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biginteger.js
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biginteger.js
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/*
JavaScript BigInteger library version 0.9
http://silentmatt.com/biginteger/
Copyright (c) 2009 Matthew Crumley <email@matthewcrumley.com>
Copyright (c) 2010,2011 by John Tobey <jtobey@john-edwin-tobey.org>
Licensed under the MIT license.
Support for arbitrary internal representation base was added by
Vitaly Magerya.
*/
/*
File: biginteger.js
Exports:
<BigInteger>
*/
/*
Class: BigInteger
An arbitrarily-large integer.
<BigInteger> objects should be considered immutable. None of the "built-in"
methods modify *this* or their arguments. All properties should be
considered private.
All the methods of <BigInteger> instances can be called "statically". The
static versions are convenient if you don't already have a <BigInteger>
object.
As an example, these calls are equivalent.
> BigInteger(4).multiply(5); // returns BigInteger(20);
> BigInteger.multiply(4, 5); // returns BigInteger(20);
> var a = 42;
> var a = BigInteger.toJSValue("0b101010"); // Not completely useless...
*/
// IE doesn't support Array.prototype.map
if (!Array.prototype.map) {
Array.prototype.map = function(fun /*, thisp*/) {
var len = this.length >>> 0;
if (typeof fun !== "function") {
throw new TypeError();
}
var res = new Array(len);
var thisp = arguments[1];
for (var i = 0; i < len; i++) {
if (i in this) {
res[i] = fun.call(thisp, this[i], i, this);
}
}
return res;
};
}
/*
Constructor: BigInteger()
Convert a value to a <BigInteger>.
Although <BigInteger()> is the constructor for <BigInteger> objects, it is
best not to call it as a constructor. If *n* is a <BigInteger> object, it is
simply returned as-is. Otherwise, <BigInteger()> is equivalent to <parse>
without a radix argument.
> var n0 = BigInteger(); // Same as <BigInteger.ZERO>
> var n1 = BigInteger("123"); // Create a new <BigInteger> with value 123
> var n2 = BigInteger(123); // Create a new <BigInteger> with value 123
> var n3 = BigInteger(n2); // Return n2, unchanged
The constructor form only takes an array and a sign. *n* must be an
array of numbers in little-endian order, where each digit is between 0
and BigInteger.base. The second parameter sets the sign: -1 for
negative, +1 for positive, or 0 for zero. The array is *not copied and
may be modified*. If the array contains only zeros, the sign parameter
is ignored and is forced to zero.
> new BigInteger([5], -1): create a new BigInteger with value -5
Parameters:
n - Value to convert to a <BigInteger>.
Returns:
A <BigInteger> value.
See Also:
<parse>, <BigInteger>
*/
function BigInteger(n, s) {
if (!(this instanceof BigInteger)) {
if (n instanceof BigInteger) {
return n;
}
else if (typeof n === "undefined") {
return BigInteger.ZERO;
}
return BigInteger.parse(n);
}
n = n || []; // Provide the nullary constructor for subclasses.
while (n.length && !n[n.length - 1]) {
--n.length;
}
this._d = n;
this._s = n.length ? (s || 1) : 0;
}
// Base-10 speedup hacks in parse, toString, exp10 and log functions
// require base to be a power of 10. 10^7 is the largest such power
// that won't cause a precision loss when digits are multiplied.
BigInteger.base = 10000000;
BigInteger.base_log10 = 7;
BigInteger.init = function() {
// Constant: ZERO
// <BigInteger> 0.
BigInteger.ZERO = new BigInteger([], 0);
// Constant: ONE
// <BigInteger> 1.
BigInteger.ONE = new BigInteger([1], 1);
// Constant: M_ONE
// <BigInteger> -1.
BigInteger.M_ONE = new BigInteger(BigInteger.ONE._d, -1);
// Constant: _0
// Shortcut for <ZERO>.
BigInteger._0 = BigInteger.ZERO;
// Constant: _1
// Shortcut for <ONE>.
BigInteger._1 = BigInteger.ONE;
/*
Constant: small
Array of <BigIntegers> from 0 to 36.
These are used internally for parsing, but useful when you need a "small"
<BigInteger>.
