Generic base version

lib/big-number.js

@@ -11,6 +11,67 @@
 !(function() {
     'use strict';
 
+    /**
+     * The base to which we hold the big number
+     *
+     * This must be set to no more than the square root of the maximum native integer that JS can represent accurately
+     * It can be any number from base 2 up this maximum
+     * The larger the number the more efficient the big number arithmetic will be
+     * However, for division, the choice of number also has a significant impact. Infact for maximum efficiency
+     * the base should have as many common factors as possible. This is maximised by having the base to be a power
+     * of 2. JS typically holds integers accurately to at least up to 2^50 so a value of 2^25 would be the optimal
+     * base to use. Setting the base to a prime number will have the worst effect on performance so far as division
+     * is concerned.
+     *
+     * The original version was hard coded with a base of 10. If we set our base to this now it will have similar
+     * performance, though the division routine has been optimised such that even with this base it could be up
+     * to twice as fast.
+     *
+     * Even with the base set to 2 (i.e. the numbers are held in binary!) the arithmetic is surprisingly fast.
+     * But nothing like as fast as with a base of 2^25
+     *
+     * @type {number}
+     */
+    var base=Math.pow(2,25);
+
+    /*
+     * Factorise the base - these then become the most efficient way to segment the division process
+     * Essentially, the most efficient base is the one with the most factors
+     * Which means that a base of a power of 2 is the best choice
+     *
+     * This is somewhat over the top, but is a good exercise to ensure that division is performed
+     * with maximum efficiency whatever the base :-)
+     */
+    var factors=(function() {
+        var result=[];
+        var primes=[];
+        var factorisingbase=base;
+        var lastprime=2;
+
+        // Check number for prime (assumes previous primes have been found)
+        function isprime(prime) {
+            // Only need to test divisors so far as the square root of the number under test
+            for (var i=0; i<primes.length && primes[i]<Math.sqrt(prime); i++) {
+                if (prime%primes[i]==0) return false;
+            }
+            return true;
+        }
+
+        while (factorisingbase>1) {
+            if (factorisingbase%lastprime==0) {
+                result.push(lastprime);
+                factorisingbase=Math.floor(factorisingbase/lastprime);
+            }
+            else {
+                do {
+                    lastprime=(lastprime==2 ? 3 : lastprime+2);
+                } while (!isprime(lastprime));
+                primes.push(lastprime);
+            }
+        }
+        return result;
+    })();
+
     // Helper function which tests if a given character is a digit
     var testDigit = function(digit) {
         return (/^\d$/.test(digit));
@@ -33,68 +94,112 @@
     };
 
     var errors = {
-        "invalid": "Invalid Number",
-        "division by zero": "Invalid Number - Division By Zero"
+        'invalid': 'Invalid Number',
+        'division by zero': 'Invalid Number - Division By Zero'
     };
 
-    // Constructor function which creates a new BigNumber object
-    // from an integer, a string, an array or other BigNumber object
+
+    /*
+     * Constructor function which creates a new BigNumber object from an integer, a string, an array or other BigNumber object
+     *
+     * IMPORTANT NOTE: Once we allow for any base, the routines to initiate a number from a string of decimal numbers
+     * and to output a number in decimal form (.val()) now use big number arithmetic. This has potential to get
+     * us into trouble if we are to build numbers from within internal routines that are already manipulating numbers. It can cause
+     * a stack overflow. This situation is now avoided by optimising the initialisation to handle more specifically
+     * (and efficiently) numbers constructed from existing BigNumber objects or native numbers.
+     */
     function BigNumber(initialNumber) {
-        var index;
 
         if (!(this instanceof BigNumber)) {
             return new BigNumber(initialNumber);
         }
 
-        this.number = []
+        this.number = [];
         this.sign = 1;
         this.rest = 0;
 
+        // No parameter initialises the number to zero
         if (!initialNumber) {
             this.number = [0];
             return;
         }
 
