892 lines
20 KiB
JavaScript
892 lines
20 KiB
JavaScript
/**
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* @license Fraction.js v4.2.0 05/03/2022
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* https://www.xarg.org/2014/03/rational-numbers-in-javascript/
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*
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* Copyright (c) 2021, Robert Eisele (robert@xarg.org)
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* Dual licensed under the MIT or GPL Version 2 licenses.
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**/
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/**
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*
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* This class offers the possibility to calculate fractions.
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* You can pass a fraction in different formats. Either as array, as double, as string or as an integer.
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*
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* Array/Object form
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* [ 0 => <nominator>, 1 => <denominator> ]
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* [ n => <nominator>, d => <denominator> ]
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*
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* Integer form
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* - Single integer value
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*
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* Double form
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* - Single double value
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*
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* String form
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* 123.456 - a simple double
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* 123/456 - a string fraction
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* 123.'456' - a double with repeating decimal places
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* 123.(456) - synonym
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* 123.45'6' - a double with repeating last place
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* 123.45(6) - synonym
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*
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* Example:
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*
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* var f = new Fraction("9.4'31'");
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* f.mul([-4, 3]).div(4.9);
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*
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*/
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(function(root) {
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"use strict";
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// Maximum search depth for cyclic rational numbers. 2000 should be more than enough.
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// Example: 1/7 = 0.(142857) has 6 repeating decimal places.
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// If MAX_CYCLE_LEN gets reduced, long cycles will not be detected and toString() only gets the first 10 digits
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var MAX_CYCLE_LEN = 2000;
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// Parsed data to avoid calling "new" all the time
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var P = {
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"s": 1,
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"n": 0,
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"d": 1
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};
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function assign(n, s) {
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if (isNaN(n = parseInt(n, 10))) {
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throw Fraction['InvalidParameter'];
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}
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return n * s;
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}
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// Creates a new Fraction internally without the need of the bulky constructor
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function newFraction(n, d) {
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if (d === 0) {
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throw Fraction['DivisionByZero'];
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}
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var f = Object.create(Fraction.prototype);
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f["s"] = n < 0 ? -1 : 1;
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n = n < 0 ? -n : n;
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var a = gcd(n, d);
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f["n"] = n / a;
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f["d"] = d / a;
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return f;
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}
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function factorize(num) {
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var factors = {};
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var n = num;
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var i = 2;
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var s = 4;
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while (s <= n) {
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while (n % i === 0) {
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n/= i;
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factors[i] = (factors[i] || 0) + 1;
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}
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s+= 1 + 2 * i++;
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}
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if (n !== num) {
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if (n > 1)
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factors[n] = (factors[n] || 0) + 1;
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} else {
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factors[num] = (factors[num] || 0) + 1;
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}
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return factors;
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}
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var parse = function(p1, p2) {
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var n = 0, d = 1, s = 1;
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var v = 0, w = 0, x = 0, y = 1, z = 1;
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var A = 0, B = 1;
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var C = 1, D = 1;
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var N = 10000000;
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var M;
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if (p1 === undefined || p1 === null) {
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/* void */
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} else if (p2 !== undefined) {
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n = p1;
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d = p2;
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s = n * d;
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if (n % 1 !== 0 || d % 1 !== 0) {
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throw Fraction['NonIntegerParameter'];
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}
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} else
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switch (typeof p1) {
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case "object":
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{
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if ("d" in p1 && "n" in p1) {
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n = p1["n"];
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d = p1["d"];
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if ("s" in p1)
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n*= p1["s"];
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} else if (0 in p1) {
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n = p1[0];
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if (1 in p1)
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d = p1[1];
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} else {
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throw Fraction['InvalidParameter'];
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}
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s = n * d;
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break;
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}
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case "number":
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{
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if (p1 < 0) {
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s = p1;
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p1 = -p1;
