/**
* @license Fraction.js v4.2.0 23/05/2021
* https://www.xarg.org/2014/03/rational-numbers-in-javascript/
*
* Copyright (c) 2021, Robert Eisele (robert@xarg.org)
* Dual licensed under the MIT or GPL Version 2 licenses.
**/
/**
*
* This class offers the possibility to calculate fractions.
* You can pass a fraction in different formats. Either as array, as double, as string or as an integer.
*
* Array/Object form
* [ 0 => <nominator>, 1 => <denominator> ]
* [ n => <nominator>, d => <denominator> ]
*
* Integer form
* - Single integer value
*
* Double form
* - Single double value
*
* String form
* 123.456 - a simple double
* 123/456 - a string fraction
* 123.'456' - a double with repeating decimal places
* 123.(456) - synonym
* 123.45'6' - a double with repeating last place
* 123.45(6) - synonym
*
* Example:
*
* let f = new Fraction("9.4'31'");
* f.mul([-4, 3]).div(4.9);
*
*/
(function(root) {
"use strict";
// Set Identity function to downgrade BigInt to Number if needed
if (!BigInt) BigInt = function(n) { if (isNaN(n)) throw new Error(""); return n; };
const C_ONE = BigInt(1);
const C_ZERO = BigInt(0);
const C_TEN = BigInt(10);
const C_TWO = BigInt(2);
const C_FIVE = BigInt(5);
// Maximum search depth for cyclic rational numbers. 2000 should be more than enough.
// Example: 1/7 = 0.(142857) has 6 repeating decimal places.
// If MAX_CYCLE_LEN gets reduced, long cycles will not be detected and toString() only gets the first 10 digits
const MAX_CYCLE_LEN = 2000;
// Parsed data to avoid calling "new" all the time
const P = {
"s": C_ONE,
"n": C_ZERO,
"d": C_ONE
};
function assign(n, s) {
try {
n = BigInt(n);
} catch (e) {
throw Fraction['InvalidParameter'];
}
return n * s;
}
// Creates a new Fraction internally without the need of the bulky constructor
function newFraction(n, d) {
if (d === C_ZERO) {
throw Fraction['DivisionByZero'];
}
const f = Object.create(Fraction.prototype);
f["s"] = n < C_ZERO ? -C_ONE : C_ONE;
n = n < C_ZERO ? -n : n;
const a = gcd(n, d);
f["n"] = n / a;
f["d"] = d / a;
return f;
}
function factorize(num) {
const factors = {};
let n = num;
let i = C_TWO;
let s = C_FIVE - C_ONE;
while (s <= n) {
while (n % i === C_ZERO) {
n/= i;
factors[i] = (factors[i] || C_ZERO) + C_ONE;
}
s+= C_ONE + C_TWO * i++;
}
if (n !== num) {
if (n > 1)
factors[n] = (factors[n] || C_ZERO) + C_ONE;
} else {
factors[num] = (factors[num] || C_ZERO) + C_ONE;
}
return factors;
}
const parse = function(p1, p2) {
let n = C_ZERO, d = C_ONE, s = C_ONE;
if (p1 === undefined || p1 === null) {
/* void */
} else if (p2 !== undefined) {
n = BigInt(p1);
d = BigInt(p2);
s = n * d;
if (n % C_ONE !== C_ZERO || d % C_ONE !== C_ZERO) {
throw Fraction['NonIntegerParameter'];
}
} else if (typeof p1 === "object") {
if ("d" in p1 && "n" in p1) {
n = BigInt(p1["n"]);
d = BigInt(p1["d"]);
if ("s" in p1)
n*= BigInt(p1["s"]);
} else if (0 in p1) {
n = BigInt(p1[0]);
if (1 in p1)
d = BigInt(p1[1]);
} else if (p1 instanceof BigInt) {
n = BigInt(p1);
} else {
throw Fraction['InvalidParameter'];
}
s = n * d;
} else if (typeof p1 === "bigint") {
n = p1;
s = p1;
d = BigInt(1);
} else if (typeof p1 === "number") {
if (isNaN(p1)) {