+        // A big number parameter initialises this number to the same value
+        if (initialNumber instanceof BigNumber) {
+            this.number = initialNumber.number.slice();
+            this.sign = initialNumber.sign;
+            this.rest = initialNumber.rest;
+            return;
+        }
+
+        // A native number makes for a fast initialisation of the number
+        if (typeof initialNumber=='number') {
+            this.sign=initialNumber<0 ? -1 : 1;
+            initialNumber=Math.abs(initialNumber);
+            while (initialNumber>0) {
+                this.number.push(initialNumber%base);
+                initialNumber=Math.floor(initialNumber/base);
+            }
+            return;
+        }
+
+        // Otherwise, now create the number from a String or Array
+        var index;
+        var multipleoftens=BigNumber(1);
+        var ten=BigNumber(10);
+
+        // Method removed from public API and made local within this context
+        function addDigit(digit) {
+            if (testDigit(digit)) {
+                this.add(BigNumber(digit).mult(multipleoftens));
+                multipleoftens.mult(ten);
+                return true;
+            }
+            return false;
+        }
+
         // The initial number can be an array or object
         // e.g. array     : [3,2,1], ['+',3,2,1], ['-',3,2,1]
         //      number    : 312
         //      string    : '321', '+321', -321'
         //      BigNumber : BigNumber(321)
         // Every character except the first must be a digit
-
+        var sign=1;
         if (isArray(initialNumber)) {
             if (initialNumber.length && initialNumber[0] === '-' || initialNumber[0] === '+') {
-                this.sign = initialNumber[0] === '+' ? 1 : -1;
+                sign = initialNumber[0] === '+' ? 1 : -1;
                 initialNumber.shift(0);
             }
             for (index = initialNumber.length - 1; index >= 0; index--) {
-                if (!this.addDigit(initialNumber[index]))
+                if (!addDigit.call(this,initialNumber[index])) {
+                    this.number = errors['invalid'];
                     return;
+                }
             }
         } else {
             initialNumber = initialNumber.toString();
             if (initialNumber.charAt(0) === '-' || initialNumber.charAt(0) === '+') {
-                this.sign = initialNumber.charAt(0) === '+' ? 1 : -1;
+                sign = initialNumber.charAt(0) === '+' ? 1 : -1;
                 initialNumber = initialNumber.substring(1);
             }
 
             for (index = initialNumber.length - 1; index >= 0; index--) {
-                if (!this.addDigit(parseInt(initialNumber.charAt(index), 10))) {
+                if (!addDigit.call(this,parseInt(initialNumber.charAt(index), 10))) {
+                    this.number = errors['invalid'];
                     return;
                 }
             }
         }
-    };
-
-    BigNumber.prototype.addDigit = function(digit) {
-        if (testDigit(digit)) {
-            this.number.push(digit);
-        } else {
-            this.number = errors['invalid'];
-            return false;
-        }
+        this.sign=sign;
+    }
 
-        return this;
+    // Statistics on the number of operations performed (useful for checking performance)
+    BigNumber.stats={
+        divs: {count: 0,iterations: 0},
+        mults: {count: 0,iterations: 0},
+        adds: {count: 0,iterations: 0},
+        subs: {count: 0,iterations: 0},
+        pows: {count: 0,iterations: 0}
     };
 
     // returns:
@@ -210,14 +315,16 @@
     // adds two positive BigNumbers
     BigNumber._add = function(a, b) {
         var index;
+
         var remainder = 0;
         var length = Math.max(a.number.length, b.number.length);
 
         for (index = 0; index < length || remainder > 0; index++) {
-            a.number[index] = (remainder += (a.number[index] || 0) + (b.number[index] || 0)) % 10;
-            remainder = Math.floor(remainder / 10);
+            a.number[index] = (remainder += (a.number[index] || 0) + (b.number[index] || 0)) % base;
+            remainder = Math.floor(remainder / base);
+            BigNumber.stats.adds.iterations++;
         }
-
+        BigNumber.stats.adds.count++;
         return a.number;
     };
 
@@ -230,7 +337,8 @@
 
         for (index = 0; index < length; index++) {
             a.number[index] -= (b.number[index] || 0) + remainder;
-            a.number[index] += (remainder = (a.number[index] < 0) ? 1 : 0) * 10;
+            a.number[index] += (remainder = (a.number[index] < 0) ? 1 : 0) * base;
+            BigNumber.stats.subs.iterations++;
         }
         // Count the zeroes which will be removed
         index = 0;
@@ -241,6 +349,7 @@
         if (index > 0) {
             a.number.splice(-index);
         }
+        BigNumber.stats.subs.count++;
         return a.number;
     };
 
@@ -264,12 +373,14 @@
         // multiply the numbers
         for (index = 0; index < this.number.length; index++) {
             for (remainder = 0, givenNumberIndex = 0; givenNumberIndex < bigNumber.number.length || remainder > 0; givenNumberIndex++) {
-                result[index + givenNumberIndex] = (remainder += (result[index + givenNumberIndex] || 0) + this.number[index] * (bigNumber.number[givenNumberIndex] || 0)) % 10;
-                remainder = Math.floor(remainder / 10);
+                result[index + givenNumberIndex] = (remainder += (result[index + givenNumberIndex] || 0) + this.number[index] * (bigNumber.number[givenNumberIndex] || 0)) % base;
+                remainder = Math.floor(remainder / base);
+                BigNumber.stats.mults.iterations++;
             }
         }
 
         this.number = result;
+        BigNumber.stats.mults.count++;
         return this;
     };
 
@@ -284,6 +395,8 @@
         var length;
         var result = [];
         var rest = BigNumber();
+        var nativebigNumber=bigNumber.number.length==1 ? bigNumber.number[0] : 0;
+        var nativerest=0;
 