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}
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if (p1 % 1 === 0) {
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n = p1;
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} else if (p1 > 0) { // check for != 0, scale would become NaN (log(0)), which converges really slow
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if (p1 >= 1) {
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z = Math.pow(10, Math.floor(1 + Math.log(p1) / Math.LN10));
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p1/= z;
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}
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// Using Farey Sequences
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// http://www.johndcook.com/blog/2010/10/20/best-rational-approximation/
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while (B <= N && D <= N) {
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M = (A + C) / (B + D);
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if (p1 === M) {
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if (B + D <= N) {
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n = A + C;
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d = B + D;
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} else if (D > B) {
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n = C;
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d = D;
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} else {
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n = A;
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d = B;
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}
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break;
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} else {
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if (p1 > M) {
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A+= C;
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B+= D;
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} else {
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C+= A;
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D+= B;
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}
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if (B > N) {
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n = C;
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d = D;
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} else {
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n = A;
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d = B;
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}
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}
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}
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n*= z;
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} else if (isNaN(p1) || isNaN(p2)) {
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d = n = NaN;
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}
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break;
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}
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case "string":
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{
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B = p1.match(/\d+|./g);
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if (B === null)
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throw Fraction['InvalidParameter'];
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if (B[A] === '-') {// Check for minus sign at the beginning
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s = -1;
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A++;
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} else if (B[A] === '+') {// Check for plus sign at the beginning
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A++;
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}
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if (B.length === A + 1) { // Check if it's just a simple number "1234"
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w = assign(B[A++], s);
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} else if (B[A + 1] === '.' || B[A] === '.') { // Check if it's a decimal number
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if (B[A] !== '.') { // Handle 0.5 and .5
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v = assign(B[A++], s);
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}
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A++;
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// Check for decimal places
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if (A + 1 === B.length || B[A + 1] === '(' && B[A + 3] === ')' || B[A + 1] === "'" && B[A + 3] === "'") {
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w = assign(B[A], s);
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y = Math.pow(10, B[A].length);
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A++;
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}
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// Check for repeating places
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if (B[A] === '(' && B[A + 2] === ')' || B[A] === "'" && B[A + 2] === "'") {
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x = assign(B[A + 1], s);
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z = Math.pow(10, B[A + 1].length) - 1;
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A+= 3;
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}
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} else if (B[A + 1] === '/' || B[A + 1] === ':') { // Check for a simple fraction "123/456" or "123:456"
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w = assign(B[A], s);
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y = assign(B[A + 2], 1);
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A+= 3;
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} else if (B[A + 3] === '/' && B[A + 1] === ' ') { // Check for a complex fraction "123 1/2"
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v = assign(B[A], s);
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w = assign(B[A + 2], s);
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y = assign(B[A + 4], 1);
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A+= 5;
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}
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if (B.length <= A) { // Check for more tokens on the stack
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d = y * z;
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s = /* void */
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n = x + d * v + z * w;
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break;
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}
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/* Fall through on error */
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}
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default:
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throw Fraction['InvalidParameter'];
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}
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if (d === 0) {
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throw Fraction['DivisionByZero'];
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}
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P["s"] = s < 0 ? -1 : 1;
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P["n"] = Math.abs(n);
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P["d"] = Math.abs(d);
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};
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function modpow(b, e, m) {
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var r = 1;
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for (; e > 0; b = (b * b) % m, e >>= 1) {
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if (e & 1) {
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r = (r * b) % m;
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}
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}
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return r;
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}
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function cycleLen(n, d) {
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for (; d % 2 === 0;
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d/= 2) {
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}
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for (; d % 5 === 0;
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d/= 5) {
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}
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if (d === 1) // Catch non-cyclic numbers
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return 0;
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// If we would like to compute really large numbers quicker, we could make use of Fermat's little theorem:
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// 10^(d-1) % d == 1
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// However, we don't need such large numbers and MAX_CYCLE_LEN should be the capstone,
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// as we want to translate the numbers to strings.
