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900 lines
21 KiB
900 lines
21 KiB
/**
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* @license Fraction.js v4.2.1 20/08/2023
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* https://www.xarg.org/2014/03/rational-numbers-in-javascript/
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*
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* Copyright (c) 2023, Robert Eisele (robert@raw.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 => <numerator>, 1 => <denominator> ]
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* [ n => <numerator>, 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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* let 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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// Set Identity function to downgrade BigInt to Number if needed
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if (typeof BigInt === 'undefined') BigInt = function(n) { if (isNaN(n)) throw new Error(""); return n; };
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const C_ONE = BigInt(1);
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const C_ZERO = BigInt(0);
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const C_TEN = BigInt(10);
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const C_TWO = BigInt(2);
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const C_FIVE = BigInt(5);
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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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const MAX_CYCLE_LEN = 2000;
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// Parsed data to avoid calling "new" all the time
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const P = {
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"s": C_ONE,
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"n": C_ZERO,
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"d": C_ONE
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};
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function assign(n, s) {
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try {
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n = BigInt(n);
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} catch (e) {
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throw 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 === C_ZERO) {
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throw DivisionByZero();
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}
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const f = Object.create(Fraction.prototype);
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f["s"] = n < C_ZERO ? -C_ONE : C_ONE;
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n = n < C_ZERO ? -n : n;
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const 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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const factors = {};
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let n = num;
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let i = C_TWO;
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let s = C_FIVE - C_ONE;
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while (s <= n) {
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while (n % i === C_ZERO) {
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n/= i;
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factors[i] = (factors[i] || C_ZERO) + C_ONE;
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}
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s+= C_ONE + C_TWO * 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] || C_ZERO) + C_ONE;
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} else {
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factors[num] = (factors[num] || C_ZERO) + C_ONE;
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}
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return factors;
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}
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const parse = function(p1, p2) {
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let n = C_ZERO, d = C_ONE, s = C_ONE;
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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 = BigInt(p1);
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d = BigInt(p2);
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s = n * d;
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if (n % C_ONE !== C_ZERO || d % C_ONE !== C_ZERO) {
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throw NonIntegerParameter();
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}
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} else if (typeof p1 === "object") {
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if ("d" in p1 && "n" in p1) {
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n = BigInt(p1["n"]);
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d = BigInt(p1["d"]);
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if ("s" in p1)
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n*= BigInt(p1["s"]);
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} else if (0 in p1) {
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n = BigInt(p1[0]);
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if (1 in p1)
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d = BigInt(p1[1]);
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} else if (p1 instanceof BigInt) {
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n = BigInt(p1);
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} else {
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throw InvalidParameter();
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}
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s = n * d;
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} else if (typeof p1 === "bigint") {
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n = p1;
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s = p1;
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d = C_ONE;
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} else if (typeof p1 === "number") {
