bigrat/lib/rational.js

rational.js: tools and libraries using rational numbers.

/*
 * rational.js - Javascript tools and libraries based around rational numbers.
 * Copyright (C) 2013 Dylan Ferris
 *
 * This file is part of rational.js.
 *
 * rational.js is free software: you may redistribute it and/or modify
 * it under the terms of the GNU General Public License as published by
 * the Free Software Foundation, either version 3 of the License, or
 * (at your option) any later version.
 *
 * rational.js is distributed in the hope that it will be useful,
 * but WITHOUT ANY WARRANTY; without even the implied warranty of
 * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
 * GNU General Public License for more details.
 *
 * You should have received a copy of the GNU General Public License
 * along with rational.js.  If not, see <http://www.gnu.org/licenses/>.
 */

(function(_global) {
	"use strict";
	var shim = {};
	if (typeof(exports) === 'undefined') {
		if(typeof define === 'function' && typeof define.amd === 'object' && define.amd) {
			shim.exports = {};
			define(function() {
				return shim.exports;
			});
		} else {
			// browser, define in global
			shim.exports = typeof(window) !== 'undefined' ? window : _global;
		}
	}
	else {
		// commonjs, define in exports
		shim.exports = exports;
	}
	(function(exports) {
var integer={greatest_common_divisor:function(a,b){if(1===b||1===a)return 1;for(var c;0!==b;){c=b;b=a%b;if(!isFinite(b))return 0;a=c}return a}};integer.gcd=integer.greatest_common_divisor;integer.fromRandom=function(a){return Math.random()*(1<<a)>>>0};integer.fromMillitime=function(){return+new Date};integer.fromUnixtime=function(){return 0.001*integer.fromMillitime()|0};"undefined"!==typeof exports&&(exports.integer=integer);(function(a){function b(a,d,e){if(e!==c)return a instanceof b?a:"undefined"===typeof a?f:b.parse(a);for(a=a||[];a.length&&!a[a.length-1];)--a.length;this._d=a;this._s=a.length?d||1:0}var c={};b._construct=function(a,d){return new b(a,d,c)};var d=1E7;b.base=d;b.base_log10=7;var f=new b([],0,c);b.ZERO=f;var e=new b([1],1,c);b.ONE=e;var g=new b(e._d,-1,c);b.M_ONE=g;b._0=f;b._1=e;b.small=[f,e,new b([2],1,c),new b([3],1,c),new b([4],1,c),new b([5],1,c),new b([6],1,c),new b([7],1,c),new b([8],1,c),new b([9],
1,c),new b([10],1,c),new b([11],1,c),new b([12],1,c),new b([13],1,c),new b([14],1,c),new b([15],1,c),new b([16],1,c),new b([17],1,c),new b([18],1,c),new b([19],1,c),new b([20],1,c),new b([21],1,c),new b([22],1,c),new b([23],1,c),new b([24],1,c),new b([25],1,c),new b([26],1,c),new b([27],1,c),new b([28],1,c),new b([29],1,c),new b([30],1,c),new b([31],1,c),new b([32],1,c),new b([33],1,c),new b([34],1,c),new b([35],1,c),new b([36],1,c)];b.digits="0123456789ABCDEFGHIJKLMNOPQRSTUVWXYZ".split("");b.prototype.toString=
function(a){a=+a||10;if(2>a||36<a)throw Error("illegal radix "+a+".");if(0===this._s)return"0";if(10===a){a=0>this._s?"-":"";a+=this._d[this._d.length-1].toString();for(var c=this._d.length-2;0<=c;c--){for(var d=this._d[c].toString();7>d.length;)d="0"+d;a+=d}return a}c=b.digits;a=b.small[a];for(var d=this._s,f=this.abs(),e=[],m;0!==f._s;)m=f.divRem(a),f=m[0],m=m[1],e.push(c[m.valueOf()]);return(0>d?"-":"")+e.reverse().join("")};b.radixRegex=[/^$/,/^$/,/^[01]*$/,/^[012]*$/,/^[0-3]*$/,/^[0-4]*$/,/^[0-5]*$/,
/^[0-6]*$/,/^[0-7]*$/,/^[0-8]*$/,/^[0-9]*$/,/^[0-9aA]*$/,/^[0-9abAB]*$/,/^[0-9abcABC]*$/,/^[0-9a-dA-D]*$/,/^[0-9a-eA-E]*$/,/^[0-9a-fA-F]*$/,/^[0-9a-gA-G]*$/,/^[0-9a-hA-H]*$/,/^[0-9a-iA-I]*$/,/^[0-9a-jA-J]*$/,/^[0-9a-kA-K]*$/,/^[0-9a-lA-L]*$/,/^[0-9a-mA-M]*$/,/^[0-9a-nA-N]*$/,/^[0-9a-oA-O]*$/,/^[0-9a-pA-P]*$/,/^[0-9a-qA-Q]*$/,/^[0-9a-rA-R]*$/,/^[0-9a-sA-S]*$/,/^[0-9a-tA-T]*$/,/^[0-9a-uA-U]*$/,/^[0-9a-vA-V]*$/,/^[0-9a-wA-W]*$/,/^[0-9a-xA-X]*$/,/^[0-9a-yA-Y]*$/,/^[0-9a-zA-Z]*$/];b.parse=function(a,d){function e(a){a=
a.replace(/\s*[*xX]\s*10\s*(\^|\*\*)\s*/,"e");return a.replace(/^([+\-])?(\d+)\.?(\d*)[eE]([+\-]?\d+)$/,function(a,b,c,d,p){p=+p;var f=0>p,e=c.length+p;a=(f?c:d).length;p=(p=Math.abs(p))>=a?p-a+f:0;a=Array(p+1).join("0");c+=d;return(b||"")+(f?c=a+c:c+=a).substr(0,e+=f?a.length:0)+(e<c.length?"."+c.substr(e):"")})}a=a.toString();if("undefined"===typeof d||10===+d)a=e(a);var n=RegExp("^([+\\-]?)("+("undefined"===typeof d?"0[xcb]":16==d?"0x":8==d?"0c":2==d?"0b":"")+")?([0-9a-z]*)(?:\\.\\d*)?$","i").exec(a);
if(n){var q=n[1]||"+",m=n[2]||"",n=n[3]||"";if("undefined"===typeof d)d="0x"===m||"0X"===m?16:"0c"===m||"0C"===m?8:"0b"===m||"0B"===m?2:10;else if(2>d||36<d)throw Error("Illegal radix "+d+".");d=+d;if(!b.radixRegex[d].test(n))throw Error("Bad digit for radix "+d);n=n.replace(/^0+/,"").split("");if(0===n.length)return f;q="-"===q?-1:1;if(10==d){for(m=[];7<=n.length;)m.push(parseInt(n.splice(n.length-b.base_log10,b.base_log10).join(""),10));m.push(parseInt(n.join(""),10));return new b(m,q,c)}m=f;d=
b.small[d];for(var k=b.small,h=0;h<n.length;h++)m=m.multiply(d).add(k[parseInt(n[h],36)]);return new b(m._d,q,c)}throw Error("Invalid BigInteger format: "+a);};b.prototype.add=function(a){if(0===this._s)return b(a);a=b(a);if(0===a._s)return this;if(this._s!==a._s)return a=a.negate(),this.subtract(a);var f=this._d;a=a._d;for(var e=f.length,n=a.length,q=Array(Math.max(e,n)+1),m=Math.min(e,n),k=0,h=0;h<m;h++)k=f[h]+a[h]+k,q[h]=k%d,k=k/d|0;n>e&&(f=a,e=n);for(h=m;k&&h<e;h++)k=f[h]+k,q[h]=k%d,k=k/d|0;for(k&&