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var rem = 10 % d;
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var t = 1;
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for (; rem !== 1; t++) {
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rem = rem * 10 % d;
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if (t > MAX_CYCLE_LEN)
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return 0; // Returning 0 here means that we don't print it as a cyclic number. It's likely that the answer is `d-1`
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}
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return t;
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}
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function cycleStart(n, d, len) {
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var rem1 = 1;
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var rem2 = modpow(10, len, d);
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for (var t = 0; t < 300; t++) { // s < ~log10(Number.MAX_VALUE)
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// Solve 10^s == 10^(s+t) (mod d)
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if (rem1 === rem2)
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return t;
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rem1 = rem1 * 10 % d;
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rem2 = rem2 * 10 % d;
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}
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return 0;
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}
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function gcd(a, b) {
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if (!a)
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return b;
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if (!b)
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return a;
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while (1) {
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a%= b;
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if (!a)
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return b;
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b%= a;
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if (!b)
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return a;
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}
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};
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/**
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* Module constructor
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*
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* @constructor
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* @param {number|Fraction=} a
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* @param {number=} b
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*/
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function Fraction(a, b) {
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parse(a, b);
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if (this instanceof Fraction) {
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a = gcd(P["d"], P["n"]); // Abuse variable a
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this["s"] = P["s"];
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this["n"] = P["n"] / a;
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this["d"] = P["d"] / a;
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} else {
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return newFraction(P['s'] * P['n'], P['d']);
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}
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}
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Fraction['DivisionByZero'] = new Error("Division by Zero");
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Fraction['InvalidParameter'] = new Error("Invalid argument");
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Fraction['NonIntegerParameter'] = new Error("Parameters must be integer");
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Fraction.prototype = {
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"s": 1,
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"n": 0,
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"d": 1,
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/**
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* Calculates the absolute value
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*
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* Ex: new Fraction(-4).abs() => 4
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**/
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"abs": function() {
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return newFraction(this["n"], this["d"]);
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},
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/**
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* Inverts the sign of the current fraction
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*
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* Ex: new Fraction(-4).neg() => 4
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**/
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"neg": function() {
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return newFraction(-this["s"] * this["n"], this["d"]);
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},
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/**
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* Adds two rational numbers
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*
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* Ex: new Fraction({n: 2, d: 3}).add("14.9") => 467 / 30
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**/
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"add": function(a, b) {
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parse(a, b);
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return newFraction(
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this["s"] * this["n"] * P["d"] + P["s"] * this["d"] * P["n"],
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this["d"] * P["d"]
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);
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},
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/**
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* Subtracts two rational numbers
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*
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* Ex: new Fraction({n: 2, d: 3}).add("14.9") => -427 / 30
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**/
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"sub": function(a, b) {
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parse(a, b);
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return newFraction(
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this["s"] * this["n"] * P["d"] - P["s"] * this["d"] * P["n"],
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this["d"] * P["d"]
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);
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},
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/**
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* Multiplies two rational numbers
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*
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* Ex: new Fraction("-17.(345)").mul(3) => 5776 / 111
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**/
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"mul": function(a, b) {
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parse(a, b);
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return newFraction(
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this["s"] * P["s"] * this["n"] * P["n"],
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this["d"] * P["d"]
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);
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},
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/**
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* Divides two rational numbers
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*
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* Ex: new Fraction("-17.(345)").inverse().div(3)
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**/
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"div": function(a, b) {
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parse(a, b);
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return newFraction(
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this["s"] * P["s"] * this["n"] * P["d"],
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this["d"] * P["n"]
