js-numbers.js

// Scheme numbers.


var __PLTNUMBERS_TOP__;
if (typeof(exports) !== 'undefined') {
    __PLTNUMBERS_TOP__ = exports;
} else {
    if (! this['jsnums']) {
 	this['jsnums'] = {};
    }
    __PLTNUMBERS_TOP__  = this['jsnums'];
}

//var jsnums = {};


// The numeric tower has the following levels:
//     integers
//     rationals
//     floats
//     complex numbers
//
// with the representations:
//     integers: fixnum or BigInteger [level=0]
//     rationals: Rational [level=1]
//     floats: FloatPoint [level=2]
//     complex numbers: Complex [level=3]

// We try to stick with the unboxed fixnum representation for
// integers, since that's what scheme programs commonly deal with, and
// we want that common type to be lightweight.


// A boxed-scheme-number is either BigInteger, Rational, FloatPoint, or Complex.
// An integer-scheme-number is either fixnum or BigInteger.


(function() {
    'use strict';
    // Abbreviation
    var Numbers = __PLTNUMBERS_TOP__;
    //var Numbers = jsnums;


    // makeNumericBinop: (fixnum fixnum -> any) (scheme-number scheme-number -> any) -> (scheme-number scheme-number) X
    // Creates a binary function that works either on fixnums or boxnums.
    // Applies the appropriate binary function, ensuring that both scheme numbers are
    // lifted to the same level.
    var makeNumericBinop = function(onFixnums, onBoxednums, options) {
	options = options || {};
	return function(x, y) {
	    if (options.isXSpecialCase && options.isXSpecialCase(x))
		return options.onXSpecialCase(x, y);
	    if (options.isYSpecialCase && options.isYSpecialCase(y))
		return options.onYSpecialCase(x, y);

	    if (typeof(x) === 'number' &&
		typeof(y) === 'number') {
		return onFixnums(x, y);
	    }
	    if (typeof(x) === 'number') {
		x = liftFixnumInteger(x, y);
	    }
	    if (typeof(y) === 'number') {
		y = liftFixnumInteger(y, x);
	    }

	    if (x.level < y.level) x = x.liftTo(y);
	    if (y.level < x.level) y = y.liftTo(x);
	    return onBoxednums(x, y);
	};
    }
    
    
    // fromFixnum: fixnum -> scheme-number
    var fromFixnum = function(x) {
	if (isNaN(x) || (! isFinite(x))) {
	    return FloatPoint.makeInstance(x);
	}
	var nf = Math.floor(x);
	if (nf === x) {
            if (isOverflow(nf)) {
		return makeBignum(expandExponent(x+''));
            } else {
		return nf;
	    }
	} else {
            return FloatPoint.makeInstance(x);
	}
    };

    var expandExponent = function(s) {
	var match = s.match(scientificPattern), mantissaChunks, exponent;
	if (match) {
	    mantissaChunks = match[1].match(/^([^.]*)(.*)$/);
	    exponent = Number(match[2]);

	    if (mantissaChunks[2].length === 0) {
		return mantissaChunks[1] + zfill(exponent);
	    }

	    if (exponent >= mantissaChunks[2].length - 1) {
		return (mantissaChunks[1] + 
			mantissaChunks[2].substring(1) + 
			zfill(exponent - (mantissaChunks[2].length - 1)));
	    } else {
		return (mantissaChunks[1] +
			mantissaChunks[2].substring(1, 1+exponent));
	    }
	} else {
	    return s;
	}
    };

    // zfill: integer -> string
    // builds a string of "0"'s of length n.
    var zfill = function(n) {
	var buffer = [];
	buffer.length = n;
	for (var i = 0; i < n; i++) {
	    buffer[i] = '0';
	}
	return buffer.join('');
    };
    

    
    // liftFixnumInteger: fixnum-integer boxed-scheme-number -> boxed-scheme-number
    // Lifts up fixnum integers to a boxed type.
    var liftFixnumInteger = function(x, other) {
	switch(other.level) {
	case 0: // BigInteger
	    return makeBignum(x);
	case 1: // Rational
	    return new Rational(x, 1);
	case 2: // FloatPoint
	    return new FloatPoint(x);
	case 3: // Complex
	    return new Complex(x, 0);
	default:
	    throwRuntimeError("IMPOSSIBLE: cannot lift fixnum integer to " + other.toString(), x, other);
	}
    };
    
