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diff --git a/node_modules/fraction.js/README.md b/node_modules/fraction.js/README.md new file mode 100644 index 0000000..7d3f31a --- /dev/null +++ b/node_modules/fraction.js/README.md @@ -0,0 +1,466 @@ +# Fraction.js - ℚ in JavaScript + +[](https://npmjs.org/package/fraction.js "View this project on npm") +[](http://opensource.org/licenses/MIT) + + +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: + +```javascript +1 / 98 * 98 // = 0.9999999999999999 +``` + +If you need more precision or just want a fraction as a result, just include *Fraction.js*: + +```javascript +var Fraction = require('fraction.js'); +// or +import Fraction from 'fraction.js'; +``` + +and give it a trial: + +```javascript +Fraction(1).div(98).mul(98) // = 1 +``` + +Internally, numbers are represented as *numerator / denominator*, which adds just a little overhead. However, the library is written with performance and accuracy in mind, which makes it the perfect basis for [Polynomial.js](https://github.com/infusion/Polynomial.js) and [Math.js](https://github.com/josdejong/mathjs). + +Convert decimal to fraction +=== +The simplest job for fraction.js is to get a fraction out of a decimal: +```javascript +var x = new Fraction(1.88); +var res = x.toFraction(true); // String "1 22/25" +``` + +Examples / Motivation +=== +A simple example might be + +```javascript +var f = new Fraction("9.4'31'"); // 9.4313131313131... +f.mul([-4, 3]).mod("4.'8'"); // 4.88888888888888... +``` +The result is + +```javascript +console.log(f.toFraction()); // -4154 / 1485 +``` +You could of course also access the sign (s), numerator (n) and denominator (d) on your own: +```javascript +f.s * f.n / f.d = -1 * 4154 / 1485 = -2.797306... +``` + +If you would try to calculate it yourself, you would come up with something like: + +```javascript +(9.4313131 * (-4 / 3)) % 4.888888 = -2.797308133... +``` + +Quite okay, but yea - not as accurate as it could be. + + +Laplace Probability +=== +Simple example. What's the probability of throwing a 3, and 1 or 4, and 2 or 4 or 6 with a fair dice? + +P({3}): +```javascript +var p = new Fraction([3].length, 6).toString(); // 0.1(6) +``` + +P({1, 4}): +```javascript +var p = new Fraction([1, 4].length, 6).toString(); // 0.(3) +``` + +P({2, 4, 6}): +```javascript +var p = new Fraction([2, 4, 6].length, 6).toString(); // 0.5 +``` + +Convert degrees/minutes/seconds to precise rational representation: +=== + +57+45/60+17/3600 +```javascript +var deg = 57; // 57° +var min = 45; // 45 Minutes +var sec = 17; // 17 Seconds + +new Fraction(deg).add(min, 60).add(sec, 3600).toString() // -> 57.7547(2) +``` + + +Rational approximation of irrational numbers +=== + +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*. + +Then the following algorithm will generate the rational number besides the binary representation. + +```javascript +var x = "/", s = ""; + +var a = new Fraction(0), + b = new Fraction(1); +for (var n = 0; n <= 10; n++) { + + var c = a.add(b).div(2); + + console.log(n + "\t" + a + "\t" + b + "\t" + c + "\t" + x); + + if (c.add(2).pow(2) < 5) { + a = c; + x = "1"; + } else { + b = c; + x = "0"; + } + s+= x; +} +console.log(s) +``` + +The result is + +``` +n a[n] b[n] c[n] x[n] +0 0/1 1/1 1/2 / +1 0/1 1/2 1/4 0 +2 0/1 1/4 1/8 0 +3 1/8 1/4 3/16 1 +4 3/16 1/4 7/32 1 +5 7/32 1/4 15/64 1 +6 15/64 1/4 31/128 1 +7 15/64 31/128 61/256 0 +8 15/64 61/256 121/512 0 +9 15/64 121/512 241/1024 0 +10 241/1024 121/512 483/2048 1 +``` +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)) + + +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). + + +Get the exact fractional part of a number +--- +```javascript +var f = new Fraction("-6.(3416)"); +console.log("" + f.mod(1).abs()); // 0.(3416) +``` + +Mathematical correct modulo +--- +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: + +```javascript +var a = -1; +var b = 10.99; + +console.log(new Fraction(a) + .mod(b)); // Not correct, usual Modulo + +console.log(new Fraction(a) + .mod(b).add(b).mod(b)); // Correct! Mathematical Modulo +``` + +fmod() impreciseness circumvented +--- +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. + +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. + + +Parser +=== + +Any function (see below) as well as the constructor of the *Fraction* class parses its input and reduce it to the smallest term. + +You can pass either Arrays, Objects, Integers, Doubles or Strings. + +Arrays / Objects +--- +```javascript +new Fraction(numerator, denominator); +new Fraction([numerator, denominator]); +new Fraction({n: numerator, d: denominator}); +``` + +Integers +--- +```javascript +new Fraction(123); +``` + +Doubles +--- +```javascript +new Fraction(55.4); +``` + +**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. + +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. + + +Strings +--- +```javascript +new Fraction("123.45"); +new Fraction("123/45"); // A rational number represented as two decimals, separated by a slash +new Fraction("123:45"); // A rational number represented as two decimals, separated by a colon +new Fraction("4 123/45"); // A rational number represented as a whole number and a fraction +new Fraction("123.'456'"); // Note the quotes, see below! +new Fraction("123.