(q[h]=k);h<e;h++)q[h]=f[h];return new b(q,this._s,c)};b.prototype.negate=function(){return new b(this._d,-this._s|0,c)};b.prototype.abs=function(){return 0>this._s?this.negate():this};b.prototype.subtract=function(a){if(0===this._s)return b(a).negate();a=b(a);if(0===a._s)return this;if(this._s!==a._s)return a=a.negate(),this.add(a);var e=this;0>this._s&&(e=new b(a._d,1,c),a=new b(this._d,1,c));var s=e.compareAbs(a);if(0===s)return f;if(0>s){var n=a;a=e;e=n}e=e._d;a=a._d;var n=e.length,q=a.length,
m=Array(n),k=0,h,g;for(h=0;h<q;h++)g=e[h]-k-a[h],0>g?(g+=d,k=1):k=0,m[h]=g;for(h=q;h<n;h++){g=e[h]-k;if(0>g)g+=d;else{m[h++]=g;break}m[h]=g}for(;h<n;h++)m[h]=e[h];return new b(m,s,c)};(function(){function a(p,e){for(var f=p._d,r=f.slice(),k=0;;){var h=(f[k]||0)+1;r[k]=h%d;if(h<=d-1)break;++k}return new b(r,e,c)}function f(a,p){for(var e=a._d,r=e.slice(),k=0;;){var h=(e[k]||0)-1;if(0>h)r[k]=h+d;else{r[k]=h;break}++k}return new b(r,p,c)}b.prototype.next=function(){switch(this._s){case 0:return e;case -1:return f(this,
-1);default:return a(this,1)}};b.prototype.prev=function(){switch(this._s){case 0:return g;case -1:return a(this,-1);default:return f(this,1)}}})();b.prototype.compareAbs=function(a){if(this===a)return 0;if(!(a instanceof b)){if(!isFinite(a))return isNaN(a)?a:-1;a=b(a)}if(0===this._s)return 0!==a._s?-1:0;if(0===a._s)return 1;var c=this._d.length,d=a._d.length;if(c<d)return-1;if(c>d)return 1;d=this._d;a=a._d;for(c-=1;0<=c;c--)if(d[c]!==a[c])return d[c]<a[c]?-1:1;return 0};b.prototype.compare=function(a){if(this===
a)return 0;a=b(a);return 0===this._s?-a._s:this._s===a._s?this.compareAbs(a)*this._s:this._s};b.prototype.isUnit=function(){return this===e||this===g||1===this._d.length&&1===this._d[0]};b.prototype.multiply=function(a){if(0===this._s)return f;a=b(a);if(0===a._s)return f;if(this.isUnit())return 0>this._s?a.negate():a;if(a.isUnit())return 0>a._s?this.negate():this;if(this===a)return this.square();var e=this._d.length>=a._d.length,g=(e?this:a)._d,e=(e?a:this)._d,n=g.length,q=e.length,m=n+q,k=Array(m),
h;for(h=0;h<m;h++)k[h]=0;for(h=0;h<q;h++){for(var m=0,l=e[h],w=n+h,v,u=h;u<w;u++)v=k[u]+l*g[u-h]+m,m=v/d|0,k[u]=v%d|0;m&&(v=k[u]+m,k[u]=v%d)}return new b(k,this._s*a._s,c)};b.prototype.multiplySingleDigit=function(a){if(0===a||0===this._s)return f;if(1===a)return this;var e;if(1===this._d.length)return e=this._d[0]*a,e>=d?new b([e%d|0,e/d|0],1,c):new b([e],1,c);if(2===a)return this.add(this);if(this.isUnit())return new b([a],1,c);var g=this._d,n=g.length;e=n+1;for(var q=Array(e),m=0;m<e;m++)q[m]=
0;for(var k=m=0;k<n;k++)e=a*g[k]+m,m=e/d|0,q[k]=e%d|0;m&&(q[k]=m);return new b(q,1,c)};b.prototype.square=function(){if(0===this._s)return f;if(this.isUnit())return e;var a=this._d,g=a.length,l=Array(g+g+1),n,q,m,k;for(k=0;k<g;k++)m=2*k,n=a[k]*a[k],q=n/d|0,l[m]=n%d,l[m+1]=q;for(k=0;k<g;k++){q=0;m=2*k+1;for(var h=k+1;h<g;h++,m++)n=2*a[h]*a[k]+l[m]+q,q=n/d|0,l[m]=n%d;m=g+k;n=q+l[m];q=n/d|0;l[m]=n%d;l[m+1]+=q}return new b(l,1,c)};b.prototype.quotient=function(a){return this.divRem(a)[0]};b.prototype.divide=
b.prototype.quotient;b.prototype.remainder=function(a){return this.divRem(a)[1]};b.prototype.divRem=function(a){a=b(a);if(0===a._s)throw Error("Divide by zero");if(0===this._s)return[f,f];if(1===a._d.length)return this.divRemSmall(a._s*a._d[0]);switch(this.compareAbs(a)){case 0:return[this._s===a._s?e:g,f];case -1:return[f,this]}var l=this._s*a._s,s=a.abs(),n=this._d,q=n.length,m=[],k,h=new b([],0,c);for(h._s=1;q;)if(h._d.unshift(n[--q]),0>h.compareAbs(a))m.push(0);else{if(0===h._s)k=0;else{var t=
h._d.length;k=s._d.length;t=h._d[t-1]*d+h._d[t-2];k=s._d[k-1]*d+s._d[k-2];h._d.length>s._d.length&&(t=(t+1)*d);k=Math.ceil(t/k)}do{t=s.multiplySingleDigit(k);if(0>=t.compareAbs(h))break;k--}while(k);m.push(k);k&&(k=h.subtract(t),h._d=k._d.slice(),0===h._d.length&&(h._s=0))}return[new b(m.reverse(),l,c),new b(h._d,this._s,c)]};b.prototype.divRemSmall=function(a){a=+a;if(0===a)throw Error("Divide by zero");var e=this._s*(0>a?-1:1);a=Math.abs(a);if(1>a||a>=d)throw Error("Argument out of range");if(0===
this._s)return[f,f];if(1===a||-1===a)return[1===e?this.abs():new b(this._d,e,c),f];if(1===this._d.length){var g=new b([this._d[0]/a|0],1,c);a=new b([this._d[0]%a|0],1,c);0>e&&(g=g.negate());0>this._s&&(a=a.negate());return[g,a]}for(var n=this._d.slice(),g=Array(n.length),l=0,m=0,k=0,h;n.length;)l=l*d+n[n.length-1],l<a?(g[k++]=0,n.pop(),m=d*m+l):(h=0===l?0:l/a|0,m=l-a*h,(g[k++]=h)?(n.pop(),l=m):n.pop());a=new b([m],1,c);0>this._s&&(a=a.negate());return[new b(g.reverse(),e,c),a]};b.prototype.isEven=
function(){var a=this._d;return 0===this._s||0===a.length||0===a[0]%2};b.prototype.isOdd=function(){return!this.isEven()};b.prototype.sign=function(){return this._s};b.prototype.isPositive=function(){return 0<this._s};b.prototype.isNegative=function(){return 0>this._s};b.prototype.isZero=function(){return 0===this._s};b.prototype.exp10=function(a){a=+a;if(0===a)return this;if(Math.abs(a)>Number(l))throw Error("exponent too large in BigInteger.exp10");if(0<a){for(var d=new b(this._d.slice(),this._s,