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if (isNaN(p1)) {
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throw InvalidParameter();
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}
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if (p1 < 0) {
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s = -C_ONE;
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p1 = -p1;
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}
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if (p1 % 1 === 0) {
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n = BigInt(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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let z = 1;
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let A = 0, B = 1;
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let C = 1, D = 1;
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let N = 10000000;
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if (p1 >= 1) {
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z = 10 ** Math.floor(1 + Math.log10(p1));
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p1/= z;
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}
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// Using Farey Sequences
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while (B <= N && D <= N) {
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let 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 = BigInt(n) * BigInt(z);
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d = BigInt(d);
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}
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} else if (typeof p1 === "string") {
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let ndx = 0;
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let v = C_ZERO, w = C_ZERO, x = C_ZERO, y = C_ONE, z = C_ONE;
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let match = p1.match(/\d+|./g);
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if (match === null)
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throw InvalidParameter();
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if (match[ndx] === '-') {// Check for minus sign at the beginning
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s = -C_ONE;
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ndx++;
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} else if (match[ndx] === '+') {// Check for plus sign at the beginning
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ndx++;
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}
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if (match.length === ndx + 1) { // Check if it's just a simple number "1234"
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w = assign(match[ndx++], s);
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} else if (match[ndx + 1] === '.' || match[ndx] === '.') { // Check if it's a decimal number
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if (match[ndx] !== '.') { // Handle 0.5 and .5
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v = assign(match[ndx++], s);
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}
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ndx++;
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// Check for decimal places
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if (ndx + 1 === match.length || match[ndx + 1] === '(' && match[ndx + 3] === ')' || match[ndx + 1] === "'" && match[ndx + 3] === "'") {
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w = assign(match[ndx], s);
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y = C_TEN ** BigInt(match[ndx].length);
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ndx++;
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}
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// Check for repeating places
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if (match[ndx] === '(' && match[ndx + 2] === ')' || match[ndx] === "'" && match[ndx + 2] === "'") {
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x = assign(match[ndx + 1], s);
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z = C_TEN ** BigInt(match[ndx + 1].length) - C_ONE;
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ndx+= 3;
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}
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} else if (match[ndx + 1] === '/' || match[ndx + 1] === ':') { // Check for a simple fraction "123/456" or "123:456"
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w = assign(match[ndx], s);
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y = assign(match[ndx + 2], C_ONE);
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ndx+= 3;
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} else if (match[ndx + 3] === '/' && match[ndx + 1] === ' ') { // Check for a complex fraction "123 1/2"
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v = assign(match[ndx], s);
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w = assign(match[ndx + 2], s);
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y = assign(match[ndx + 4], C_ONE);
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ndx+= 5;
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}
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if (match.length <= ndx) { // 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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} else {
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throw InvalidParameter();
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}
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} else {
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throw InvalidParameter();
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}
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if (d === C_ZERO) {
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throw DivisionByZero();
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}
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P["s"] = s < C_ZERO ? -C_ONE : C_ONE;
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P["n"] = n < C_ZERO ? -n : n;
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P["d"] = d < C_ZERO ? -d : d;
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};
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function modpow(b, e, m) {