    
    // throwRuntimeError: string (scheme-number | undefined) (scheme-number | undefined) -> void
    // Throws a runtime error with the given message string.
    var throwRuntimeError = function(msg, x, y) {
	Numbers['onThrowRuntimeError'](msg, x, y);
    };



    // onThrowRuntimeError: string (scheme-number | undefined) (scheme-number | undefined) -> void
    // By default, will throw a new Error with the given message.
    // Override Numbers['onThrowRuntimeError'] if you need to do something special.
    var onThrowRuntimeError = function(msg, x, y) {
	throw new Error(msg);
    };


    // isSchemeNumber: any -> boolean
    // Returns true if the thing is a scheme number.
    var isSchemeNumber = function(thing) {
	return (typeof(thing) === 'number'
		|| (thing instanceof Rational ||
		    thing instanceof FloatPoint ||
		    thing instanceof Complex ||
		    thing instanceof BigInteger));
    };


    // isRational: scheme-number -> boolean
    var isRational = function(n) {
	return (typeof(n) === 'number' ||
		(isSchemeNumber(n) && n.isRational()));
    };

    // isReal: scheme-number -> boolean
    var isReal = function(n) {
	return (typeof(n) === 'number' ||
		(isSchemeNumber(n) && n.isReal()));
    };

    // isExact: scheme-number -> boolean
    var isExact = function(n) {
	return (typeof(n) === 'number' || 
		(isSchemeNumber(n) && n.isExact()));
    };

    // isExact: scheme-number -> boolean
    var isInexact = function(n) {
	if (typeof(n) === 'number') {
	    return false;
	} else {
	    return (isSchemeNumber(n) && n.isInexact());
	}
    };

    // isInteger: scheme-number -> boolean
    var isInteger = function(n) {
	return (typeof(n) === 'number' ||
		(isSchemeNumber(n) && n.isInteger()));
    };

    // isExactInteger: scheme-number -> boolean
    var isExactInteger = function(n) {
	return (typeof(n) === 'number' ||
		(isSchemeNumber(n) && 
		 n.isInteger() && 
		 n.isExact()));
    }



    // toFixnum: scheme-number -> javascript-number
    var toFixnum = function(n) {
	if (typeof(n) === 'number')
	    return n;
	return n.toFixnum();
    };

    // toExact: scheme-number -> scheme-number
    var toExact = function(n) {
	if (typeof(n) === 'number')
	    return n;
	return n.toExact();
    };


    // toExact: scheme-number -> scheme-number
    var toInexact = function(n) {
	if (typeof(n) === 'number')
	    return FloatPoint.makeInstance(n);
	return n.toInexact();
    };



    //////////////////////////////////////////////////////////////////////


    // add: scheme-number scheme-number -> scheme-number
    var add = function(x, y) {
        var sum;
        if (typeof(x) === 'number' && typeof(y) === 'number') {
            sum = x + y;
            if (isOverflow(sum)) {
		return (makeBignum(x)).add(makeBignum(y));
            }
        }
        if (x instanceof FloatPoint && y instanceof FloatPoint) {
            return x.add(y);
        }
        return addSlow(x, y);        
    };

    var addSlow = makeNumericBinop(
	function(x, y) {
	    var sum = x + y;
	    if (isOverflow(sum)) {
		return (makeBignum(x)).add(makeBignum(y));
	    } else {
		return sum;
	    }
	},
	function(x, y) {
	    return x.add(y);
	},
	{isXSpecialCase: function(x) { 
	    return isExactInteger(x) && _integerIsZero(x) },
	 onXSpecialCase: function(x, y) { return y; },
	 isYSpecialCase: function(y) { 
	     return isExactInteger(y) && _integerIsZero(y) },
	 onYSpecialCase: function(x, y) { return x; }
	});