         // test if one of the numbers is zero
         if (bigNumber.isZero()) {
@@ -301,12 +414,31 @@
             return this;
 
         for (index = this.number.length - 1; index >= 0; index--) {
-            rest.multiply(10);
-            rest.number[0] = this.number[index];
-            result[index] = 0;
-            while (bigNumber.lte(rest)) {
-                result[index]++;
-                rest.subtract(bigNumber);
+
+            if (nativebigNumber) {
+                // If we are dividing by a small number (less than our base) then we can do this division step natively
+                // which is much faster
+                nativerest*=base;
+                nativerest+=this.number[index];
+                result[index]=Math.floor(nativerest/nativebigNumber);
+                nativerest-=(result[index]*nativebigNumber);
+            } else {
+                // Otherwise we need to use BigNumber arithmetic
+                var digit=this.number[index];
+                result[index] = 0;
+                // Segment the division by the factors of the base
+                var divisor=base;
+                for (var i=0; i<factors.length; i++) {
+                    divisor/=factors[i];
+                    rest.multiply(factors[i]);
+                    rest.add(Math.floor(digit/divisor));
+                    digit=digit%divisor;
+                    while (bigNumber.lte(rest)) {
+                        result[index]+=divisor;
+                        rest.subtract(bigNumber);
+                        BigNumber.stats.divs.iterations++;
+                    }
+                }
             }
         }
 
@@ -318,9 +450,10 @@
         if (index > 0) {
             result.splice(-index);
         }
-
-        this.rest = rest;
+        if (nativebigNumber) this.rest=BigNumber(nativerest);
+        else this.rest = rest;
         this.number = result;
+        BigNumber.stats.divs.count++;
         return this;
     };
 
@@ -346,6 +479,7 @@
 
         this.number = [1];
         while (number > 0) {
+            BigNumber.stats.pows.iterations++;
             if (number % 2 === 1) {
                 this.multiply(bigNumber);
                 number--;
@@ -354,7 +488,7 @@
             bigNumber.multiply(bigNumber);
             number = Math.floor(number / 2);
         }
-
+        BigNumber.stats.pows.count++;
         return this;
     };
 
@@ -376,22 +510,37 @@
         return true;
     };
 
+    // Gets a single decimal digit value of the number assuming it is less than 10 (used by the toString() function)
+    BigNumber.prototype._singleDigit=function() {
+        var digit=0;
+        var multiplier=1;
+        for (var i=0; i<this.number.length && digit<10; i++) {
+            digit+=(multiplier*this.number[i]);
+            multiplier*=base;
+        }
+        if (digit<10) return digit.toString();
+        return '';
+    };
+
     // this.number.toString()
+    // Due to the number not necessarily being in base 10, we need to use BigNumber division
+    // to extract the decimal digits
     BigNumber.prototype.toString = function() {
-        var index;
         var str = '';
-        if (typeof this.number === "string") {
+        if (typeof this.number === 'string') {
             return this.number;
         }
 
-        for (index = this.number.length - 1; index >= 0; index--) {
-            str += this.number[index];
+        var ten=BigNumber(10);
+        var clone=BigNumber(this).abs();
+        while(!clone.isZero()) {
+            str=clone.div(ten).rest._singleDigit()+str;
         }
-
+        if (str.length==0) str='0';
         return (this.sign > 0) ? str : ('-' + str);
     };
 
-    // Use shorcuts for functions names
+    // Use shortcuts for functions names
     BigNumber.prototype.plus = BigNumber.prototype.add;
     BigNumber.prototype.minus = BigNumber.prototype.subtract;
     BigNumber.prototype.div = BigNumber.prototype.divide;

test/test.js

@@ -292,4 +292,21 @@ describe('BigNumber.js', function () {
             BigNumber(5).plus(97).minus(53).plus(434).multiply(5435423).add(321453).multiply(21).div(2).val().should.equal("27569123001");
         });
     });
+
+    describe('very big numbers', function () {
+        it('should test main arithmetic on very large integers', function () {
+            var a=BigNumber("43216789895328312675790036721362674782378923750950234212368965373910384654");
+            var b=BigNumber("85764636478697019837453424264758699685764562524345456576868");
+            var x = BigNumber(a);
+            x.add(b).val().should.equal("43216789895328398440426515418382512235803188509649919976931489719366961522");
+            x = BigNumber(a);
+            x.minus(b).val().should.equal("43216789895328226911153558024342837328954658992250548447806441028453807786");
+            x = BigNumber(a);
+            x.multiply(b).val().should.equal("3706472275149059366968352499608225426876935061337924498923340460238964318011366196170940767807131779369992004353943009853967998583672");
+            x = BigNumber(a);
+            x.divide(b).val().should.equal("503899878431396");
+            x = BigNumber(a);
+            x.divide(b).rest.val().should.equal("13662369696833870468897163678348569097398678349966371836926");
+        });
+    });
 });