throw Fraction['InvalidParameter'];
}
if (p1 < 0) {
s = -C_ONE;
p1 = -p1;
}
if (p1 % 1 === 0) {
n = BigInt(p1);
} else if (p1 > 0) { // check for != 0, scale would become NaN (log(0)), which converges really slow
let z = 1;
let A = 0, B = 1;
let C = 1, D = 1;
let N = 10000000;
if (p1 >= 1) {
z = 10 ** Math.floor(1 + Math.log10(p1));
p1/= z;
}
// Using Farey Sequences
while (B <= N && D <= N) {
let M = (A + C) / (B + D);
if (p1 === M) {
if (B + D <= N) {
n = A + C;
d = B + D;
} else if (D > B) {
n = C;
d = D;
} else {
n = A;
d = B;
}
break;
} else {
if (p1 > M) {
A+= C;
B+= D;
} else {
C+= A;
D+= B;
}
if (B > N) {
n = C;
d = D;
} else {
n = A;
d = B;
}
}
}
n = BigInt(n) * BigInt(z);
d = BigInt(d);
}
} else if (typeof p1 === "string") {
let ndx = 0;
let v = C_ZERO, w = C_ZERO, x = C_ZERO, y = C_ONE, z = C_ONE;
let match = p1.match(/\d+|./g);
if (match === null)
throw Fraction['InvalidParameter'];
if (match[ndx] === '-') {// Check for minus sign at the beginning
s = -C_ONE;
ndx++;
} else if (match[ndx] === '+') {// Check for plus sign at the beginning
ndx++;
}
if (match.length === ndx + 1) { // Check if it's just a simple number "1234"
w = assign(match[ndx++], s);
} else if (match[ndx + 1] === '.' || match[ndx] === '.') { // Check if it's a decimal number
if (match[ndx] !== '.') { // Handle 0.5 and .5
v = assign(match[ndx++], s);
}
ndx++;
// Check for decimal places
if (ndx + 1 === match.length || match[ndx + 1] === '(' && match[ndx + 3] === ')' || match[ndx + 1] === "'" && match[ndx + 3] === "'") {
w = assign(match[ndx], s);
y = C_TEN ** BigInt(match[ndx].length);
ndx++;
}
// Check for repeating places
if (match[ndx] === '(' && match[ndx + 2] === ')' || match[ndx] === "'" && match[ndx + 2] === "'") {
x = assign(match[ndx + 1], s);
z = C_TEN ** BigInt(match[ndx + 1].length) - C_ONE;
ndx+= 3;
}
} else if (match[ndx + 1] === '/' || match[ndx + 1] === ':') { // Check for a simple fraction "123/456" or "123:456"
w = assign(match[ndx], s);
y = assign(match[ndx + 2], C_ONE);
ndx+= 3;
} else if (match[ndx + 3] === '/' && match[ndx + 1] === ' ') { // Check for a complex fraction "123 1/2"
v = assign(match[ndx], s);
w = assign(match[ndx + 2], s);
y = assign(match[ndx + 4], C_ONE);
ndx+= 5;
}
if (match.length <= ndx) { // Check for more tokens on the stack
d = y * z;
s = /* void */
n = x + d * v + z * w;
} else {
throw Fraction['InvalidParameter'];
}
} else {
throw Fraction['InvalidParameter'];
}
if (d === C_ZERO) {
throw Fraction['DivisionByZero'];
}
P["s"] = s < C_ZERO ? -C_ONE : C_ONE;
P["n"] = n < C_ZERO ? -n : n;
P["d"] = d < C_ZERO ? -d : d;
};
function modpow(b, e, m) {
let r = C_ONE;
for (; e > C_ZERO; b = (b * b) % m, e >>= C_ONE) {
if (e & C_ONE) {
r = (r * b) % m;
}
}
return r;
}
function cycleLen(n, d) {
for (; d % C_TWO === C_ZERO;
d/= C_TWO) {
}
for (; d % C_FIVE === C_ZERO;
d/= C_FIVE) {
}
if (d === C_ONE) // Catch non-cyclic numbers
return C_ZERO;
// If we would like to compute really large numbers quicker, we could make use of Fermat's little theorem:
// 10^(d-1) % d == 1
// However, we don't need such large numbers and MAX_CYCLE_LEN should be the capstone,
// as we want to translate the numbers to strings.