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);
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},
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/**
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* Clones the actual object
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*
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* Ex: new Fraction("-17.(345)").clone()
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**/
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"clone": function() {
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return newFraction(this['s'] * this['n'], this['d']);
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},
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/**
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* Calculates the modulo of two rational numbers - a more precise fmod
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*
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* Ex: new Fraction('4.(3)').mod([7, 8]) => (13/3) % (7/8) = (5/6)
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**/
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"mod": function(a, b) {
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if (isNaN(this['n']) || isNaN(this['d'])) {
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return new Fraction(NaN);
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}
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if (a === undefined) {
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return newFraction(this["s"] * this["n"] % this["d"], 1);
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}
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parse(a, b);
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if (0 === P["n"] && 0 === this["d"]) {
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throw Fraction['DivisionByZero'];
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}
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/*
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* First silly attempt, kinda slow
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*
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return that["sub"]({
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"n": num["n"] * Math.floor((this.n / this.d) / (num.n / num.d)),
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"d": num["d"],
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"s": this["s"]
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});*/
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/*
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* New attempt: a1 / b1 = a2 / b2 * q + r
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* => b2 * a1 = a2 * b1 * q + b1 * b2 * r
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* => (b2 * a1 % a2 * b1) / (b1 * b2)
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*/
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return newFraction(
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this["s"] * (P["d"] * this["n"]) % (P["n"] * this["d"]),
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P["d"] * this["d"]
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);
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},
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/**
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* Calculates the fractional gcd of two rational numbers
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*
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* Ex: new Fraction(5,8).gcd(3,7) => 1/56
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*/
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"gcd": function(a, b) {
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parse(a, b);
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// gcd(a / b, c / d) = gcd(a, c) / lcm(b, d)
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return newFraction(gcd(P["n"], this["n"]) * gcd(P["d"], this["d"]), P["d"] * this["d"]);
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},
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/**
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* Calculates the fractional lcm of two rational numbers
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*
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* Ex: new Fraction(5,8).lcm(3,7) => 15
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*/
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"lcm": function(a, b) {
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parse(a, b);
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// lcm(a / b, c / d) = lcm(a, c) / gcd(b, d)
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if (P["n"] === 0 && this["n"] === 0) {
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return newFraction(0, 1);
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}
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return newFraction(P["n"] * this["n"], gcd(P["n"], this["n"]) * gcd(P["d"], this["d"]));
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},
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/**
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* Calculates the ceil of a rational number
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*
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* Ex: new Fraction('4.(3)').ceil() => (5 / 1)
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**/
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"ceil": function(places) {
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places = Math.pow(10, places || 0);
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if (isNaN(this["n"]) || isNaN(this["d"])) {
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return new Fraction(NaN);
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}
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return newFraction(Math.ceil(places * this["s"] * this["n"] / this["d"]), places);
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},
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/**
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* Calculates the floor of a rational number
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*
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* Ex: new Fraction('4.(3)').floor() => (4 / 1)
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**/
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"floor": function(places) {
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places = Math.pow(10, places || 0);
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if (isNaN(this["n"]) || isNaN(this["d"])) {
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return new Fraction(NaN);
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}
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return newFraction(Math.floor(places * this["s"] * this["n"] / this["d"]), places);
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},
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/**
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* Rounds a rational numbers
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*
|
|
* Ex: new Fraction('4.(3)').round() => (4 / 1)
|
|
**/
|
|
"round": function(places) {
|
|
|
|
places = Math.pow(10, places || 0);
|
|
|
|
if (isNaN(this["n"]) || isNaN(this["d"])) {
|
|
return new Fraction(NaN);
|
|
}
|
|
return newFraction(Math.round(places * this["s"] * this["n"] / this["d"]), places);
|
|
},
|
|
|
|
/**
|
|
* Gets the inverse of the fraction, means numerator and denominator are exchanged
|
|
*
|
|
* Ex: new Fraction([-3, 4]).inverse() => -4 / 3
|
|
**/
|
|
"inverse": function() {
|
|
|
|
return newFraction(this["s"] * this["d"], this["n"]);
|
|
},
|
|
|
|
/**
|
|
* Calculates the fraction to some rational exponent, if possible
|
|
*
|
|
* Ex: new Fraction(-1,2).pow(-3) => -8
|
|
*/
|
|
"pow": function(a, b) {
|
|
|
|
parse(a, b);
|
|
|
|
// Trivial case when exp is an integer
|
|
|
|
if (P['d'] === 1) {
|
|
|
|
if (P['s'] < 0) {
|
|
return newFraction(Math.pow(this['s'] * this["d"], P['n']), Math.pow(this["n"], P['n']));
|
|
} else {
|
|
return newFraction(Math.pow(this['s'] * this["n"], P['n']), Math.pow(this["d"], P['n']));
|
|
}
|
|
}
|
|
|
|
// Negative roots become complex
|
|
// (-a/b)^(c/d) = x
|
|
// <=> (-1)^(c/d) * (a/b)^(c/d) = x
|
|
// <=> (cos(pi) + i*sin(pi))^(c/d) * (a/b)^(c/d) = x # rotate 1 by 180°
|
|
// <=> (cos(c*pi/d) + i*sin(c*pi/d)) * (a/b)^(c/d) = x # DeMoivre's formula in Q ( https://proofwiki.org/wiki/De_Moivre%27s_Formula/Rational_Index )
|
|
// From which follows that only for c=0 the root is non-complex. c/d is a reduced fraction, so that sin(c/dpi)=0 occurs for d=1, which is handled by our trivial case.
