diff options
Diffstat (limited to 'node_modules/fraction.js/README.md')
| -rw-r--r-- | node_modules/fraction.js/README.md | 466 |
1 files changed, 0 insertions, 466 deletions
diff --git a/node_modules/fraction.js/README.md b/node_modules/fraction.js/README.md deleted file mode 100644 index 7d3f31a4..00000000 --- a/node_modules/fraction.js/README.md +++ /dev/null @@ -1,466 +0,0 @@ -# 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. |