c);7<=a;a-=7)d._d.unshift(0);if(0==a)return d;d._s=1;d=d.multiplySingleDigit(Math.pow(10,a));return 0>this._s?d.negate():d}if(-a>=7*this._d.length)return f;d=new b(this._d.slice(),this._s,c);for(a=-a;7<=a;a-=7)d._d.shift();return 0==a?d:d.divRemSmall(Math.pow(10,a))[0]};b.prototype.pow=function(a){if(this.isUnit())return 0<this._s?this:b(a).isOdd()?this:this.negate();a=b(a);if(0===a._s)return e;if(0>a._s){if(0===this._s)throw Error("Divide by zero");return f}if(0===this._s)return f;if(a.isUnit())return this;
if(0<a.compareAbs(l))throw Error("exponent too large in BigInteger.pow");for(var c=this,d=e,g=b.small[2];a.isPositive()&&(!a.isOdd()||(d=d.multiply(c),!a.isUnit()));)c=c.square(),a=a.quotient(g);return d};b.prototype.modPow=function(a,c){for(var d=e,f=this;a.isPositive();)a.isOdd()&&(d=d.multiply(f).remainder(c)),a=a.quotient(b.small[2]),a.isPositive()&&(f=f.square().remainder(c));return d};b.prototype.log=function(){switch(this._s){case 0:return-Infinity;case -1:return NaN}var a=this._d.length;if(30>
7*a)return Math.log(this.valueOf());var e=Math.ceil(30/7),f=this._d.slice(a-e);return Math.log((new b(f,1,c)).valueOf())+(a-e)*Math.log(d)};b.prototype.valueOf=function(){return parseInt(this.toString(),10)};b.prototype.toJSValue=function(){return parseInt(this.toString(),10)};var l=b(2147483647);b.MAX_EXP=l;(function(){function a(c){return function(a){return c.call(b(a))}}function c(a){return function(c,d){return a.call(b(c),b(d))}}function d(a){return function(c,d,e){return a.call(b(c),b(d),b(e))}}
(function(){var e,f,g="toJSValue isEven isOdd sign isZero isNegative abs isUnit square negate isPositive toString next prev log".split(" "),k="compare remainder divRem subtract add quotient divide multiply pow compareAbs".split(" "),h=["modPow"];for(e=0;e<g.length;e++)f=g[e],b[f]=a(b.prototype[f]);for(e=0;e<k.length;e++)f=k[e],b[f]=c(b.prototype[f]);for(e=0;e<h.length;e++)f=h[e],b[f]=d(b.prototype[f]);b.exp10=function(a,c){return b(a).exp10(c)}})()})();a.BigInteger=b})("undefined"!==typeof exports?
exports:this);if("function"!==typeof BigInteger)var BigInteger=require("../src/biginteger.js").BigInteger;
var bigint={greatest_common_divisor:function(a,b){if(b.isUnit()||a.isUnit())return BigInteger.ONE;for(var c;!b.isZero();)c=b,b=a.remainder(b),a=c;return a},create:function(){return[0]},clone:function(a){for(var b=[],c=0,d=a.length;c<d;c++)b[c]=a[c];return b},copy:function(a,b){a=[];for(var c=0,d=b.length;c<d;c++)a[c]=b[c];return a},abs:function(a,b){a[0]=Math.abs(b[0]);return a},str:function(a){return a[0].toString()},toInteger:function(a){return out[0]},fromInteger:function(a){return[parseInt(a)]}};
bigint.ZERO=bigint.fromInteger(0);bigint.ONE=bigint.fromInteger(1);"undefined"!==typeof exports&&(exports.bigint=bigint);"object"!==typeof integer&&(integer=require("../src/integer.js").integer);if(!RAT_ARRAY_TYPE)var RAT_ARRAY_TYPE=Array;if(!RAT_INFINITESIMAL_PRECISION)var RAT_INFINITESIMAL_PRECISION=Math.pow(2,56);if(!RAT_MAX_LOOPS)var RAT_MAX_LOOPS=16777216;var rat={EPSILON:2E-16};rat.MAX_LOOPS=RAT_MAX_LOOPS;rat.create=function(){var a=new RAT_ARRAY_TYPE(2);a[0]=0;a[1]=1;return a};rat.clone=function(a){var b=new RAT_ARRAY_TYPE(2);b[0]=a[0];b[1]=a[1];return b};
rat.fromValues=function(a,b){var c=new RAT_ARRAY_TYPE(2);c[0]=a;c[1]=b;return rat.normalize(c,c)};rat.copy=function(a,b){a[0]=b[0];a[1]=b[1];return a};rat.set=function(a,b,c){a[0]=b;a[1]=c;return rat.normalize(a,a)};rat.abs=function(a,b){a[0]=Math.abs(b[0]);a[1]=b[1];return a};rat.invert=function(a,b){var c=b[0];a[0]=b[1];a[1]=c;return a};rat.reciprocal=rat.invert;rat.add=function(a,b,c){b[1]===c[1]?(a[0]=b[0]+c[0],a[1]=b[1]):(a[0]=b[0]*c[1]+c[0]*b[1],a[1]=b[1]*c[1]);return rat.normalize(a,a)};
rat.subtract=function(a,b,c){b[1]===c[1]?(a[0]=b[0]-c[0],a[1]=b[1]):(a[0]=b[0]*c[1]-c[0]*b[1],a[1]=b[1]*c[1]);return rat.normalize(a,a)};rat.sub=rat.subtract;rat.multiply=function(a,b,c){a[0]=b[0]*c[0];a[1]=b[1]*c[1];return rat.normalize(a,a)};rat.mul=rat.multiply;rat.mediant=function(a,b,c){a[0]=b[0]+c[0];a[1]=b[1]+c[1];return rat.normalize(a,a)};rat.divide=function(a,b,c){a[0]=b[0]*c[1];a[1]=b[1]*c[0];return rat.normalize(a,a)};rat.div=rat.divide;
rat.equals=function(a,b){return 0===a[0]&&0===b[0]||0===a[1]&&0===b[1]?!0:a[0]===b[0]&&a[1]===b[1]};rat.approximates=function(a,b){if(rat.equals(a,b))return!0;var c=rat.create();rat.sub(c,a,b);rat.abs(c,c);return rat.isLessThan(c,rat.INFINITESIMAL)};rat.isGreaterThan=function(a,b){return rat.equals(a,b)?!1:a[0]*b[1]>b[0]*a[1]};rat.isLessThan=function(a,b){return rat.equals(a,b)?!1:a[0]*b[1]<b[0]*a[1]};rat.isNegative=function(a){return 0>a[0]};
rat.min=function(a,b,c){rat.isLessThan(b,c)?(a[0]=b[0],a[1]=b[1]):(a[0]=c[0],a[1]=c[1]);return a};rat.max=function(a,b,c){rat.isGreaterThan(b,c)?(a[0]=b[0],a[1]=b[1]):(a[0]=c[0],a[1]=c[1]);return a};rat.scalar_multiply=function(a,b,c){a[0]=b[0]*c;a[1]=b[1];return rat.normalize(a,a)};rat.scalar_divide=function(a,b,c){a[0]=b[0];a[1]=b[1]*c;return rat.normalize(a,a)};