|
|
if (this['s'] < 0) return null;
|
|
|
|
// Now prime factor n and d
|
|
var N = factorize(this['n']);
|
|
var D = factorize(this['d']);
|
|
|
|
// Exponentiate and take root for n and d individually
|
|
var n = 1;
|
|
var d = 1;
|
|
for (var k in N) {
|
|
if (k === '1') continue;
|
|
if (k === '0') {
|
|
n = 0;
|
|
break;
|
|
}
|
|
N[k]*= P['n'];
|
|
|
|
if (N[k] % P['d'] === 0) {
|
|
N[k]/= P['d'];
|
|
} else return null;
|
|
n*= Math.pow(k, N[k]);
|
|
}
|
|
|
|
for (var k in D) {
|
|
if (k === '1') continue;
|
|
D[k]*= P['n'];
|
|
|
|
if (D[k] % P['d'] === 0) {
|
|
D[k]/= P['d'];
|
|
} else return null;
|
|
d*= Math.pow(k, D[k]);
|
|
}
|
|
|
|
if (P['s'] < 0) {
|
|
return newFraction(d, n);
|
|
}
|
|
return newFraction(n, d);
|
|
},
|
|
|
|
/**
|
|
* Check if two rational numbers are the same
|
|
*
|
|
* Ex: new Fraction(19.6).equals([98, 5]);
|
|
**/
|
|
"equals": function(a, b) {
|
|
|
|
parse(a, b);
|
|
return this["s"] * this["n"] * P["d"] === P["s"] * P["n"] * this["d"]; // Same as compare() === 0
|
|
},
|
|
|
|
/**
|
|
* Check if two rational numbers are the same
|
|
*
|
|
* Ex: new Fraction(19.6).equals([98, 5]);
|
|
**/
|
|
"compare": function(a, b) {
|
|
|
|
parse(a, b);
|
|
var t = (this["s"] * this["n"] * P["d"] - P["s"] * P["n"] * this["d"]);
|
|
return (0 < t) - (t < 0);
|
|
},
|
|
|
|
"simplify": function(eps) {
|
|
|
|
if (isNaN(this['n']) || isNaN(this['d'])) {
|
|
return this;
|
|
}
|
|
|
|
eps = eps || 0.001;
|
|
|
|
var thisABS = this['abs']();
|
|
var cont = thisABS['toContinued']();
|
|
|
|
for (var i = 1; i < cont.length; i++) {
|
|
|
|
var s = newFraction(cont[i - 1], 1);
|
|
for (var k = i - 2; k >= 0; k--) {
|
|
s = s['inverse']()['add'](cont[k]);
|
|
}
|
|
|
|
if (s['sub'](thisABS)['abs']().valueOf() < eps) {
|
|
return s['mul'](this['s']);
|
|
}
|
|
}
|
|
return this;
|
|
},
|
|
|
|
/**
|
|
* Check if two rational numbers are divisible
|
|
*
|
|
* Ex: new Fraction(19.6).divisible(1.5);
|
|
*/
|
|
"divisible": function(a, b) {
|
|
|
|
parse(a, b);
|
|
return !(!(P["n"] * this["d"]) || ((this["n"] * P["d"]) % (P["n"] * this["d"])));
|
|
},
|
|
|
|
/**
|
|
* Returns a decimal representation of the fraction
|
|
*
|
|
* Ex: new Fraction("100.'91823'").valueOf() => 100.91823918239183
|
|
**/
|
|
'valueOf': function() {
|
|
|
|
return this["s"] * this["n"] / this["d"];
|
|
},
|
|
|
|
/**
|
|
* Returns a string-fraction representation of a Fraction object
|
|
*
|
|
* Ex: new Fraction("1.'3'").toFraction(true) => "4 1/3"
|
|
**/
|
|
'toFraction': function(excludeWhole) {
|
|
|
|
var whole, str = "";
|
|
var n = this["n"];
|
|
var d = this["d"];
|
|
if (this["s"] < 0) {
