495 lines
15 KiB
Markdown
495 lines
15 KiB
Markdown
# Fraction.js - ℚ in JavaScript
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[](https://npmjs.org/package/fraction.js)
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[](https://travis-ci.org/infusion/Fraction.js)
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[](http://opensource.org/licenses/MIT)
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Tired of inprecise numbers represented by doubles, which have to store rational and irrational numbers like PI or sqrt(2) the same way? Obviously the following problem is preventable:
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```javascript
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1 / 98 * 98 // = 0.9999999999999999
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```
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If you need more precision or just want a fraction as a result, have a look at *Fraction.js*:
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```javascript
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var Fraction = require('fraction.js');
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Fraction(1).div(98).mul(98) // = 1
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```
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Internally, numbers are represented as *numerator / denominator*, which adds just a little overhead. However, the library is written with performance in mind and outperforms any other implementation, as you can see [here](http://jsperf.com/convert-a-rational-number-to-a-babylonian-fractions/28). This basic data-type makes it the perfect basis for [Polynomial.js](https://github.com/infusion/Polynomial.js) and [Math.js](https://github.com/josdejong/mathjs).
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Convert decimal to fraction
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===
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The simplest job for fraction.js is to get a fraction out of a decimal:
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```javascript
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var x = new Fraction(1.88);
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var res = x.toFraction(true); // String "1 22/25"
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```
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Examples / Motivation
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===
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A simple example might be
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```javascript
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var f = new Fraction("9.4'31'"); // 9.4313131313131...
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f.mul([-4, 3]).mod("4.'8'"); // 4.88888888888888...
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```
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The result is
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```javascript
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console.log(f.toFraction()); // -4154 / 1485
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```
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You could of course also access the sign (s), numerator (n) and denominator (d) on your own:
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```javascript
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f.s * f.n / f.d = -1 * 4154 / 1485 = -2.797306...
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```
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If you would try to calculate it yourself, you would come up with something like:
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```javascript
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(9.4313131 * (-4 / 3)) % 4.888888 = -2.797308133...
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```
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Quite okay, but yea - not as accurate as it could be.
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Laplace Probability
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===
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Simple example. What's the probability of throwing a 3, and 1 or 4, and 2 or 4 or 6 with a fair dice?
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P({3}):
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```javascript
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var p = new Fraction([3].length, 6).toString(); // 0.1(6)
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```
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P({1, 4}):
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```javascript
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var p = new Fraction([1, 4].length, 6).toString(); // 0.(3)
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```
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P({2, 4, 6}):
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```javascript
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var p = new Fraction([2, 4, 6].length, 6).toString(); // 0.5
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```
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Convert degrees/minutes/seconds to precise rational representation:
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===
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57+45/60+17/3600
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```javascript
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var deg = 57; // 57°
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var min = 45; // 45 Minutes
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var sec = 17; // 17 Seconds
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new Fraction(deg).add(min, 60).add(sec, 3600).toString() // -> 57.7547(2)
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```
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Rational approximation of irrational numbers
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===
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Now it's getting messy ;d To approximate a number like *sqrt(5) - 2* with a numerator and denominator, you can reformat the equation as follows: *pow(n / d + 2, 2) = 5*.
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Then the following algorithm will generate the rational number besides the binary representation.
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```javascript
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var x = "/", s = "";
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var a = new Fraction(0),
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b = new Fraction(1);
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for (var n = 0; n <= 10; n++) {
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var c = new Fraction(a).add(b).div(2);
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console.log(n + "\t" + a + "\t" + b + "\t" + c + "\t" + x);
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if (c.add(2).pow(2) < 5) {
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a = c;
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x = "1";
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} else {
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b = c;
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x = "0";
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}
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s+= x;
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}
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console.log(s)
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```
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The result is
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```
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n a[n] b[n] c[n] x[n]
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0 0/1 1/1 1/2 /
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1 0/1 1/2 1/4 0
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2 0/1 1/4 1/8 0
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3 1/8 1/4 3/16 1
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4 3/16 1/4 7/32 1
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5 7/32 1/4 15/64 1
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6 15/64 1/4 31/128 1
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7 15/64 31/128 61/256 0
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8 15/64 61/256 121/512 0
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9 15/64 121/512 241/1024 0
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10 241/1024 121/512 483/2048 1
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```
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Thus the approximation after 11 iterations of the bisection method is *483 / 2048* and the binary representation is 0.00111100011 (see [WolframAlpha](http://www.wolframalpha.com/input/?i=sqrt%285%29-2+binary))
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I published another example on how to approximate PI with fraction.js on my [blog](http://www.xarg.org/2014/03/precise-calculations-in-javascript/) (Still not the best idea to approximate irrational numbers, but it illustrates the capabilities of Fraction.js perfectly).