    // subtract: scheme-number scheme-number -> scheme-number
    var subtract = makeNumericBinop(
	function(x, y) {
	    var diff = x - y;
	    if (isOverflow(diff)) {
		return (makeBignum(x)).subtract(makeBignum(y));
	    } else {
		return diff;
	    }
	},
	function(x, y) {
	    return x.subtract(y);
	},
	{isXSpecialCase: function(x) { 
	    return isExactInteger(x) && _integerIsZero(x) },
	 onXSpecialCase: function(x, y) { return negate(y); },
	 isYSpecialCase: function(y) { 
	     return isExactInteger(y) && _integerIsZero(y) },
	 onYSpecialCase: function(x, y) { return x; }
	});


    // mulitply: scheme-number scheme-number -> scheme-number
    var multiply = function(x, y) {
        var prod;
        if (typeof(x) === 'number' && typeof(y) === 'number') {
	    prod = x * y;
	    if (isOverflow(prod)) {
		return (makeBignum(x)).multiply(makeBignum(y));
            } else {
                return prod;
            }
        }
        if (x instanceof FloatPoint && y instanceof FloatPoint) {
            return x.multiply(y);
        }
        return multiplySlow(x, y);
    };
    var multiplySlow = makeNumericBinop(
	function(x, y) {
	    var prod = x * y;
	    if (isOverflow(prod)) {
		return (makeBignum(x)).multiply(makeBignum(y));
	    } else {
		return prod;
	    }
	},
	function(x, y) {
	    return x.multiply(y);
	},
	{isXSpecialCase: function(x) { 
	    return (isExactInteger(x) && 
		    (_integerIsZero(x) || _integerIsOne(x) || _integerIsNegativeOne(x))) },
	 onXSpecialCase: function(x, y) { 
	     if (_integerIsZero(x))
		 return 0;
	     if (_integerIsOne(x))
		 return y;
	     if (_integerIsNegativeOne(x))
		 return negate(y);
	 },
	 isYSpecialCase: function(y) { 
	     return (isExactInteger(y) && 
		     (_integerIsZero(y) || _integerIsOne(y) || _integerIsNegativeOne(y)))},
	 onYSpecialCase: function(x, y) { 
	     if (_integerIsZero(y))
		 return 0;
	     if (_integerIsOne(y))
		 return x;
	     if (_integerIsNegativeOne(y)) 
		 return negate(x);
	 }
	});

    
    // divide: scheme-number scheme-number -> scheme-number
    var divide = makeNumericBinop(
	function(x, y) {
	    if (_integerIsZero(y))
		throwRuntimeError("/: division by zero", x, y);
	    var div = x / y;
	    if (isOverflow(div)) {
		return (makeBignum(x)).divide(makeBignum(y));
	    } else if (Math.floor(div) !== div) {
		return Rational.makeInstance(x, y);
	    } else {
		return div;
	    }
	},
	function(x, y) {
	    return x.divide(y);
	},
	{ isXSpecialCase: function(x) {
	    return (eqv(x, 0));
	},
	  onXSpecialCase: function(x, y) {
	      if (eqv(y, 0)) {
		  throwRuntimeError("/: division by zero", x, y);
	      }
	      return 0;
	  },
	  isYSpecialCase: function(y) { 
	    return (eqv(y, 0)); },
	  onYSpecialCase: function(x, y) {
	      throwRuntimeError("/: division by zero", x, y);
	  }
	});
    
    
    // equals: scheme-number scheme-number -> boolean
    var equals = makeNumericBinop(
	function(x, y) {
	    return x === y;
	},
	function(x, y) {
	    return x.equals(y);
	});


    // eqv: scheme-number scheme-number -> boolean
    var eqv = function(x, y) {
	if (x === y)
	    return true;
	if (typeof(x) === 'number' && typeof(y) === 'number')
	    return x === y;
	if (x === NEGATIVE_ZERO || y === NEGATIVE_ZERO)
	    return x === y;
	if (x instanceof Complex || y instanceof Complex) {
	    return (eqv(realPart(x), realPart(y)) &&
		    eqv(imaginaryPart(x), imaginaryPart(y)));
	}
	var ex = isExact(x), ey = isExact(y);
	return (((ex && ey) || (!ex && !ey)) && equals(x, y));
    };

    // approxEqual: scheme-number scheme-number scheme-number -> boolean
    var approxEquals = function(x, y, delta) {
	return lessThan(abs(subtract(x, y)),
                        delta);
    };