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let r = C_ONE;
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for (; e > C_ZERO; b = (b * b) % m, e >>= C_ONE) {
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if (e & C_ONE) {
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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 % C_TWO === C_ZERO;
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d/= C_TWO) {
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}
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for (; d % C_FIVE === C_ZERO;
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d/= C_FIVE) {
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}
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if (d === C_ONE) // Catch non-cyclic numbers
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return C_ZERO;
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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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let rem = C_TEN % d;
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let t = 1;
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for (; rem !== C_ONE; t++) {
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rem = rem * C_TEN % d;
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if (t > MAX_CYCLE_LEN)
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return C_ZERO; // 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 BigInt(t);
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}
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function cycleStart(n, d, len) {
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let rem1 = C_ONE;
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let rem2 = modpow(C_TEN, len, d);
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for (let 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 BigInt(t);
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rem1 = rem1 * C_TEN % d;
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rem2 = rem2 * C_TEN % 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 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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var DivisionByZero = function() {return new Error("Division by Zero");};
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var InvalidParameter = function() {return new Error("Invalid argument");};
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var NonIntegerParameter = function() {return new Error("Parameters must be integer");};
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Fraction.prototype = {
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"s": C_ONE,
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"n": C_ZERO,
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"d": C_ONE,
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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 (a === undefined) {
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return newFraction(this["s"] * this["n"] % this["d"], C_ONE);
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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 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"] === C_ZERO && this["n"] === C_ZERO) {
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return newFraction(C_ZERO, C_ONE);
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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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* 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 integer exponent
|
|
*
|
|
* 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'] === C_ONE) {
|
|
|
|
if (P['s'] < C_ZERO) {
|
|
return newFraction((this['s'] * this["d"]) ** P['n'], this["n"] ** P['n']);
|
|
} else {
|
|
return newFraction((this['s'] * this["n"]) ** P['n'], 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
|
|
// <=> (cos(c*pi/d) + i*sin(c*pi/d)) * (a/b)^(c/d) = x # DeMoivre's formula
|
|
// From which follows that only for c=0 the root is non-complex
|
|
if (this['s'] < C_ZERO) return null;
|
|
|
|
// Now prime factor n and d
|
|
let N = factorize(this['n']);
|
|
let D = factorize(this['d']);
|
|
|
|
// Exponentiate and take root for n and d individually
|
|
let n = C_ONE;
|
|
let d = C_ONE;
|
|
for (let k in N) {
|
|
if (k === '1') continue;
|
|
if (k === '0') {
|
|
n = C_ZERO;
|
|
break;
|
|
}
|
|
N[k]*= P['n'];
|
|
|
|
if (N[k] % P['d'] === C_ZERO) {
|
|
N[k]/= P['d'];
|
|
} else return null;
|
|
n*= BigInt(k) ** N[k];
|
|
}
|
|
|
|
for (let k in D) {
|
|
if (k === '1') continue;
|
|
D[k]*= P['n'];
|
|
|
|
if (D[k] % P['d'] === C_ZERO) {
|
|
D[k]/= P['d'];
|
|
} else return null;
|
|
d*= BigInt(k) ** D[k];
|
|
}
|
|
|
|
if (P['s'] < C_ZERO) {
|
|
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);
|
|
let t = (this["s"] * this["n"] * P["d"] - P["s"] * P["n"] * this["d"]);
|
|
|
|
return (C_ZERO < t) - (t < C_ZERO);
|
|
},
|
|
|
|
/**
|
|
* Calculates the ceil of a rational number
|
|
*
|
|
* Ex: new Fraction('4.(3)').ceil() => (5 / 1)
|
|
**/
|
|
"ceil": function(places) {
|
|
|
|
places = C_TEN ** BigInt(places || 0);
|
|
|
|
return newFraction(this["s"] * places * this["n"] / this["d"] +
|
|
(places * this["n"] % this["d"] > C_ZERO && this["s"] >= C_ZERO ? C_ONE : C_ZERO),
|
|
places);
|
|
},
|
|
|
|
/**
|
|
* Calculates the floor of a rational number
|
|
*
|
|
* Ex: new Fraction('4.(3)').floor() => (4 / 1)
|
|
**/
|
|
"floor": function(places) {
|
|
|
|
places = C_TEN ** BigInt(places || 0);
|
|
|
|
return newFraction(this["s"] * places * this["n"] / this["d"] -
|
|
(places * this["n"] % this["d"] > C_ZERO && this["s"] < C_ZERO ? C_ONE : C_ZERO),
|
|
places);
|
|
},
|
|
|
|
/**
|
|
* Rounds a rational numbers
|
|
*
|
|
* Ex: new Fraction('4.(3)').round() => (4 / 1)
|
|
**/
|
|
"round": function(places) {
|
|
|
|
places = C_TEN ** BigInt(places || 0);
|
|
|
|
/* Derivation:
|
|
|
|
s >= 0:
|
|
round(n / d) = trunc(n / d) + (n % d) / d >= 0.5 ? 1 : 0
|
|
= trunc(n / d) + 2(n % d) >= d ? 1 : 0
|
|
s < 0:
|
|
round(n / d) =-trunc(n / d) - (n % d) / d > 0.5 ? 1 : 0
|
|
=-trunc(n / d) - 2(n % d) > d ? 1 : 0
|
|
|
|
=>:
|
|
|
|
round(s * n / d) = s * trunc(n / d) + s * (C + 2(n % d) > d ? 1 : 0)
|
|
where C = s >= 0 ? 1 : 0, to fix the >= for the positve case.