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Get the exact fractional part of a number
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---
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```javascript
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var f = new Fraction("-6.(3416)");
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console.log("" + f.mod(1).abs()); // Will print 0.(3416)
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```
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Mathematical correct modulo
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---
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The behaviour on negative congruences is different to most modulo implementations in computer science. Even the *mod()* function of Fraction.js behaves in the typical way. To solve the problem of having the mathematical correct modulo with Fraction.js you could come up with this:
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```javascript
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var a = -1;
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var b = 10.99;
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console.log(new Fraction(a)
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.mod(b)); // Not correct, usual Modulo
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console.log(new Fraction(a)
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.mod(b).add(b).mod(b)); // Correct! Mathematical Modulo
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```
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fmod() impreciseness circumvented
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---
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It turns out that Fraction.js outperforms almost any fmod() implementation, including JavaScript itself, [php.js](http://phpjs.org/functions/fmod/), C++, Python, Java and even Wolframalpha due to the fact that numbers like 0.05, 0.1, ... are infinite decimal in base 2.
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The equation *fmod(4.55, 0.05)* gives *0.04999999999999957*, wolframalpha says *1/20*. The correct answer should be **zero**, as 0.05 divides 4.55 without any remainder.
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Parser
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===
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Any function (see below) as well as the constructor of the *Fraction* class parses its input and reduce it to the smallest term.
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You can pass either Arrays, Objects, Integers, Doubles or Strings.
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Arrays / Objects
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---
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```javascript
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new Fraction(numerator, denominator);
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new Fraction([numerator, denominator]);
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new Fraction({n: numerator, d: denominator});
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```
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Integers
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---
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```javascript
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new Fraction(123);
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```
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Doubles
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---
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```javascript
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new Fraction(55.4);
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```
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**Note:** If you pass a double as it is, Fraction.js will perform a number analysis based on Farey Sequences. If you concern performance, cache Fraction.js objects and pass arrays/objects.
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The method is really precise, but too large exact numbers, like 1234567.9991829 will result in a wrong approximation. If you want to keep the number as it is, convert it to a string, as the string parser will not perform any further observations. If you have problems with the approximation, in the file `examples/approx.js` is a different approximation algorithm, which might work better in some more specific use-cases.
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Strings
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---
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```javascript
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new Fraction("123.45");
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new Fraction("123/45"); // A rational number represented as two decimals, separated by a slash
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new Fraction("123:45"); // A rational number represented as two decimals, separated by a colon
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new Fraction("4 123/45"); // A rational number represented as a whole number and a fraction
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new Fraction("123.'456'"); // Note the quotes, see below!
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new Fraction("123.(456)"); // Note the brackets, see below!
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new Fraction("123.45'6'"); // Note the quotes, see below!
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new Fraction("123.45(6)"); // Note the brackets, see below!