rat.normalize=function(a,b){if(isNaN(b[0])||isNaN(b[1])||0===b[0]&&0===b[1])return a[0]=0,a[1]=0,a;if(0===b[0])return a[0]=0,a[1]=1,a;if(0===b[1])return a[0]=1,a[1]=0,a;if(b[0]===b[1])return a[0]=1,a[1]=1,a;0<b[1]?(a[0]=b[0],a[1]=b[1]):(a[0]=-b[0],a[1]=-b[1]);var c=integer.greatest_common_divisor(Math.abs(a[0]),a[1]);1<c&&(a[0]/=c,a[1]/=c);return a};rat.opposite=function(a,b){a[0]=-b[0];a[1]=b[1];return a};rat.negative=rat.opposite;rat.neg=rat.opposite;
rat.power=function(a,b,c){2===c?(a[0]=b[0]*b[0],a[1]=b[1]*b[1]):0<c?(a[0]=Math.pow(b[0],c),a[1]=Math.pow(b[1],c)):0>c?(c=Math.abs(c),a[0]=Math.pow(b[1],c),a[1]=Math.pow(b[0],c)):rat.copy(a,rat.ONE);return rat.normalize(a,a)};rat.pow=rat.power;rat.sqrt=function(a,b){return rat.nthRoot(a,b,2)};
rat.nthRoot=function(a,b,c){if(rat.equals(b,rat.ZERO))return rat.copy(a,rat.ZERO);if(rat.equals(b,rat.ONE))return rat.copy(a,rat.ONE);if(rat.equals(b,rat.INFINITY))return rat.copy(a,rat.INFINITY);if(rat.equals(b,rat.INFINULL))return rat.copy(a,rat.INFINULL);var d=rat.isNegative(b);d&&(b[0]=-b[0]);a=rat.copy(a,rat.ONE);for(var f=[1,0,0,1],e=rat.clone(rat.ONE),g=rat.MAX_LOOPS;!rat.approximates(b,e)&&g--;)rat.isLessThan(b,e)?(f[0]+=f[1],f[2]+=f[3]):(f[1]+=f[0],f[3]+=f[2]),a[0]=f[0]+f[1],a[1]=f[2]+f[3],
rat.pow(e,a,c);d&&(b[0]=-b[0],1===c%2&&rat.neg(a,a));return a};rat.dot=function(a,b){return a[0]*b[0]+a[1]*b[1]};rat.str=function(a){return 1===a[1]?a[0]:a[0]+"/"+a[1]};rat.toDecimal=function(a){return a[0]/a[1]};rat.dec=rat.toDecimal;rat.toInteger=function(a){return Math.round(rat.toDecimal(a))};rat.round=rat.toInteger;rat.floor=function(a){return Math.floor(rat.toDecimal(a))};rat.ceil=function(a){return Math.ceil(rat.toDecimal(a))};rat.fromInteger_copy=function(a,b){a[0]=parseInt(b);a[1]=1;return a};
rat.fromInteger=function(a){return rat.fromInteger_copy(rat.create(),a)};rat.fromIntegerInverse_copy=function(a,b){a[0]=1;a[1]=parseInt(b);0>a[1]&&(a[0]=-a[0],a[1]=-a[1]);return a};rat.fromIntegerInverse=function(a){return rat.fromIntegerInverse_copy(rat.create(),a)};rat.fromDecimal=function(a){return rat.fromDecimal_copy(rat.create(),a)};
rat.fromDecimal_copy=function(a,b){b=parseFloat(b);if(isNaN(b))return rat.copy(a,rat.INFINULL);if(Infinity===b)return rat.copy(a,rat.INFINITY);if(Math.abs(b)<rat.EPSILON)return rat.copy(a,rat.ZERO);if(Math.abs(b-1)<rat.EPSILON)return rat.copy(a,rat.ONE);if(Math.abs(b%1)<rat.EPSILON)return rat.fromInteger_copy(a,b);if(Math.abs(1/b%1)<rat.EPSILON)return rat.fromIntegerInverse_copy(a,Math.round(1/b));rat.copy(a,rat.ONE);for(var c=[1,0,0,1],d=b,f=1,e=rat.MAX_LOOPS;e--&&Math.abs(b-rat.toDecimal(a))>rat.EPSILON;)f=
Math.floor(d),a[0]=f*c[0]+c[2],a[1]=f*c[1]+c[3],d=1/(d-f),c[2]=c[0],c[3]=c[1],c[0]=a[0],c[1]=a[1];return a};rat.fromRandom=function(a){a[0]=0xfffffffffffff*Math.random()<<0;a[1]=Math.abs(0xfffffffffffff*Math.random()<<0);return rat.normalize(a,a)};rat.sin=function(a,b){if(0===b[1])return rat.copy(a,rat.ZERO);rat.scalar_multiply(a,b,2);var c=rat.create();rat.pow(c,b,2);rat.add(c,c,rat.ONE);rat.divide(a,a,c);return a};
rat.cos=function(a,b){if(0===b[1])return rat.neg(a,rat.ONE);var c=rat.create();rat.pow(c,b,2);rat.sub(a,rat.ONE,c);var d=rat.create();rat.add(d,rat.ONE,c);rat.divide(a,a,d);return a};rat.tan=function(a,b){rat.scalar_multiply(a,b,2);var c=rat.create();rat.pow(c,b,2);rat.scalar_multiply(c,c,2);rat.add(a,a,c);rat.pow(c,b,4);rat.sub(c,rat.ONE,c);rat.divide(a,a,c);return a};
rat.toEgyptian=function(a){a=rat.clone(a);rat.abs(a,a);var b=rat.floor(a);b&&rat.sub(a,a,rat.fromInteger(b));if(!a[0])return b.toString();b||(b="");for(var c=1,d=rat.create();1!==a[0];)c++,d=rat.fromValues(1,c),rat.isGreaterThan(a,d)&&(b&&(b+=" + "),b+=rat.str(d),rat.sub(a,a,d));return a?b?b+" + "+rat.str(a):rat.str(a):b?b:"0"};
rat.toBabylonian=function(a){var b="",c=rat.toDecimal(a);a=parseInt(c);for(var c=c-a,d=0,f=0;0<a;)(d=a%60)&&(b=d+" * 60^"+f+(b?" + ":"")+b),a=(a-d)/60,f++;for(f=-1;0<c;)c*=60,d=parseInt(c+1E-13),c-=d,-1E-13>c||(d&&(b+=(b?" + ":"")+d+" * 60^"+f),f--);return b?b:"0"};
rat.traceSternBrocot=function(a){var b="";if(rat.equals(a,rat.ZERO)||rat.equals(a,rat.ONE)||rat.equals(a,rat.INFINITY)||rat.equals(a,rat.INFINULL))return b;if(rat.equals(a,rat.ZERO))return rat.copy(out,rat.ZERO);if(rat.equals(a,rat.ONE))return rat.copy(out,rat.ONE);if(rat.equals(a,rat.INFINITY))return rat.copy(out,rat.INFINITY);if(rat.equals(a,rat.INFINULL))return rat.copy(out,rat.INFINULL);a=rat.clone(a);rat.isNegative(a)&&(a[0]=-a[0]);for(var c=rat.clone(rat.ONE),d=[1,0,0,1],f=0,e=0,g=rat.MAX_LOOPS;!rat.approximates(a,
c)&&g--;)rat.isLessThan(a,c)?(d[0]+=d[1],d[2]+=d[3],e++,f&&(b+="R",1!==f&&(b+=f),f=0,b+=" ")):(d[1]+=d[0],d[3]+=d[2],f++,e&&(b+="L",1!==e&&(b+=e),e=0,b+=" ")),c[0]=d[0]+d[1],c[1]=d[2]+d[3];e?(b+="L",1!==e&&(b+=e)):f&&(b+="R",1!==f&&(b+=f));0>g&&(b+="...");return b};