See Also:
<ZERO>, <ONE>, <_0>, <_1>
*/
BigInteger.small = [
BigInteger.ZERO,
BigInteger.ONE,
/* Assuming BigInteger.base > 36 */
new BigInteger( [2], 1),
new BigInteger( [3], 1),
new BigInteger( [4], 1),
new BigInteger( [5], 1),
new BigInteger( [6], 1),
new BigInteger( [7], 1),
new BigInteger( [8], 1),
new BigInteger( [9], 1),
new BigInteger([10], 1),
new BigInteger([11], 1),
new BigInteger([12], 1),
new BigInteger([13], 1),
new BigInteger([14], 1),
new BigInteger([15], 1),
new BigInteger([16], 1),
new BigInteger([17], 1),
new BigInteger([18], 1),
new BigInteger([19], 1),
new BigInteger([20], 1),
new BigInteger([21], 1),
new BigInteger([22], 1),
new BigInteger([23], 1),
new BigInteger([24], 1),
new BigInteger([25], 1),
new BigInteger([26], 1),
new BigInteger([27], 1),
new BigInteger([28], 1),
new BigInteger([29], 1),
new BigInteger([30], 1),
new BigInteger([31], 1),
new BigInteger([32], 1),
new BigInteger([33], 1),
new BigInteger([34], 1),
new BigInteger([35], 1),
new BigInteger([36], 1)
];
}
BigInteger.init();
// Used for parsing/radix conversion
BigInteger.digits = "0123456789ABCDEFGHIJKLMNOPQRSTUVWXYZ".split("");
/*
Method: toString
Convert a <BigInteger> to a string.
When *base* is greater than 10, letters are upper case.
Parameters:
base - Optional base to represent the number in (default is base 10).
Must be between 2 and 36 inclusive, or an Error will be thrown.
Returns:
The string representation of the <BigInteger>.
*/
BigInteger.prototype.toString = function(base) {
base = +base || 10;
if (base < 2 || base > 36) {
throw new Error("illegal radix " + base + ".");
}
if (this._s === 0) {
return "0";
}
if (base === 10) {
var str = this._s < 0 ? "-" : "";
str += this._d[this._d.length - 1].toString();
for (var i = this._d.length - 2; i >= 0; i--) {
var group = this._d[i].toString();
while (group.length < BigInteger.base_log10) group = '0' + group;
str += group;
}
return str;
}
else {
var numerals = BigInteger.digits;
base = BigInteger.small[base];
var sign = this._s;
var n = this.abs();
var digits = [];
var digit;
while (n._s !== 0) {
var divmod = n.divRem(base);
n = divmod[0];
digit = divmod[1];
// TODO: This could be changed to unshift instead of reversing at the end.
// Benchmark both to compare speeds.
digits.push(numerals[digit.valueOf()]);
}
return (sign < 0 ? "-" : "") + digits.reverse().join("");
}
};
// Verify strings for parsing
BigInteger.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]*$/
];
/*
Function: parse
Parse a string into a <BigInteger>.
*base* is optional but, if provided, must be from 2 to 36 inclusive. If
*base* is not provided, it will be guessed based on the leading characters
of *s* as follows:
- "0x" or "0X": *base* = 16
- "0c" or "0C": *base* = 8
- "0b" or "0B": *base* = 2
- else: *base* = 10
If no base is provided, or *base* is 10, the number can be in exponential
form. For example, these are all valid:
> BigInteger.parse("1e9"); // Same as "1000000000"
> BigInteger.parse("1.234*10^3"); // Same as 1234
> BigInteger.parse("56789 * 10 ** -2"); // Same as 567
If any characters fall outside the range defined by the radix, an exception
will be thrown.
Parameters:
s - The string to parse.
base - Optional radix (default is to guess based on *s*).
Returns:
a <BigInteger> instance.
*/
BigInteger.parse = function(s, base) {
// Expands a number in exponential form to decimal form.