    // greaterThanOrEqual: scheme-number scheme-number -> boolean
    var greaterThanOrEqual = makeNumericBinop(
	function(x, y) {
	    return x >= y;
	},
	function(x, y) {
	    if (!(isReal(x) && isReal(y)))
		throwRuntimeError(
		    ">=: couldn't be applied to complex number", x, y);
	    return x.greaterThanOrEqual(y);
	});


    // lessThanOrEqual: scheme-number scheme-number -> boolean
    var lessThanOrEqual = makeNumericBinop(
	function(x, y){

	    return x <= y;
	},
	function(x, y) {
	    if (!(isReal(x) && isReal(y)))
		throwRuntimeError("<=: couldn't be applied to complex number", x, y);
	    return x.lessThanOrEqual(y);
	});


    // greaterThan: scheme-number scheme-number -> boolean
    var greaterThan = makeNumericBinop(
	function(x, y){
	    return x > y;
	},
	function(x, y) {
	    if (!(isReal(x) && isReal(y)))
		throwRuntimeError(">: couldn't be applied to complex number", x, y);
	    return x.greaterThan(y);
	});


    // lessThan: scheme-number scheme-number -> boolean
    var lessThan = makeNumericBinop(
	function(x, y){

	    return x < y;
	},
	function(x, y) {
	    if (!(isReal(x) && isReal(y)))
		throwRuntimeError("<: couldn't be applied to complex number", x, y);
	    return x.lessThan(y);
	});



    // expt: scheme-number scheme-number -> scheme-number
    var expt = (function() {
	var _expt = makeNumericBinop(
	    function(x, y){
		var pow = Math.pow(x, y);
		if (isOverflow(pow)) {
		    return (makeBignum(x)).expt(makeBignum(y));
		} else {
		    return pow;
		}
	    },
	    function(x, y) {
		if (equals(y, 0)) {
		    return add(y, 1);
		} else {
		    return x.expt(y);
		}
	    });
	return function(x, y) {
	    if (equals(y, 0)) 
		return add(y, 1);
	    if (isReal(y) && lessThan(y, 0)) {
		return _expt(divide(1, x), negate(y));
	    }
	    return _expt(x, y);
	};
    })();


    // exp: scheme-number -> scheme-number
    var exp = function(n) {
	if ( eqv(n, 0) ) {
		return 1;
	}
	if (typeof(n) === 'number') {
	    return FloatPoint.makeInstance(Math.exp(n));
	}
	return n.exp();
    };


    // modulo: scheme-number scheme-number -> scheme-number
    var modulo = function(m, n) {
	if (! isInteger(m)) {
	    throwRuntimeError('modulo: the first argument '
			      + m + " is not an integer.", m, n);
	}
	if (! isInteger(n)) {
	    throwRuntimeError('modulo: the second argument '
			      + n + " is not an integer.", m, n);
	}
	var result;
	if (typeof(m) === 'number') {
	    result = m % n;
	    if (n < 0) {
		if (result <= 0)
		    return result;
		else
		    return result + n;
	    } else {
		if (result < 0)
		    return result + n;
		else
		    return result;
	    }
	}
	result = _integerModulo(floor(m), floor(n));
	// The sign of the result should match the sign of n.
	if (lessThan(n, 0)) {
	    if (lessThanOrEqual(result, 0)) {
		return result;
	    }
	    return add(result, n);

	} else {
	    if (lessThan(result, 0)) {
		return add(result, n);
	    }
	    return result;
	}
    };



    // numerator: scheme-number -> scheme-number
    var numerator = function(n) {
	if (typeof(n) === 'number')
	    return n;
	return n.numerator();
    };


    // denominator: scheme-number -> scheme-number
    var denominator = function(n) {
	if (typeof(n) === 'number')
	    return 1;
	return n.denominator();
    };

    // sqrt: scheme-number -> scheme-number
    var sqrt = function(n) {
	if (typeof(n) === 'number') {
	    if (n >= 0) {
		var result = Math.sqrt(n);
		if (Math.floor(result) === result) {
		    return result;
		} else {
		    return FloatPoint.makeInstance(result);
		}
	    } else {
		return (Complex.makeInstance(0, sqrt(-n)));
	    }
	}
	return n.sqrt();
    };