rat.toContinuedFraction=function(a,b){b="undefined"===typeof b?65536:parseInt(b);if(rat.equals(a,rat.ZERO))return[0];if(rat.equals(a,rat.ONE))return[1];if(rat.equals(a,rat.NEGONE))return[-1];if(rat.equals(a,rat.INFINITY))return[1,0];if(rat.equals(a,rat.INFINULL))return[0,0];var c=rat.clone(a),d=rat.isNegative(c);d&&(c[0]=-c[0]);for(var f=rat.clone(rat.ONE),e=[1,0,0,1],g=1,l=[0],p=l.length-1;!rat.equals(c,f)&&b--;)rat.isLessThan(c,f)?(-1===g?l[p]++:(g=-1,l.push(1),p++),e[0]+=e[1],e[2]+=e[3]):(1===
g?l[p]++:(g=1,l.push(1),p++),e[1]+=e[0],e[3]+=e[2]),f[0]=e[0]+e[1],f[1]=e[2]+e[3];0>b?l.push(0):l[p]++;if(d)for(var r in l)l[r]=-l[r];return l};rat.fromContinuedFraction=function(a,b){rat.fromInteger_copy(a,b[b.length-1]);for(var c=b.length-2;-1<c;c--)rat.invert(a,a),rat.add(a,rat.fromInteger(b[c]),a);return a};rat.dump=function(a){rat.create();return"rat\t"+rat.str(a)+"\n~\t"+rat.toDecimal(a)+"\nCF:\t["+rat.toContinuedFraction(a)+"]\n"};rat.ZERO=rat.fromInteger(0);rat.ONE=rat.fromInteger(1);
rat.NEGONE=rat.fromInteger(-1);rat.INFINITY=rat.fromValues(1,0);rat.INFINULL=rat.fromValues(0,0);rat.INFINITESIMAL=rat.clone([1,RAT_INFINITESIMAL_PRECISION]);rat.PI=rat.fromValues(1320192667429,420230377710);"undefined"!==typeof exports&&(exports.rat=rat);"function"!==typeof BigInteger&&(BigInteger=require("../src/biginteger.js").BigInteger);"object"!==typeof bigint&&(bigint=require("../src/bigint.js").bigint);if(!BIGRAT_INFINITESIMAL_PRECISION)var BIGRAT_INFINITESIMAL_PRECISION=new BigInteger("0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF");if(!BIGRAT_MAX_LOOPS)var BIGRAT_MAX_LOOPS=1073741824;var bigrat={EPSILON:2E-16};bigrat.MAX_LOOPS=BIGRAT_MAX_LOOPS;bigrat.create=function(){var a=[];a[0]=BigInteger.ZERO;a[1]=BigInteger.ONE;return a};
bigrat.clone=function(a){var b=[];b[0]=a[0];b[1]=a[1];return b};bigrat.fromValues=function(a,b){var c=[];c[0]=new BigInteger(a);c[1]=new BigInteger(b);return bigrat.normalize(c,c)};bigrat.copy=function(a,b){a[0]=b[0];a[1]=b[1];return a};bigrat.set=function(a,b,c){a[0]=b;a[1]=c;return bigrat.normalize(a,a)};bigrat.toRat=function(a,b){return rat.set(a,b[0].toJSValue(),b[1].toJSValue())};bigrat.fromRat=function(a,b){return bigrat.set(a,BigInteger(b[0]),BigInteger(b[1]))};
bigrat.abs=function(a,b){a[0]=b[0].abs();a[1]=b[1];return a};bigrat.invert=function(a,b){var c=b[0];a[0]=b[1];a[1]=c;return a};bigrat.reciprocal=bigrat.invert;bigrat.add=function(a,b,c){0===b[1].compare(c[1])?(a[0]=b[0].add(c[0]),a[1]=b[1]):(a[0]=b[0].multiply(c[1]).add(c[0].multiply(b[1])),a[1]=b[1].multiply(c[1]));return bigrat.normalize(a,a)};
bigrat.subtract=function(a,b,c){0===b[1].compare(c[1])?(a[0]=b[0].subtract(c[0]),a[1]=b[1]):(a[0]=b[0].multiply(c[1]).subtract(c[0].multiply(b[1])),a[1]=b[1].multiply(c[1]));return bigrat.normalize(a,a)};bigrat.sub=bigrat.subtract;bigrat.multiply=function(a,b,c){a[0]=b[0].multiply(c[0]);a[1]=b[1].multiply(c[1]);return bigrat.normalize(a,a)};bigrat.mul=bigrat.multiply;bigrat.mediant=function(a,b,c){a[0]=b[0].add(c[0]);a[1]=b[1].add(c[1]);return bigrat.normalize(a,a)};
bigrat.divide=function(a,b,c){a[0]=b[0].multiply(c[1]);a[1]=b[1].multiply(c[0]);return bigrat.normalize(a,a)};bigrat.div=bigrat.divide;bigrat.equals=function(a,b){return a[0].isZero()&&b[0].isZero()||a[1].isZero()&&b[1].isZero()?!0:0===a[0].compare(b[0])&&0===a[1].compare(b[1])};bigrat.approximates=function(a,b){if(bigrat.equals(a,b))return!0;var c=bigrat.create();bigrat.sub(c,a,b);bigrat.abs(c,c);return bigrat.isLessThan(c,bigrat.INFINITESIMAL)};
bigrat.isGreaterThan=function(a,b){return bigrat.equals(a,b)?!1:0<a[0].multiply(b[1]).compare(b[0].multiply(a[1]))};bigrat.isLessThan=function(a,b){return bigrat.equals(a,b)?!1:0>a[0].multiply(b[1]).compare(b[0].multiply(a[1]))};bigrat.isNegative=function(a){return a[0].isNegative()};bigrat.min=function(a,b,c){bigrat.isLessThan(b,c)?(a[0]=b[0],a[1]=b[1]):(a[0]=c[0],a[1]=c[1]);return a};bigrat.max=function(a,b,c){bigrat.isGreaterThan(b,c)?(a[0]=b[0],a[1]=b[1]):(a[0]=c[0],a[1]=c[1]);return a};
bigrat.scalar_multiply=function(a,b,c){a[0]=b[0].multiply(c);a[1]=b[1];return bigrat.normalize(a,a)};bigrat.scalar_divide=function(a,b,c){a[0]=b[0];a[1]=b[1].multiply(c);return bigrat.normalize(a,a)};
bigrat.normalize=function(a,b){if(b[0].isZero()||b[1].isZero())return b;if(0===b[0].compare(b[1]))return bigrat.clone(bigrat.ONE);if(b[1].isNegative())a[0]=b[0].negate(),a[1]=b[1].negate();else if(a[0]=b[0],a[1]=b[1],a[1].isZero())return a;var c=bigint.greatest_common_divisor(a[0].abs(),a[1]);0<c.compare(BigInteger.ONE)&&(a[0]=a[0].quotient(c),a[1]=a[1].quotient(c));return a};bigrat.opposite=function(a,b){a[0]=b[0].negate();a[1]=b[1];return a};bigrat.negative=bigrat.opposite;bigrat.neg=bigrat.opposite;
bigrat.power=function(a,b,c){2===c?(a[0]=b[0].square(),a[1]=b[1].square()):0<c?(a[0]=b[0].pow(c),a[1]=b[1].pow(c)):0>c?(c=Math.abs(c),a[0]=b[1].pow(c),a[1]=b[0].pow(c)):bigrat.copy(a,bigrat.ONE);return a};bigrat.pow=bigrat.power;bigrat.sqrt=function(a,b){return bigrat.nthRoot(a,b,2)};
bigrat.nthRoot=function(a,b,c){if(bigrat.equals(b,bigrat.ZERO))return bigrat.copy(a,bigrat.ZERO);if(bigrat.equals(b,bigrat.ONE))return bigrat.copy(a,bigrat.ONE);if(bigrat.equals(b,bigrat.INFINITY))return bigrat.copy(a,bigrat.INFINITY);if(bigrat.equals(b,bigrat.INFINULL))return bigrat.copy(a,bigrat.INFINULL);var d=bigrat.isNegative(b);d&&(b[0]=b[0].negate());bigrat.copy(a,bigrat.ONE);for(var f=[BigInteger(1),BigInteger(0),BigInteger(0),BigInteger(1)],e=bigrat.clone(bigrat.ONE),g=BIGRAT_MAX_LOOPS;!bigrat.approximates(b,