// expandExponential("-13.441*10^5") === "1344100";
// expandExponential("1.12300e-1") === "0.112300";
// expandExponential(1000000000000000000000000000000) === "1000000000000000000000000000000";
function expandExponential(str) {
str = str.replace(/\s*[*xX]\s*10\s*(\^|\*\*)\s*/, "e");
return str.replace(/^([+\-])?(\d+)\.?(\d*)[eE]([+\-]?\d+)$/, function(x, s, n, f, c) {
c = +c;
var l = c < 0;
var i = n.length + c;
x = (l ? n : f).length;
c = ((c = Math.abs(c)) >= x ? c - x + l : 0);
var z = (new Array(c + 1)).join("0");
var r = n + f;
return (s || "") + (l ? r = z + r : r += z).substr(0, i += l ? z.length : 0) + (i < r.length ? "." + r.substr(i) : "");
});
}
s = s.toString();
if (typeof base === "undefined" || +base === 10) {
s = expandExponential(s);
}
var parts = /^([+\-]?)(0[xXcCbB])?([0-9A-Za-z]*)(?:\.\d*)?$/.exec(s);
if (parts) {
var sign = parts[1] || "+";
var baseSection = parts[2] || "";
var digits = parts[3] || "";
if (typeof base === "undefined") {
// Guess base
if (baseSection === "0x" || baseSection === "0X") { // Hex
base = 16;
}
else if (baseSection === "0c" || baseSection === "0C") { // Octal
base = 8;
}
else if (baseSection === "0b" || baseSection === "0B") { // Binary
base = 2;
}
else {
base = 10;
}
}
else if (base < 2 || base > 36) {
throw new Error("Illegal radix " + base + ".");
}
base = +base;
// Check for digits outside the range
if (!(BigInteger.radixRegex[base].test(digits))) {
throw new Error("Bad digit for radix " + base);
}
// Strip leading zeros, and convert to array
digits = digits.replace(/^0+/, "").split("");
if (digits.length === 0) {
return BigInteger.ZERO;
}
// Get the sign (we know it's not zero)
sign = (sign === "-") ? -1 : 1;
// Optimize 10
if (base == 10) {
var d = [];
while (digits.length >= BigInteger.base_log10) {
d.push(parseInt(digits.splice(-BigInteger.base_log10).join(''), 10));
}
d.push(parseInt(digits.join(''), 10));
return new BigInteger(d, sign);
}
// Optimize base
if (base === BigInteger.base) {
return new BigInteger(digits.map(Number).reverse(), sign);
}
// Do the conversion
var d = BigInteger.ZERO;
base = BigInteger.small[base];
var small = BigInteger.small;
for (var i = 0; i < digits.length; i++) {
d = d.multiply(base).add(small[parseInt(digits[i], 36)]);
}
return new BigInteger(d._d, sign);
}
else {
throw new Error("Invalid BigInteger format: " + s);
}
};
/*
Function: add
Add two <BigIntegers>.
Parameters:
n - The number to add to *this*. Will be converted to a <BigInteger>.
Returns:
The numbers added together.
See Also:
<subtract>, <multiply>, <quotient>, <next>
*/
BigInteger.prototype.add = function(n) {
if (this._s === 0) {
return BigInteger(n);
}
n = BigInteger(n);
if (n._s === 0) {
return this;
}
if (this._s !== n._s) {
n = n.negate();
return this.subtract(n);
}
var a = this._d;
var b = n._d;
var al = a.length;
var bl = b.length;
var sum = new Array(Math.max(al, bl) + 1);
var size = Math.min(al, bl);
var carry = 0;
var digit;
for (var i = 0; i < size; i++) {
digit = a[i] + b[i] + carry;
sum[i] = digit % BigInteger.base;
carry = (digit / BigInteger.base) | 0;
}
if (bl > al) {
a = b;
al = bl;
}
for (i = size; carry && i < al; i++) {
digit = a[i] + carry;
sum[i] = digit % BigInteger.base;
carry = (digit / BigInteger.base) | 0;
}
if (carry) {
sum[i] = carry;
}
for ( ; i < al; i++) {
sum[i] = a[i];
}
return new BigInteger(sum, this._s);
};
/*
Function: negate
Get the additive inverse of a <BigInteger>.
Returns:
A <BigInteger> with the same magnatude, but with the opposite sign.
See Also:
<abs>
*/
BigInteger.prototype.negate = function() {
return new BigInteger(this._d, -this._s);
};
/*
Function: abs
Get the absolute value of a <BigInteger>.
Returns:
A <BigInteger> with the same magnatude, but always positive (or zero).
See Also:
<negate>
*/
BigInteger.prototype.abs = function() {
return (this._s < 0) ? this.negate() : this;
};
/*
Function: subtract
Subtract two <BigIntegers>.
Parameters:
n - The number to subtract from *this*. Will be converted to a <BigInteger>.
Returns:
The *n* subtracted from *this*.