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```
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Two arguments
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---
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```javascript
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new Fraction(3, 2); // 3/2 = 1.5
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```
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Repeating decimal places
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---
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*Fraction.js* can easily handle repeating decimal places. For example *1/3* is *0.3333...*. There is only one repeating digit. As you can see in the examples above, you can pass a number like *1/3* as "0.'3'" or "0.(3)", which are synonym. There are no tests to parse something like 0.166666666 to 1/6! If you really want to handle this number, wrap around brackets on your own with the function below for example: 0.1(66666666)
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Assume you want to divide 123.32 / 33.6(567). [WolframAlpha](http://www.wolframalpha.com/input/?i=123.32+%2F+%2812453%2F370%29) states that you'll get a period of 1776 digits. *Fraction.js* comes to the same result. Give it a try:
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```javascript
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var f = new Fraction("123.32");
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console.log("Bam: " + f.div("33.6(567)"));
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```
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To automatically make a number like "0.123123123" to something more Fraction.js friendly like "0.(123)", I hacked this little brute force algorithm in a 10 minutes. Improvements are welcome...
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```javascript
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function formatDecimal(str) {
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var comma, pre, offset, pad, times, repeat;
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if (-1 === (comma = str.indexOf(".")))
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return str;
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pre = str.substr(0, comma + 1);
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str = str.substr(comma + 1);
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for (var i = 0; i < str.length; i++) {
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offset = str.substr(0, i);
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for (var j = 0; j < 5; j++) {
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pad = str.substr(i, j + 1);
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times = Math.ceil((str.length - offset.length) / pad.length);
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repeat = new Array(times + 1).join(pad); // Silly String.repeat hack
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if (0 === (offset + repeat).indexOf(str)) {
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return pre + offset + "(" + pad + ")";
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}
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}
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}
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return null;
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}
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var f, x = formatDecimal("13.0123123123"); // = 13.0(123)
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if (x !== null) {
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f = new Fraction(x);
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}
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```
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Attributes
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===
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The Fraction object allows direct access to the numerator, denominator and sign attributes. It is ensured that only the sign-attribute holds sign information so that a sign comparison is only necessary against this attribute.
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```javascript
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var f = new Fraction('-1/2');
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console.log(f.n); // Numerator: 1
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console.log(f.d); // Denominator: 2
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console.log(f.s); // Sign: -1
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```
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Functions
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===
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Fraction abs()
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---
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Returns the actual number without any sign information
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Fraction neg()
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---
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Returns the actual number with flipped sign in order to get the additive inverse
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Fraction add(n)
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---
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Returns the sum of the actual number and the parameter n
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Fraction sub(n)
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---
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Returns the difference of the actual number and the parameter n
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Fraction mul(n)
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---
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Returns the product of the actual number and the parameter n
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Fraction div(n)
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---
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Returns the quotient of the actual number and the parameter n
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Fraction pow(exp)
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---
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Returns the power of the actual number, raised to an integer exponent.
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*Note:* Rational exponents are planned, but would slow down the function a lot, because of a kinda slow root finding algorithm, whether the result will become irrational. So for now, only integer exponents are implemented.
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Fraction mod(n)
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---
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Returns the modulus (rest of the division) of the actual object and n (this % n). It's a much more precise [fmod()](#fmod-impreciseness-circumvented) if you will. Please note that *mod()* is just like the modulo operator of most programming languages. If you want a mathematical correct modulo, see [here](#mathematical-correct-modulo).
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Fraction mod()
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---
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Returns the modulus (rest of the division) of the actual object (numerator mod denominator)
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Fraction gcd(n)
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---
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Returns the fractional greatest common divisor
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Fraction lcm(n)
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---
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Returns the fractional least common multiple
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Fraction ceil([places=0-16])
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---
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Returns the ceiling of a rational number with Math.ceil
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Fraction floor([places=0-16])
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---
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Returns the floor of a rational number with Math.floor
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Fraction round([places=0-16])
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---
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Returns the rational number rounded with Math.round
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Fraction inverse()
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---
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Returns the multiplicative inverse of the actual number (n / d becomes d / n) in order to get the reciprocal
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Fraction simplify([eps=0.001])
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---
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Simplifies the rational number under a certain error threshold. Ex. `0.333` will be `1/3` with `eps=0.001`
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boolean equals(n)
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---
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Check if two numbers are equal
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int compare(n)
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---
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Compare two numbers.