e)&&g--;)bigrat.isLessThan(b,e)?(f[0]=f[0].add(f[1]),f[2]=f[2].add(f[3])):(f[1]=f[1].add(f[0]),f[3]=f[3].add(f[2])),a[0]=f[0].add(f[1]),a[1]=f[2].add(f[3]),bigrat.pow(e,a,c);d&&(b[0]=b[0].negate(),1===c%2&&bigrat.neg(a,a));return a};bigrat.dot=function(a,b){return a[0].multiply(b[0]).add(a[1].multiply(b[1]))};bigrat.str=function(a){return 0===a[1].compare(BigInteger.ONE)?a[0].toString():a[0].toString()+"/"+a[1].toString()};bigrat.toDecimal=function(a){return a[0].toJSValue()/a[1].toJSValue()};
bigrat.dec=bigrat.toDecimal;bigrat.toInteger=function(a){return a[0].quotient(a[1]).toJSValue()};bigrat.round=bigrat.toInteger;bigrat.toBigInteger=function(a){return a[0].quotient(a[1])};bigrat.floor=function(a){return Math.floor(bigrat.toDecimal(a))};bigrat.ceil=function(a){return Math.ceil(bigrat.toDecimal(a))};bigrat.fromInteger_copy=function(a,b){a[0]=BigInteger(b);a[1]=BigInteger.ONE;return a};bigrat.fromInteger=function(a){return bigrat.fromInteger_copy(bigrat.create(),a)};
bigrat.fromIntegerInverse_copy=function(a,b){a[0]=BigInteger.ONE;a[1]=BigInteger(b);a[1].isNegative()&&(a[0]=a[0].negate(),a[1]=a[1].negate());return a};bigrat.fromIntegerInverse=function(a){return bigrat.fromIntegerInverse_copy(bigrat.create(),a)};bigrat.fromDecimal=function(a){return bigrat.fromDecimal_copy(bigrat.create(),a)};
bigrat.fromDecimal_copy=function(a,b){b=parseFloat(b);if(isNaN(b))return bigrat.copy(a,bigrat.INFINULL);if(Infinity===b)return bigrat.copy(a,bigrat.INFINITY);if(Math.abs(b)<bigrat.EPSILON)return bigrat.copy(a,bigrat.ZERO);if(Math.abs(b-1)<bigrat.EPSILON)return bigrat.copy(a,bigrat.ONE);if(Math.abs(b%1)<bigrat.EPSILON)return bigrat.fromInteger_copy(a,b);if(Math.abs(1/b%1)<bigrat.EPSILON)return bigrat.fromIntegerInverse_copy(a,Math.round(1/b));bigrat.copy(a,bigrat.ONE);for(var c=[BigInteger(1),BigInteger(0),
BigInteger(0),BigInteger(1)],d=b,f=1,e=bigrat.MAX_LOOPS;e--&&Math.abs(b-bigrat.toDecimal(a))>bigrat.EPSILON;)f=Math.floor(d),a[0]=BigInteger(f*c[0]+c[2]),a[1]=BigInteger(f*c[1]+c[3]),d=1/(d-f),c[2]=c[0],c[3]=c[1],c[0]=a[0],c[1]=a[1];return a};bigrat.fromRandom=function(a){a[0]=BigInteger(0xfffffffffffff*Math.random()<<0);a[1]=BigInteger(Math.abs(0xfffffffffffff*Math.random()<<0));return bigrat.normalize(a,a)};
bigrat.sin=function(a,b){if(b[1].isZero())return bigrat.copy(a,bigrat.ZERO);bigrat.scalar_multiply(a,b,2);var c=bigrat.create();bigrat.pow(c,b,2);bigrat.add(c,c,bigrat.ONE);bigrat.divide(a,a,c);return a};bigrat.cos=function(a,b){if(b[1].isZero())return bigrat.neg(a,bigrat.ONE);var c=bigrat.create();bigrat.pow(c,b,2);bigrat.sub(a,bigrat.ONE,c);var d=bigrat.create();bigrat.add(d,bigrat.ONE,c);bigrat.divide(a,a,d);return a};
bigrat.tan=function(a,b){bigrat.scalar_multiply(a,b,2);var c=bigrat.create();bigrat.pow(c,b,2);bigrat.scalar_multiply(c,c,2);bigrat.add(a,a,c);bigrat.pow(c,b,4);bigrat.sub(c,bigrat.ONE,c);bigrat.divide(a,a,c);return a};
bigrat.toEgyptian=function(a){a=bigrat.clone(a);bigrat.abs(a,a);var b=bigrat.floor(a);b&&bigrat.sub(a,a,bigrat.fromInteger(b));if(!a[0])return b.toString();b||(b="");for(var c=1,d=bigrat.create();1!==a[0];)c++,d=bigrat.fromValues(1,c),bigrat.isGreaterThan(a,d)&&(b&&(b+=" + "),b+=bigrat.str(d),bigrat.sub(a,a,d));return a?b?b+" + "+bigrat.str(a):bigrat.str(a):b?b:"0"};
bigrat.toBabylonian=function(a){var b="",c=bigrat.toDecimal(a);a=parseInt(c);for(var c=c-a,d=0,f=0;0<a;)(d=a%60)&&(b=d+" * 60^"+f+(b?" + ":"")+b),a=(a-d)/60,f++;for(f=-1;0<c;)c*=60,d=parseInt(c+1E-13),c-=d,-1E-13>c||(d&&(b+=(b?" + ":"")+d+" * 60^"+f),f--);return b?b:"0"};
bigrat.traceSternBrocot=function(a){var b="";if(bigrat.equals(a,bigrat.ZERO)||bigrat.equals(a,bigrat.ONE)||bigrat.equals(a,bigrat.INFINITY)||bigrat.equals(a,bigrat.INFINULL))return b;a=bigrat.clone(a);bigrat.isNegative(a)&&(a[0]=a[0].negate());for(var c=bigrat.clone(bigrat.ONE),d=[BigInteger(1),BigInteger(0),BigInteger(0),BigInteger(1)],f=0,e=0,g=65536;!bigrat.equals(a,c)&&g--;)bigrat.isLessThan(a,c)?(d[0]=d[0].add(d[1]),d[2]=d[2].add(d[3]),e++,f&&(b+="R",1!==f&&(b+=f),f=0,b+=" ")):(d[1]=d[1].add(d[0]),
d[3]=d[3].add(d[2]),f++,e&&(b+="L",1!==e&&(b+=e),e=0,b+=" ")),c[0]=d[0].add(d[1]),c[1]=d[2].add(d[3]);e?(b+="L",1!==e&&(b+=e)):f&&(b+="R",1!==f&&(b+=f));0>g&&(b+="...");return b};
bigrat.toContinuedFraction=function(a,b){b="undefined"===typeof b?65536:parseInt(b);if(bigrat.equals(a,bigrat.ZERO))return[0];if(bigrat.equals(a,bigrat.ONE))return[1];if(bigrat.equals(a,bigrat.NEGONE))return[-1];if(bigrat.equals(a,bigrat.INFINITY))return[1,0];if(bigrat.equals(a,bigrat.INFINULL))return[0,0];var c=bigrat.clone(a),d=bigrat.isNegative(c);d&&(c[0]=c[0].negate());for(var f=bigrat.clone(bigrat.ONE),e=[BigInteger(1),BigInteger(0),BigInteger(0),BigInteger(1)],g=1,l=[0],p=l.length-1;!bigrat.equals(c,