See Also:
<add>, <multiply>, <quotient>, <prev>
*/
BigInteger.prototype.subtract = function(n) {
if (this._s === 0) {
return BigInteger(n).negate();
}
n = BigInteger(n);
if (n._s === 0) {
return this;
}
if (this._s !== n._s) {
n = n.negate();
return this.add(n);
}
var m = this;
var t;
// negative - negative => -|a| - -|b| => -|a| + |b| => |b| - |a|
if (this._s < 0) {
t = m;
m = new BigInteger(n._d, 1);
n = new BigInteger(t._d, 1);
}
// Both are positive => a - b
var sign = m.compareAbs(n);
if (sign === 0) {
return BigInteger.ZERO;
}
else if (sign < 0) {
// swap m and n
t = n;
n = m;
m = t;
}
// a > b
var a = m._d;
var b = n._d;
var al = a.length;
var bl = b.length;
var diff = new Array(al); // al >= bl since a > b
var borrow = 0;
var i;
var digit;
for (i = 0; i < bl; i++) {
digit = a[i] - borrow - b[i];
if (digit < 0) {
digit += BigInteger.base;
borrow = 1;
}
else {
borrow = 0;
}
diff[i] = digit;
}
for (i = bl; i < al; i++) {
digit = a[i] - borrow;
if (digit < 0) {
digit += BigInteger.base;
}
else {
diff[i++] = digit;
break;
}
diff[i] = digit;
}
for ( ; i < al; i++) {
diff[i] = a[i];
}
return new BigInteger(diff, sign);
};
(function() {
function addOne(n, sign) {
var a = n._d;
var sum = a.slice();
var carry = true;
var i = 0;
while (true) {
var digit = (a[i] || 0) + 1;
sum[i] = digit % BigInteger.base;
if (digit <= BigInteger.base - 1) {
break;
}
++i;
}
return new BigInteger(sum, sign);
}
function subtractOne(n, sign) {
var a = n._d;
var sum = a.slice();
var borrow = true;
var i = 0;
while (true) {
var digit = (a[i] || 0) - 1;
if (digit < 0) {
sum[i] = digit + BigInteger.base;
}
else {
sum[i] = digit;
break;
}
++i;
}
return new BigInteger(sum, sign);
}
/*
Function: next
Get the next <BigInteger> (add one).
Returns:
*this* + 1.
See Also:
<add>, <prev>
*/
BigInteger.prototype.next = function() {
switch (this._s) {
case 0:
return BigInteger.ONE;
case -1:
return subtractOne(this, -1);
// case 1:
default:
return addOne(this, 1);
}
};
/*
Function: prev
Get the previous <BigInteger> (subtract one).
Returns:
*this* - 1.
See Also:
<next>, <subtract>
*/
BigInteger.prototype.prev = function() {
switch (this._s) {
case 0:
return BigInteger.M_ONE;
case -1:
return addOne(this, -1);
// case 1:
default:
return subtractOne(this, 1);
}
};
})();
/*
Function: compareAbs
Compare the absolute value of two <BigIntegers>.
Calling <compareAbs> is faster than calling <abs> twice, then <compare>.
Parameters:
n - The number to compare to *this*. Will be converted to a <BigInteger>.
Returns:
-1, 0, or +1 if *|this|* is less than, equal to, or greater than *|n|*.
See Also:
<compare>, <abs>
*/
BigInteger.prototype.compareAbs = function(n) {
if (this === n) {
return 0;
}
if (!(n instanceof BigInteger)) {
if (!isFinite(n)) {
return(isNaN(n) ? n : -1);
}
n = BigInteger(n);
}
if (this._s === 0) {
return (n._s !== 0) ? -1 : 0;
}
if (n._s === 0) {
return 1;
}
var l = this._d.length;
var nl = n._d.length;
if (l < nl) {
return -1;
}
else if (l > nl) {
return 1;
}
var a = this._d;
var b = n._d;
for (var i = l-1; i >= 0; i--) {
if (a[i] !== b[i]) {
return a[i] < b[i] ? -1 : 1;
}
}
return 0;
};
/*
Function: compare
Compare two <BigIntegers>.
Parameters:
n - The number to compare to *this*. Will be converted to a <BigInteger>.
Returns:
-1, 0, or +1 if *this* is less than, equal to, or greater than *n*.
See Also:
<compareAbs>, <isPositive>, <isNegative>, <isUnit>
*/
BigInteger.prototype.compare = function(n) {
if (this === n) {
return 0;
}
n = BigInteger(n);
if (this._s === 0) {
return -n._s;
}
if (this._s === n._s) { // both positive or both negative
var cmp = this.compareAbs(n);
return cmp * this._s;
}
else {
return this._s;
}
};
/*
Function: isUnit
Return true iff *this* is either 1 or -1.
Returns:
true if *this* compares equal to <BigInteger.ONE> or <BigInteger.M_ONE>.
See Also:
<isZero>, <isNegative>, <isPositive>, <compareAbs>, <compare>,
<BigInteger.ONE>, <BigInteger.M_ONE>
*/
BigInteger.prototype.isUnit = function() {
return this === BigInteger.ONE ||
this === BigInteger.M_ONE ||
(this._d.length === 1 && this._d[0] === 1);
};
/*
Function: multiply
Multiply two <BigIntegers>.