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```
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result < 0: n is greater than actual number
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result > 0: n is smaller than actual number
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result = 0: n is equal to the actual number
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```
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boolean divisible(n)
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---
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Check if two numbers are divisible (n divides this)
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double valueOf()
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---
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Returns a decimal representation of the fraction
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String toString([decimalPlaces=15])
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---
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Generates an exact string representation of the actual object. For repeated decimal places all digits are collected within brackets, like `1/3 = "0.(3)"`. For all other numbers, up to `decimalPlaces` significant digits are collected - which includes trailing zeros if the number is getting truncated. However, `1/2 = "0.5"` without trailing zeros of course.
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**Note:** As `valueOf()` and `toString()` are provided, `toString()` is only called implicitly in a real string context. Using the plus-operator like `"123" + new Fraction` will call valueOf(), because JavaScript tries to combine two primitives first and concatenates them later, as string will be the more dominant type. `alert(new Fraction)` or `String(new Fraction)` on the other hand will do what you expect. If you really want to have control, you should call `toString()` or `valueOf()` explicitly!
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String toLatex(excludeWhole=false)
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---
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Generates an exact LaTeX representation of the actual object. You can see a [live demo](http://www.xarg.org/2014/03/precise-calculations-in-javascript/) on my blog.
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The optional boolean parameter indicates if you want to exclude the whole part. "1 1/3" instead of "4/3"
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String toFraction(excludeWhole=false)
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---
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Gets a string representation of the fraction
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The optional boolean parameter indicates if you want to exclude the whole part. "1 1/3" instead of "4/3"
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Array toContinued()
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---
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Gets an array of the fraction represented as a continued fraction. The first element always contains the whole part.
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```javascript
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var f = new Fraction('88/33');
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var c = f.toContinued(); // [2, 1, 2]
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```
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Fraction clone()
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---
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Creates a copy of the actual Fraction object
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Options
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===
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The library should work without configuring anything. However, there is one global option:
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```javascript
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Fraction.REDUCE = <true|false>
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```
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It tells Fraction.js whether to reduce the fraction or not.
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```javascript
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// Normal behavior
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var f = Fraction(3, 6);
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console.log(f); // 1/2
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// Disable fraction reduction
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Fraction.REDUCE = false;
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var g = Fraction(3, 6);
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console.log(g); // 3/6
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// Back to normal behavior
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Fraction.REDUCE = true;
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var h = Fraction(g);
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console.log(h); // 1/2
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```
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Exceptions
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===
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If a really hard error occurs (parsing error, division by zero), *fraction.js* throws exceptions! Please make sure you handle them correctly.
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Installation
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===
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Installing fraction.js is as easy as cloning this repo or use one of the following commands:
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```
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bower install fraction.js
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```
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or
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```
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npm install fraction.js
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```
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Using Fraction.js with the browser
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===
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```html
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<script src="fraction.js"></script>
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<script>
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console.log(Fraction("123/456"));
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</script>
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```
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Using Fraction.js with require.js
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===
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```html
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<script src="require.js"></script>
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<script>
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requirejs(['fraction.js'],
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function(Fraction) {
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console.log(Fraction("123/456"));
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});
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</script>
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```
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Coding Style
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===
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As every library I publish, fraction.js is also built to be as small as possible after compressing it with Google Closure Compiler in advanced mode. Thus the coding style orientates a little on maxing-out the compression rate. Please make sure you keep this style if you plan to extend the library.
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Precision
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===
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Fraction.js tries to circumvent floating point errors, by having an internal representation of numerator and denominator. As it relies on JavaScript, there is also a limit. The biggest number representable is `|Number.MAX_SAFE_INTEGER / 1|` and the smallest is `|1 / Number.MAX_SAFE_INTEGER|`, with `Number.MAX_SAFE_INTEGER=9007199254740991`.
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Testing
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===
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If you plan to enhance the library, make sure you add test cases and all the previous tests are passing. You can test the library with
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```
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npm test
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```
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Copyright and licensing
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===
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Copyright (c) 2014-2019, [Robert Eisele](https://www.xarg.org/)
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Dual licensed under the MIT or GPL Version 2 licenses.
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