f)&&b--;)bigrat.isLessThan(c,f)?(-1===g?l[p]++:(g=-1,l.push(1),p++),e[0]=e[0].add(e[1]),e[2]=e[2].add(e[3])):(1===g?l[p]++:(g=1,l.push(1),p++),e[1]=e[1].add(e[0]),e[3]=e[3].add(e[2])),f[0]=e[0].add(e[1]),f[1]=e[2].add(e[3]);0>b?l.push(0):l[p]++;if(d)for(var r in l)l[r]=-l[r];return l};bigrat.fromContinuedFraction=function(a,b){bigrat.fromInteger_copy(a,b[b.length-1]);for(var c=b.length-2;-1<c;c--)bigrat.invert(a,a),bigrat.add(a,bigrat.fromInteger(b[c]),a);return a};
bigrat.fromFactorial=function(a,b){if("undefined"!==typeof factorial.PREPARED[b])return bigrat.fromInteger_copy(a,factorial.PREPARED[b]);var c=factorial.PREPARED.length-1;for(bigrat.fromInteger_copy(a,factorial.PREPARED[c]);c<b;)bigrat.scalar_multiply(a,a,++c);return a};bigrat.dump=function(a){return"bigrat\t"+bigrat.str(a)+"\n~\t"+bigrat.toDecimal(a)+"\nCF:\t["+bigrat.toContinuedFraction(a)+"]\n"};bigrat.ZERO=bigrat.fromInteger(0);bigrat.ONE=bigrat.fromInteger(1);bigrat.NEGONE=bigrat.fromInteger(-1);
bigrat.INFINITY=bigrat.fromValues(1,0);bigrat.INFINULL=bigrat.fromValues(0,0);bigrat.INFINITESIMAL=bigrat.clone([new BigInteger(1),BIGRAT_INFINITESIMAL_PRECISION]);
bigrat.PI=bigrat.fromValues("3141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117067982148086513282306647093844609550582231725359408128481117450284102701938521105559644622948954930381964428810975665933446128475648233786783165271201909145648566923460348610454326648213393607260249141273724587","1000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000");
"undefined"!==typeof exports&&(exports.bigrat=bigrat);"object"!==typeof rat&&(rat=require("../src/rat.js").rat);var rational=function(a,b){this.a=rat.fromValues(parseInt(a),parseInt(b))};rational.prototype.toString=function(){return rat.str(this.a)};rational.prototype.toContinuedFraction=function(a){return rat.toContinuedFraction(this.a,a)};rational.prototype.invert=function(){return rat.invert(this.a)};rational.prototype.reciprocal=rational.prototype.invert;
rational.prototype.add=function(a){var b=rat.create();rat.add(b,this.a,a.a);return new rational(b[0],b[1])};rational.prototype.plus=rational.prototype.add;rational.prototype.subtract=function(a){var b=rat.create();rat.sub(b,this.a,a.a);return new rational(b[0],b[1])};rational.prototype.sub=rational.prototype.subtract;rational.prototype.minus=rational.prototype.subtract;rational.prototype.multiply=function(a){var b=rat.create();rat.mul(b,this.a,a.a);return new rational(b[0],b[1])};
rational.prototype.mul=rational.prototype.multiply;rational.prototype.times=rational.prototype.multiply;rational.prototype.mediant=function(a){var b=rat.create();rat.mediant(b,this.a,a.a);return new rational(b[0],b[1])};rational.prototype.divide=function(a){var b=rat.create();rat.div(b,this.a,a.a);return new rational(b[0],b[1])};rational.prototype.div=rational.prototype.divide;rational.prototype.divided_by=rational.prototype.divide;
rational.prototype.power=function(a){var b=rat.create();rat.pow(b,this.a,a);return new rational(b[0],b[1])};rational.prototype.pow=rational.prototype.power;rational.prototype.dump=function(){return rat.dump(this.a)};rational.prototype.toRat=function(){return rat.clone(this.a)};rational.prototype.toDecimal=function(){return rat.toDecimal(this.a)};rational.fromRat=function(a){var b=new rational;b.a[0]=a[0];b.a[1]=a[1];return b};rational.fromInteger=function(a){return rational.fromRat(rat.fromInteger(a))};
rational.fromIntegerInverse=function(a){return rational.fromRat(rat.fromIntegerInverse(a))};rational.fromDecimal=function(a){return rational.fromRat(rat.fromDecimal(a))};rational.fromContinuedFraction=function(a){return rational.fromRat(rat.fromContinuedFraction(rat.create(),a))};"undefined"!==typeof exports&&(exports.rational=rational);var alpha={GREEK:"\u03b1\u03b2\u03b3\u03b4\u03bc\u03c0\u03c1\u03c2\u03c4\u03c6\u03c7\u03c8\u03c9\u0398\u03a6\u03a9".split(""),iterator:function(){this.i=0;this.next=function(){return this.i<alpha.GREEK.length?alpha.GREEK[this.i++]:"?"}}};"undefined"!==typeof exports&&(exports.alpha=alpha);var physics={};physics.FINE_STRUCTURE=bigrat.fromValues(30223,4141639);physics.a=physics.FINE_STRUCTURE;physics.FINE_STRUCTURE_ALT=bigrat.fromValues(13355891,1830237867);physics.PLANK=bigrat.fromValues(662606957,1E42);physics.h=physics.PLANK;physics.hBAR=bigrat.fromValues(1054571726,1E43);physics.GRAVITATIONAL=bigrat.fromValues(667384,1E16);physics.G=physics.GRAVITATIONAL;physics.SPEED_OF_LIGHT=bigrat.fromInteger(299792458);physics.c=physics.SPEED_OF_LIGHT;
"undefined"!==typeof exports&&(exports.physics=physics);"object"!==typeof rat&&(rat=require("../src/rat.js").rat);"object"!==typeof alpha&&(alpha=require("../src/alpha.js").alpha);
var polyrat={create:function(){return[[],[]]},fromValues:function(a){var b=polyrat.create();b[0]=a.slice();a=polyrat.countDimensions(b);for(b[1]=Array(a);a--;)b[1][a]=0;return polyrat.normalize(b,b)},mergeDimension:function(a,b,c){a[0]=[b.slice()];a[1].push(0);for(var d in c)a[0].push(c[d]);return a},evaluate:function(a,b,c){bigrat.copy(a,bigrat.ZERO);var d=polyrat.countDimensions(b);if(0===d)return"0";if(1===d)for(var f in b[0]){if(b[0][f]){var d=bigrat.fromInteger(b[0][f]),e=bigrat.pow(bigrat.create(),