    // abs: scheme-number -> scheme-number
    var abs = function(n) {
	if (typeof(n) === 'number') {
	    return Math.abs(n);
	}
	return n.abs();
    };

    // floor: scheme-number -> scheme-number
    var floor = function(n) {
	if (typeof(n) === 'number')
	    return n;
	return n.floor();
    };

    // ceiling: scheme-number -> scheme-number
    var ceiling = function(n) {
	if (typeof(n) === 'number')
	    return n;
	return n.ceiling();
    };

    // conjugate: scheme-number -> scheme-number
    var conjugate = function(n) {
	if (typeof(n) === 'number')
	    return n;
	return n.conjugate();
    };

    // magnitude: scheme-number -> scheme-number
    var magnitude = function(n) {
	if (typeof(n) === 'number')
	    return Math.abs(n);
	return n.magnitude();
    };


    // log: scheme-number -> scheme-number
    var log = function(n) {
	if ( eqv(n, 1) ) {
		return 0;
	}
	if (typeof(n) === 'number') {
	    return FloatPoint.makeInstance(Math.log(n));
	}
	return n.log();
    };

    // angle: scheme-number -> scheme-number
    var angle = function(n) {
	if (typeof(n) === 'number') {
	    if (n > 0)
		return 0;
	    else
		return FloatPoint.pi;
	}
	return n.angle();
    };

    // tan: scheme-number -> scheme-number
    var tan = function(n) {
	if (eqv(n, 0)) { return 0; }
	if (typeof(n) === 'number') {
	    return FloatPoint.makeInstance(Math.tan(n));
	}
	return n.tan();
    };

    // atan: scheme-number -> scheme-number
    var atan = function(n) {
	if (eqv(n, 0)) { return 0; }
	if (typeof(n) === 'number') {
	    return FloatPoint.makeInstance(Math.atan(n));
	}
	return n.atan();
    };

    // cos: scheme-number -> scheme-number
    var cos = function(n) {
	if (eqv(n, 0)) { return 1; }
	if (typeof(n) === 'number') {
	    return FloatPoint.makeInstance(Math.cos(n));
	}
	return n.cos();
    };

    // sin: scheme-number -> scheme-number
    var sin = function(n) {
	if (eqv(n, 0)) { return 0; }
	if (typeof(n) === 'number') {
	    return FloatPoint.makeInstance(Math.sin(n));
	}
	return n.sin();
    };

    // acos: scheme-number -> scheme-number
    var acos = function(n) {
	if (eqv(n, 1)) { return 0; }
	if (typeof(n) === 'number') {
	    return FloatPoint.makeInstance(Math.acos(n));
	}
	return n.acos();
    };

    // asin: scheme-number -> scheme-number
    var asin = function(n) {
        if (eqv(n, 0)) { return 0; }
	if (typeof(n) === 'number') {
	    return FloatPoint.makeInstance(Math.asin(n));
	}
	return n.asin();
    };

    // imaginaryPart: scheme-number -> scheme-number
    var imaginaryPart = function(n) {
	if (typeof(n) === 'number') {
	    return 0;
	}
	return n.imaginaryPart();
    };

    // realPart: scheme-number -> scheme-number
    var realPart = function(n) {
	if (typeof(n) === 'number') {
	    return n;
	}
	return n.realPart();
    };

    // round: scheme-number -> scheme-number
    var round = function(n) {
	if (typeof(n) === 'number') {
	    return n;
	}
	return n.round();
    };



    // sqr: scheme-number -> scheme-number
    var sqr = function(x) {
	return multiply(x, x);
    };


    // integerSqrt: scheme-number -> scheme-number
    var integerSqrt = function(x) {
	if (! isInteger(x)) {
	    throwRuntimeError('integer-sqrt: the argument ' + x.toString() +
			      " is not an integer.", x);
	}
	if (typeof (x) === 'number') {
	    if(x < 0) {
	        return Complex.makeInstance(0,
					    Math.floor(Math.sqrt(-x)))
	    } else {
		return Math.floor(Math.sqrt(x));
	    }
	}
	return x.integerSqrt();
    };