|
|
*/
|
|
|
|
return newFraction(this["s"] * places * this["n"] / this["d"] +
|
|
this["s"] * ((this["s"] >= C_ZERO ? C_ONE : C_ZERO) + C_TWO * (places * this["n"] % this["d"]) > this["d"] ? C_ONE : C_ZERO),
|
|
places);
|
|
},
|
|
|
|
/**
|
|
* 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() {
|
|
// Best we can do so far
|
|
return Number(this["s"] * this["n"]) / Number(this["d"]);
|
|
},
|
|
|
|
/**
|
|
* Creates a string representation of a fraction with all digits
|
|
*
|
|
* Ex: new Fraction("100.'91823'").toString() => "100.(91823)"
|
|
**/
|
|
'toString': function(dec) {
|
|
|
|
let N = this["n"];
|
|
let D = this["d"];
|
|
|
|
function trunc(x) {
|
|
return typeof x === 'bigint' ? x : Math.floor(x);
|
|
}
|
|
|
|
dec = dec || 15; // 15 = decimal places when no repetition
|
|
|
|
let cycLen = cycleLen(N, D); // Cycle length
|
|
let cycOff = cycleStart(N, D, cycLen); // Cycle start
|
|
|
|
let str = this['s'] < C_ZERO ? "-" : "";
|
|
|
|
// Append integer part
|
|
str+= trunc(N / D);
|
|
|
|
N%= D;
|
|
N*= C_TEN;
|
|
|
|
if (N)
|
|
str+= ".";
|
|
|
|
if (cycLen) {
|
|
|
|
for (let i = cycOff; i--;) {
|
|
str+= trunc(N / D);
|
|
N%= D;
|
|
N*= C_TEN;
|
|
}
|
|
str+= "(";
|
|
for (let i = cycLen; i--;) {
|
|
str+= trunc(N / D);
|
|
N%= D;
|
|
N*= C_TEN;
|
|
}
|
|
str+= ")";
|
|
} else {
|
|
for (let i = dec; N && i--;) {
|
|
str+= trunc(N / D);
|
|
N%= D;
|
|
N*= C_TEN;
|
|
}
|
|
}
|
|
return str;
|
|
},
|
|
|
|
/**
|
|
* Returns a string-fraction representation of a Fraction object
|
|
*
|
|
* Ex: new Fraction("1.'3'").toFraction() => "4 1/3"
|
|
**/
|
|
'toFraction': function(excludeWhole) {
|
|
|
|
let n = this["n"];
|
|
let d = this["d"];
|
|
let str = this['s'] < C_ZERO ? "-" : "";
|
|
|
|
if (d === C_ONE) {
|
|
str+= n;
|
|
} else {
|
|
let whole = n / d;
|
|
if (excludeWhole && whole > C_ZERO) {
|
|
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) {
|
|
|
|
let n = this["n"];
|
|
let d = this["d"];
|
|
let str = this['s'] < C_ZERO ? "-" : "";
|
|
|
|
if (d === C_ONE) {
|
|
str+= n;
|
|
} else {
|
|
let whole = n / d;
|
|
if (excludeWhole && whole > C_ZERO) {
|
|
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() {
|
|
|
|
let a = this['n'];
|
|
let b = this['d'];
|
|
let res = [];
|
|
|
|
do {
|
|
res.push(a / b);
|
|
let t = a % b;
|
|
a = b;
|
|
b = t;
|
|
} while (a !== C_ONE);
|
|
|
|
return res;
|
|
},
|
|
|
|
"simplify": function(eps) {
|
|
|
|
eps = eps || 0.001;
|
|
|
|
const thisABS = this['abs']();
|
|
const cont = thisABS['toContinued']();
|
|
|
|
for (let i = 1; i < cont.length; i++) {
|
|
|
|
let s = newFraction(cont[i - 1], C_ONE);
|
|
for (let k = i - 2; k >= 0; k--) {
|
|
s = s['inverse']()['add'](cont[k]);
|
|
}
|
|
|
|
if (Math.abs(s['sub'](thisABS).valueOf()) < eps) {
|
|
return s['mul'](this['s']);
|
|
}
|
|
}
|
|
return this;
|
|
}
|
|
};
|
|
|
|
if (typeof define === "function" && define["amd"]) {
|
|
define([], function() {
|
|
return Fraction;
|
|
});
|
|
} else if (typeof exports === "object") {
|
|
Object.defineProperty(exports, "__esModule", { 'value': true });
|
|
Fraction['default'] = Fraction;
|
|
Fraction['Fraction'] = Fraction;
|
|
module['exports'] = Fraction;
|
|
} else {
|
|
root['Fraction'] = Fraction;
|
|
}
|
|
|
|
})(this);
|