bigrat.fromInteger(c[0]),b[1][0]+ +f);bigrat.mul(d,d,e);bigrat.add(a,a,d)}}else if(2===d)for(var g in b[0])for(f in b[0][g]){if(b[0][g][f]){var d=bigrat.fromInteger(b[0][g][f]),e=bigrat.pow(bigrat.create(),bigrat.fromInteger(c[0]),b[1][1]+ +f),l=bigrat.pow(bigrat.create(),bigrat.fromInteger(c[1]),b[1][0]+ +g);bigrat.mul(d,d,e);bigrat.mul(d,d,l);bigrat.add(a,a,d)}}else if(3===d)for(var p in b[0])for(g in b[0][p])for(f in b[0][p][g]){if(b[0][p][g][f]){var d=bigrat.fromInteger(b[0][p][g][f]),e=bigrat.pow(bigrat.create(),
bigrat.fromInteger(c[0]),b[1][2]+ +f),l=bigrat.pow(bigrat.create(),bigrat.fromInteger(c[1]),b[1][1]+ +g),r=bigrat.pow(bigrat.create(),bigrat.fromInteger(c[1]),b[1][0]+ +p);bigrat.mul(d,d,e);bigrat.mul(d,d,l);bigrat.mul(d,d,r);bigrat.add(a,a,d)}}else alert("unsupported dimension "+d);return a},derivatives:function(a){return[]},forEachNonZeroElement:function(a,b){for(var c=a.slice(),d=0;"object"===typeof c[0];)c=c[0].slice(),d++;for(var f=d+1;0<f;--f){print(Array(f).join("[0]"));var c=eval("(a"+Array(f).join("[0]")+
")"),e;for(e in c)for(var g=d+2;0<g;--g)print(c[e])}return""},str:function(a){var b="",c=new alpha.iterator,d=c.next(),f=c.next(),c=c.next(),e=polyrat.countDimensions(a);if(0===e)b+=a.toString();else if(1===e)for(var g in a[0]){if(a[0][g]){b?b+=" "+(0>a[0][g]?"-":"+")+" ":0>a[0][g]&&(b+="-");var e=Math.abs(a[0][g]),l=a[1][0]+ +g;if(1!==e||0===l)b+=e;0!==l&&(b+=d+(1===l?"":"<sup>"+l+"</sup>"))}}else if(2===e)for(var p in a[0])for(g in a[0][p]){if(a[0][p][g]){b?b+=" "+(0>a[0][p][g]?"-":"+")+" ":0>a[0][p][g]&&
(b+="-");var e=Math.abs(a[0][p][g]),l=a[1][1]+ +g,r=a[1][0]+ +p;if(1!==e||0===r&&0===l)b+=e;0!==l&&(b+=d+(1===l?"":"<sup>"+l+"</sup>"));0!==r&&(b+=f+(1===r?"":"<sup>"+r+"</sup>"))}}else if(3===e)for(var s in a[0])for(p in a[0][s])for(g in a[0][s][p]){if(a[0][s][p][g]){b?b+=" "+(0>a[0][s][p][g]?"-":"+")+" ":0>a[0][s][p][g]&&(b+="-");var e=Math.abs(a[0][s][p][g]),l=a[1][2]+ +g,r=a[1][1]+ +p,n=a[1][0]+ +s;if(1!==e||0===n&&0===r&&0===l)b+=e;0!==l&&(b+=d+(1===l?"":"<sup>"+l+"</sup>"));0!==r&&(b+=f+(1===
r?"":"<sup>"+r+"</sup>"));0!==n&&(b+=c+(1===n?"":"<sup>"+n+"</sup>"))}}else alert("unsupported dimension "+e);""===b&&(b="0");return b},getJSFormula:function(a){var b="",c=polyrat.countDimensions(a);if(0===c)return"0";if(1===c)for(var d in a[0])a[0][d]&&(b&&(b+="+"),b+=a[0][d],c=a[1][0]+ +d,0!==c&&(b=1===c?b+"*m[0]":b+("*Math.pow(m[0],"+c+")")));else if(2===c)for(var f in a[0])for(d in a[0][f]){if(a[0][f][d]){b&&(b+="+");var b=b+a[0][f][d],c=a[1][1]+ +d,e=a[1][0]+ +f;0!==c&&(b=1===c?b+"*m[0]":b+("*Math.pow(m[0],"+
c+")"));0!==e&&(b=1===e?b+"*m[1]":b+("*Math.pow(m[1],"+e+")"))}}else if(3===c)for(var g in a[0])for(f in a[0][g])for(d in a[0][g][f]){if(a[0][g][f][d]){b&&(b+="+");var b=b+a[0][g][f][d],c=a[1][2]+ +d,e=a[1][1]+ +f,l=a[1][0]+ +g;0!==c&&(b=1===c?b+"*m[0]":b+("*Math.pow(m[0],"+c+")"));0!==e&&(b=1===e?b+"*m[1]":b+("*Math.pow(m[1],"+e+")"));0!==l&&(b=1===l?b+"*m[2]":b+("*Math.pow(m[2],"+l+")"))}}else alert("unsupported dimension "+c);""===b&&(b="0");return b}};
String.prototype.repeat=function(a){return Array(a+1).join(this)};
polyrat.getGLSLFormula=function(a){var b="",c=polyrat.countDimensions(a);if(0===c)return"0.0";if(1===c)for(var d in a[0])a[0][d]&&(b&&(b+="+"),b+=a[0][d]+".0",c=a[1][0]+ +d,0!==c&&(b=1===c?b+"*m_0":1<c?b+"*m_0".repeat(c):b+("*pow(1.0"+"*m_0".repeat(Math.abs(c))+",-1.0)")));else if(2===c)for(var f in a[0])for(d in a[0][f]){if(a[0][f][d]){b&&(b+="+");var b=b+(a[0][f][d]+".0"),c=a[1][1]+ +d,e=a[1][0]+ +f;0!==c&&(b=1===c?b+"*m_0":1<c?b+"*m_0".repeat(c):b+("*pow(1.0"+"*m_0".repeat(Math.abs(c))+",-1.0)"));
0!==e&&(b=1===e?b+"*m_1":1<e?b+"*m_1".repeat(e):b+("*pow(1.0"+"*m_1".repeat(Math.abs(e))+",-1.0)"))}}else alert("unsupported dimension "+c);""===b&&(b="0.0");return b};polyrat.toJSON=function(a){return JSON.stringify(a)};polyrat.fromJSON=function(a){return JSON.parse(a)};polyrat.normalize=function(a,b){return b};
polyrat.add=function(a,b,c){var d=polyrat.countDimensions(b),f=polyrat.countDimensions(c);b[1][e]!==c[1][e]&&alert("adding polyrats only works if the offsets are the same (for now)");if(1===d&&1===f){for(var e in b[0])a[0][e]="undefined"===typeof c[0][e]?b[0][e]:b[0][e]+c[0][e];for(e in c[0])"undefined"===typeof b[0][e]&&(a[0][e]=c[0][e]);for(e in a[0])"undefined"===typeof b[0][e]&&"undefined"===typeof c[0][e]&&(a[0][e]=null)}else alert("unsupported dimension "+d);for(e=d;e--;)a[1][e]=b[1][e];return a};
polyrat.subtract=function(a,b,c){return a};polyrat.sub=polyrat.subtract;polyrat.multiply=function(a,b,c){return a};polyrat.mul=polyrat.multiply;polyrat.divide=function(a,b,c){return a};polyrat.div=polyrat.divide;polyrat.equals=function(a,b){return a[0]===b[0]&&a[1]===b[1]};polyrat.countDimensions=function(a){return"object"===typeof a[0]?polyrat._countDimensions(a[0],1):0};polyrat._countDimensions=function(a,b){return"object"===typeof a[0]?polyrat._countDimensions(a[0],++b):b};polyrat.ZERO=polyrat.fromValues([0]);
polyrat.IDENTITY=polyrat.fromValues([1]);polyrat.ONE=polyrat.IDENTITY;"undefined"!==typeof exports&&(exports.polyrat=polyrat);var RAT_ZERO=rat.ZERO,RAT_ONE=rat.ONE,RAT_INFINITY=rat.INFINITY,RAT_INFINULL=rat.INFINULL,RAT_INFINITESIMAL=rat.INFINITESIMAL;
	})(shim.exports);
})(this);