintain a feasible fraction (with bounded denominator) below the target and another fraction above the target. The lower bound is floor(x) / 1 and the upper bound is (floor(x) + 1) / 1. We keep track of the numerators and denominators separately:

    var n1 = Math.floor(x), d1 = 1;
    var n2 = n1+1, d2 = 1;

If x is not integral, we bisect using mediants until a denominator exceeds our target:

    if(x !== n1) while(d1 <= D && d2 <= D) {    

The mediant is the sum of the numerators divided by the sum of demoninators:

        var m = (n1 + n2) / (d1 + d2);

If we happened to stumble upon the exact value, then we choose the closer one (the mediant if the denominator is within bounds, or the bound with the larger denominator)

        if(x === m) {
            if(d1 + d2 <= D) d1+=d2, n1+=n2, d2=D+1;
            else if(d1 > d2) d2=D+1;
            else d1=D+1;
            break;
        }

Otherwise shrink the range:

        else if(x < m) n2 = n1+n2, d2 = d1+d2;
        else n1 = n1+n2, d1 = d1+d2;
    }

At this point, d1 > D or d2 > D (but not both -- keep track of how d1 and d2 change). So we merely return the desired values:

    if(d1 > D) d1 = d2, n1 = n2;
    if(!mixed) return [0, n1, d1];
    var q = Math.floor(n1/d1);
    return [q, n1 - q*d1, d1];
};

Finally we put some export jazz:

if(typeof module !== undefined) module.exports = frac;

Tests

var frac;
describe('source', function() { it('should load', function() { frac = require('./'); }); });

Miscellany

frac.js: frac.md
        voc frac.md

.PHONY: test
test: 
        mocha -R spec

Node Ilk

{
    "name": "frac",
    "version": "0.1.0",
    "author": "SheetJS",
    "description": "Rational approximation with bounded denominator",
    "keywords": [ "math", "fraction", "rational", "approximation" ],
    "main": "./frac.js",
    "dependencies": {},
    "devDependencies": {"mocha":""},
    "repository": {
        "type":"git",
        "url": "git://github.com/SheetJS/frac.git"
    },
    "scripts": {
        "test": "make test"
    },
    "bugs": { "url": "https://github.com/SheetJS/frac/issues" },
    "engines": { "node": ">=0.8" }
}