let rem = C_TEN % d;
let t = 1;
for (; rem !== C_ONE; t++) {
rem = rem * C_TEN % d;
if (t > MAX_CYCLE_LEN)
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`
}
return BigInt(t);
}
function cycleStart(n, d, len) {
let rem1 = C_ONE;
let rem2 = modpow(C_TEN, len, d);
for (let t = 0; t < 300; t++) { // s < ~log10(Number.MAX_VALUE)
// Solve 10^s == 10^(s+t) (mod d)
if (rem1 === rem2)
return BigInt(t);
rem1 = rem1 * C_TEN % d;
rem2 = rem2 * C_TEN % d;
}
return 0;
}
function gcd(a, b) {
if (!a)
return b;
if (!b)
return a;
while (1) {
a%= b;
if (!a)
return b;
b%= a;
if (!b)
return a;
}
}
/**
* Module constructor
*
* @constructor
* @param {number|Fraction=} a
* @param {number=} b
*/
function Fraction(a, b) {
parse(a, b);
if (this instanceof Fraction) {
a = gcd(P["d"], P["n"]); // Abuse a
this["s"] = P["s"];
this["n"] = P["n"] / a;
this["d"] = P["d"] / a;
} else {
return newFraction(P['s'] * P['n'], P['d']);
}
}
Fraction['DivisionByZero'] = new Error("Division by Zero");
Fraction['InvalidParameter'] = new Error("Invalid argument");
Fraction['NonIntegerParameter'] = new Error("Parameters must be integer");
Fraction.prototype = {
"s": C_ONE,
"n": C_ZERO,
"d": C_ONE,
/**
* Calculates the absolute value
*
* Ex: new Fraction(-4).abs() => 4
**/
"abs": function() {
return newFraction(this["n"], this["d"]);
},
/**
* Inverts the sign of the current fraction
*
* Ex: new Fraction(-4).neg() => 4
**/
"neg": function() {
return newFraction(-this["s"] * this["n"], this["d"]);
},
/**
* Adds two rational numbers
*
* Ex: new Fraction({n: 2, d: 3}).add("14.9") => 467 / 30
**/
"add": function(a, b) {
parse(a, b);
return newFraction(
this["s"] * this["n"] * P["d"] + P["s"] * this["d"] * P["n"],
this["d"] * P["d"]
);
},
/**
* Subtracts two rational numbers
*
* Ex: new Fraction({n: 2, d: 3}).add("14.9") => -427 / 30
**/
"sub": function(a, b) {
parse(a, b);
return newFraction(
this["s"] * this["n"] * P["d"] - P["s"] * this["d"] * P["n"],
this["d"] * P["d"]
);
},
/**
* Multiplies two rational numbers
*
* Ex: new Fraction("-17.(345)").mul(3) => 5776 / 111
**/
"mul": function(a, b) {
parse(a, b);
return newFraction(
this["s"] * P["s"] * this["n"] * P["n"],
this["d"] * P["d"]
);
},
/**
* Divides two rational numbers
*
* Ex: new Fraction("-17.(345)").inverse().div(3)
**/
"div": function(a, b) {
parse(a, b);
return newFraction(
this["s"] * P["s"] * this["n"] * P["d"],
this["d"] * P["n"]
);
},
/**
* Clones the actual object
*
* Ex: new Fraction("-17.(345)").clone()
**/
"clone": function() {
return newFraction(this['s'] * this['n'], this['d']);
},
/**
* Calculates the modulo of two rational numbers - a more precise fmod
*
* Ex: new Fraction('4.(3)').mod([7, 8]) => (13/3) % (7/8) = (5/6)
**/
"mod": function(a, b) {
if (a === undefined) {
return newFraction(this["s"] * this["n"] % this["d"], C_ONE);
}
parse(a, b);
if (0 === P["n"] && 0 === this["d"]) {
throw Fraction['DivisionByZero'];
}
/*
* First silly attempt, kinda slow
*
return that["sub"]({
"n": num["n"] * Math.floor((this.n / this.d) / (num.n / num.d)),
"d": num["d"],
"s": this["s"]
});*/
/*
* New attempt: a1 / b1 = a2 / b2 * q + r
* => b2 * a1 = a2 * b1 * q + b1 * b2 * r
* => (b2 * a1 % a2 * b1) / (b1 * b2)
*/
return newFraction(
this["s"] * (P["d"] * this["n"]) % (P["n"] * this["d"]),
P["d"] * this["d"]
);
},
/**
* Calculates the fractional gcd of two rational numbers
*
* Ex: new Fraction(5,8).gcd(3,7) => 1/56
*/
"gcd": function(a, b) {
parse(a, b);
// gcd(a / b, c / d) = gcd(a, c) / lcm(b, d)
return newFraction(gcd(P["n"], this["n"]) * gcd(P["d"], this["d"]), P["d"] * this["d"]);
},
/**
* Calculates the fractional lcm of two rational numbers
*
* Ex: new Fraction(5,8).lcm(3,7) => 15
*/
"lcm": function(a, b) {
parse(a, b);
// lcm(a / b, c / d) = lcm(a, c) / gcd(b, d)
if (P["n"] === C_ZERO && this["n"] === C_ZERO) {
return newFraction(C_ZERO, C_ONE);
}
return newFraction(P["n"] * this["n"], gcd(P["n"], this["n"]) * gcd(P["d"], this["d"]));
},
/**
* 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"];
dec = dec || 15; // 15 = decimal places when no repitation
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+= N / D;
N%= D;
N*= C_TEN;
if (N)
str+= ".";
if (cycLen) {
for (let i = cycOff; i--;) {
str+= N / D;
N%= D;
N*= C_TEN;
}
str+= "(";
for (let i = cycLen; i--;) {
str+= N / D;
N%= D;
N*= C_TEN;
}
str+= ")";
} else {
for (let i = dec; N && i--;) {
str+= 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 (s['sub'](thisABS)['abs']().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);