Parameters:
n - The number to multiply *this* by. Will be converted to a
<BigInteger>.
Returns:
The numbers multiplied together.
See Also:
<add>, <subtract>, <quotient>, <square>
*/
BigInteger.prototype.multiply = function(n) {
// TODO: Consider adding Karatsuba multiplication for large numbers
if (this._s === 0) {
return BigInteger.ZERO;
}
n = BigInteger(n);
if (n._s === 0) {
return BigInteger.ZERO;
}
if (this.isUnit()) {
if (this._s < 0) {
return n.negate();
}
return n;
}
if (n.isUnit()) {
if (n._s < 0) {
return this.negate();
}
return this;
}
if (this === n) {
return this.square();
}
var r = (this._d.length >= n._d.length);
var a = (r ? this : n)._d; // a will be longer than b
var b = (r ? n : this)._d;
var al = a.length;
var bl = b.length;
var pl = al + bl;
var partial = new Array(pl);
var i;
for (i = 0; i < pl; i++) {
partial[i] = 0;
}
for (i = 0; i < bl; i++) {
var carry = 0;
var bi = b[i];
var jlimit = al + i;
var digit;
for (var j = i; j < jlimit; j++) {
digit = partial[j] + bi * a[j - i] + carry;
carry = (digit / BigInteger.base) | 0;
partial[j] = (digit % BigInteger.base) | 0;
}
if (carry) {
digit = partial[j] + carry;
carry = (digit / BigInteger.base) | 0;
partial[j] = digit % BigInteger.base;
}
}
return new BigInteger(partial, this._s * n._s);
};
// Multiply a BigInteger by a single-digit native number
// Assumes that this and n are >= 0
// This is not really intended to be used outside the library itself
BigInteger.prototype.multiplySingleDigit = function(n) {
if (n === 0 || this._s === 0) {
return BigInteger.ZERO;
}
if (n === 1) {
return this;
}
var digit;
if (this._d.length === 1) {
digit = this._d[0] * n;
if (digit >= BigInteger.base) {
return new BigInteger([(digit % BigInteger.base)|0,
(digit / BigInteger.base)|0], 1);
}
return new BigInteger([digit], 1);
}
if (n === 2) {
return this.add(this);
}
if (this.isUnit()) {
return new BigInteger([n], 1);
}
var a = this._d;
var al = a.length;
var pl = al + 1;
var partial = new Array(pl);
for (var i = 0; i < pl; i++) {
partial[i] = 0;
}
var carry = 0;
for (var j = 0; j < al; j++) {
digit = n * a[j] + carry;
carry = (digit / BigInteger.base) | 0;
partial[j] = (digit % BigInteger.base) | 0;
}
if (carry) {
digit = carry;
carry = (digit / BigInteger.base) | 0;
partial[j] = digit % BigInteger.base;
}
return new BigInteger(partial, 1);
};
/*
Function: square
Multiply a <BigInteger> by itself.
This is slightly faster than regular multiplication, since it removes the
duplicated multiplcations.
Returns:
> this.multiply(this)
See Also:
<multiply>
*/
BigInteger.prototype.square = function() {
// Normally, squaring a 10-digit number would take 100 multiplications.
// Of these 10 are unique diagonals, of the remaining 90 (100-10), 45 are repeated.
// This procedure saves (N*(N-1))/2 multiplications, (e.g., 45 of 100 multiplies).
// Based on code by Gary Darby, Intellitech Systems Inc., www.DelphiForFun.org
if (this._s === 0) {
return BigInteger.ZERO;
}
if (this.isUnit()) {
return BigInteger.ONE;
}
var digits = this._d;
var length = digits.length;
var imult1 = new Array(length + length + 1);
var product, carry, k;
var i;
// Calculate diagonal
for (i = 0; i < length; i++) {
k = i * 2;
product = digits[i] * digits[i];
carry = (product / BigInteger.base) | 0;
imult1[k] = product % BigInteger.base;
imult1[k + 1] = carry;
}
// Calculate repeating part
for (i = 0; i < length; i++) {
carry = 0;
k = i * 2 + 1;
for (var j = i + 1; j < length; j++, k++) {
product = digits[j] * digits[i] * 2 + imult1[k] + carry;
carry = (product / BigInteger.base) | 0;
imult1[k] = product % BigInteger.base;
}
k = length + i;
var digit = carry + imult1[k];
carry = (digit / BigInteger.base) | 0;
imult1[k] = digit % BigInteger.base;
imult1[k + 1] += carry;
}
return new BigInteger(imult1, 1);
};
/*
Function: quotient
Divide two <BigIntegers> and truncate towards zero.