|
|
str+= '-';
|
|
}
|
|
|
|
if (d === 1) {
|
|
str+= n;
|
|
} else {
|
|
|
|
if (excludeWhole && (whole = Math.floor(n / d)) > 0) {
|
|
str+= whole;
|
|
str+= " ";
|
|
n%= d;
|
|
}
|
|
|
|
str+= n;
|
|
str+= '/';
|
|
str+= d;
|
|
}
|
|
return str;
|
|
},
|
|
|
|
/**
|
|
* Returns a latex representation of a Fraction object
|
|
*
|
|
* Ex: new Fraction("1.'3'").toLatex() => "\frac{4}{3}"
|
|
**/
|
|
'toLatex': function(excludeWhole) {
|
|
|
|
var whole, str = "";
|
|
var n = this["n"];
|
|
var d = this["d"];
|
|
if (this["s"] < 0) {
|
|
str+= '-';
|
|
}
|
|
|
|
if (d === 1) {
|
|
str+= n;
|
|
} else {
|
|
|
|
if (excludeWhole && (whole = Math.floor(n / d)) > 0) {
|
|
str+= whole;
|
|
n%= d;
|
|
}
|
|
|
|
str+= "\\frac{";
|
|
str+= n;
|
|
str+= '}{';
|
|
str+= d;
|
|
str+= '}';
|
|
}
|
|
return str;
|
|
},
|
|
|
|
/**
|
|
* Returns an array of continued fraction elements
|
|
*
|
|
* Ex: new Fraction("7/8").toContinued() => [0,1,7]
|
|
*/
|
|
'toContinued': function() {
|
|
|
|
var t;
|
|
var a = this['n'];
|
|
var b = this['d'];
|
|
var res = [];
|
|
|
|
if (isNaN(a) || isNaN(b)) {
|
|
return res;
|
|
}
|
|
|
|
do {
|
|
res.push(Math.floor(a / b));
|
|
t = a % b;
|
|
a = b;
|
|
b = t;
|
|
} while (a !== 1);
|
|
|
|
return res;
|
|
},
|
|
|
|
/**
|
|
* Creates a string representation of a fraction with all digits
|
|
*
|
|
* Ex: new Fraction("100.'91823'").toString() => "100.(91823)"
|
|
**/
|
|
'toString': function(dec) {
|
|
|
|
var N = this["n"];
|
|
var D = this["d"];
|
|
|
|
if (isNaN(N) || isNaN(D)) {
|
|
return "NaN";
|
|
}
|
|
|
|
dec = dec || 15; // 15 = decimal places when no repetation
|
|
|
|
var cycLen = cycleLen(N, D); // Cycle length
|
|
var cycOff = cycleStart(N, D, cycLen); // Cycle start
|
|
|
|
var str = this['s'] < 0 ? "-" : "";
|
|
|
|
str+= N / D | 0;
|
|
|
|
N%= D;
|
|
N*= 10;
|
|
|
|
if (N)
|
|
str+= ".";
|
|
|
|
if (cycLen) {
|
|
|
|
for (var i = cycOff; i--;) {
|
|
str+= N / D | 0;
|
|
N%= D;
|
|
N*= 10;
|
|
}
|
|
str+= "(";
|
|
for (var i = cycLen; i--;) {
|
|
str+= N / D | 0;
|
|
N%= D;
|
|
N*= 10;
|
|
}
|
|
str+= ")";
|
|
} else {
|
|
for (var i = dec; N && i--;) {
|
|
str+= N / D | 0;
|
|
N%= D;
|
|
N*= 10;
|
|
}
|
|
}
|
|
return str;
|
|
}
|
|
};
|
|
|
|
if (typeof define === "function" && define["amd"]) {
|
|
define([], function() {
|
|
return Fraction;
|
|
});
|
|
} else if (typeof exports === "object") {
|
|
Object.defineProperty(Fraction, "__esModule", { 'value': true });
|
|
Fraction['default'] = Fraction;
|
|
Fraction['Fraction'] = Fraction;
|
|
module['exports'] = Fraction;
|
|
} else {
|
|
root['Fraction'] = Fraction;
|
|
}
|
|
|
|
})(this);
|