(456)"); // Note the brackets, see below! +new Fraction("123.45'6'"); // Note the quotes, see below! +new Fraction("123.45(6)"); // Note the brackets, see below! +``` + +Two arguments +--- +```javascript +new Fraction(3, 2); // 3/2 = 1.5 +``` + +Repeating decimal places +--- +*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) + +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: + +```javascript +var f = new Fraction("123.32"); +console.log("Bam: " + f.div("33.6(567)")); +``` + +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... + +```javascript +function formatDecimal(str) { + + var comma, pre, offset, pad, times, repeat; + + if (-1 === (comma = str.indexOf("."))) + return str; + + pre = str.substr(0, comma + 1); + str = str.substr(comma + 1); + + for (var i = 0; i < str.length; i++) { + + offset = str.substr(0, i); + + for (var j = 0; j < 5; j++) { + + pad = str.substr(i, j + 1); + + times = Math.ceil((str.length - offset.length) / pad.length); + + repeat = new Array(times + 1).join(pad); // Silly String.repeat hack + + if (0 === (offset + repeat).indexOf(str)) { + return pre + offset + "(" + pad + ")"; + } + } + } + return null; +} + +var f, x = formatDecimal("13.0123123123"); // = 13.0(123) +if (x !== null) { + f = new Fraction(x); +} +``` + +Attributes +=== + +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. + +```javascript +var f = new Fraction('-1/2'); +console.log(f.n); // Numerator: 1 +console.log(f.d); // Denominator: 2 +console.log(f.s); // Sign: -1 +``` + + +Functions +=== + +Fraction abs() +--- +Returns the actual number without any sign information + +Fraction neg() +--- +Returns the actual number with flipped sign in order to get the additive inverse + +Fraction add(n) +--- +Returns the sum of the actual number and the parameter n + +Fraction sub(n) +--- +Returns the difference of the actual number and the parameter n + +Fraction mul(n) +--- +Returns the product of the actual number and the parameter n + +Fraction div(n) +--- +Returns the quotient of the actual number and the parameter n + +Fraction pow(exp) +--- +Returns the power of the actual number, raised to an possible rational exponent. If the result becomes non-rational the function returns `null`. + +Fraction mod(n) +--- +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 like. 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). + +Fraction mod() +--- +Returns the modulus (rest of the division) of the actual object (numerator mod denominator) + +Fraction gcd(n) +--- +Returns the fractional greatest common divisor + +Fraction lcm(n) +--- +Returns the fractional least common multiple + +Fraction ceil([places=0-16]) +--- +Returns the ceiling of a rational number with Math.ceil + +Fraction floor([places=0-16]) +--- +Returns the floor of a rational number with Math.floor + +Fraction round([places=0-16]) +--- +Returns the rational number rounded with Math.round + +Fraction roundTo(multiple) +--- +Rounds a fraction to the closest multiple of another fraction. + +Fraction inverse() +--- +Returns the multiplicative inverse of the actual number (n / d becomes d / n) in order to get the reciprocal + +Fraction simplify([eps=0.001]) +--- +Simplifies the rational number under a certain error threshold. Ex. `0.333` will be `1/3` with `eps=0.001` + +boolean equals(n) +--- +Check if two numbers are equal + +int compare(n) +--- +Compare two numbers. +``` +result < 0: n is greater than actual number +result > 0: n is smaller than actual number +result = 0: n is equal to the actual number +``` + +boolean divisible(n) +--- +Check if two numbers are divisible (n divides this) + +double valueOf() +--- +Returns a decimal representation of the fraction + +String toString([decimalPlaces=15]) +--- +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. + +**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! + +String toLatex(excludeWhole=false) +--- +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. + +The optional boolean parameter indicates if you want to exclude the whole part. "1 1/3" instead of "4/3" + +String toFraction(excludeWhole=false) +--- +Gets a string representation of the fraction + +The optional boolean parameter indicates if you want to exclude the whole part. "1 1/3" instead of "4/3" + +Array toContinued() +--- +Gets an array of the fraction represented as a continued fraction. The first element always contains the whole part. + +```javascript +var f = new Fraction('88/33'); +var c = f.toContinued(); // [2, 1, 2] +``` + +Fraction clone() +--- +Creates a copy of the actual Fraction object + + +Exceptions +=== +If a really hard error occurs (parsing error, division by zero), *fraction.js* throws exceptions! Please make sure you handle them correctly. + + + +Installation +=== +Installing fraction.js is as easy as cloning this repo or use the following command: + +``` +npm install fraction.js +``` + +Using Fraction.js with the browser +=== +```html +<script src="fraction.js"></script> +<script> + console.log(Fraction("123/456")); +</script> +``` + +Using Fraction.js with TypeScript +=== +```js +import Fraction from "fraction.js"; +console.log(Fraction("123/456")); +``` + +Coding Style +=== +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. + + +Precision +=== +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`. If this is not enough, there is `bigfraction.js` shipped experimentally, which relies on `BigInt` and should become the new Fraction.js eventually. + +Testing +=== +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 + +``` +npm test +``` + + +Copyright and licensing +=== +Copyright (c) 2023, [Robert Eisele](https://raw.org/) +Licensed under the MIT license. |