    // gcd: scheme-number [scheme-number ...] -> scheme-number
    var gcd = function(first, rest) {
	if (! isInteger(first)) {
	    throwRuntimeError('gcd: the argument ' + first.toString() +
			      " is not an integer.", first);
	}
	var a = abs(first), t, b;
	for(var i = 0; i < rest.length; i++) {
	    b = abs(rest[i]);	
	    if (! isInteger(b)) {
		throwRuntimeError('gcd: the argument ' + b.toString() +
				  " is not an integer.", b);
	    }
	    while (! _integerIsZero(b)) {
		t = a;
		a = b;
		b = _integerModulo(t, b);
	    }
	}
	return a;
    };

    // lcm: scheme-number [scheme-number ...] -> scheme-number
    var lcm = function(first, rest) {
	if (! isInteger(first)) {
	    throwRuntimeError('lcm: the argument ' + first.toString() +
			      " is not an integer.", first);
	}
	var result = abs(first);
	if (_integerIsZero(result)) { return 0; }
	for (var i = 0; i < rest.length; i++) {
	    if (! isInteger(rest[i])) {
		throwRuntimeError('lcm: the argument ' + rest[i].toString() +
				  " is not an integer.", rest[i]);
	    }
	    var divisor = _integerGcd(result, rest[i]);
	    if (_integerIsZero(divisor)) {
		return 0;
	    }
	    result = divide(multiply(result, rest[i]), divisor);
	}
	return result;
    };
    

    var quotient = function(x, y) {
 	if (! isInteger(x)) {
	    throwRuntimeError('quotient: the first argument ' + x.toString() +
			      " is not an integer.", x);
	}
	if (! isInteger(y)) {
	    throwRuntimeError('quotient: the second argument ' + y.toString() +
			      " is not an integer.", y);
	}
	return _integerQuotient(x, y);
    };

    
    var remainder = function(x, y) {
	if (! isInteger(x)) {
	    throwRuntimeError('remainder: the first argument ' + x.toString() +
			      " is not an integer.", x);
	}
	if (! isInteger(y)) {
	    throwRuntimeError('remainder: the second argument ' + y.toString() +
			      " is not an integer.", y);
	}
	return _integerRemainder(x, y);
    };


    // Implementation of the hyperbolic functions
    // http://en.wikipedia.org/wiki/Hyperbolic_cosine
    var cosh = function(x) {
	if (eqv(x, 0)) {
	    return FloatPoint.makeInstance(1.0);
	}
	return divide(add(exp(x), exp(negate(x))),
		      2);
    };
	
    var sinh = function(x) {
	return divide(subtract(exp(x), exp(negate(x))),
		      2);
    };


        
    var makeComplexPolar = function(r, theta) {
	// special case: if theta is zero, just return
	// the scalar.
	if (eqv(theta, 0)) {
	    return r;
	}
	return Complex.makeInstance(multiply(r, cos(theta)),
				    multiply(r, sin(theta)));
    };



    //////////////////////////////////////////////////////////////////////

    // Helpers


    // IsFinite: scheme-number -> boolean
    // Returns true if the scheme number is finite or not.
    var isSchemeNumberFinite = function(n) {
	if (typeof(n) === 'number') {
	    return isFinite(n);
	} else {
	    return n.isFinite();
	}
    };

    // isOverflow: javascript-number -> boolean
    // Returns true if we consider the number an overflow.
    var MIN_FIXNUM = -(9e15);
    var MAX_FIXNUM = (9e15);
    var isOverflow = function(n) {
	return (n < MIN_FIXNUM ||  MAX_FIXNUM < n);
    };


    // negate: scheme-number -> scheme-number
    // multiplies a number times -1.
    var negate = function(n) {
	if (typeof(n) === 'number') {
	    return -n;
	}
	return n.negate();
    };


    // halve: scheme-number -> scheme-number
    // Divide a number by 2.
    var halve = function(n) {
	return divide(n, 2);
    };


    // timesI: scheme-number scheme-number
    // multiplies a number times i.
    var timesI = function(x) {
	return multiply(x, plusI);
    };


    // fastExpt: computes n^k by squaring.
    // n^k = (n^2)^(k/2)
    // Assumes k is non-negative integer.
    var fastExpt = function(n, k) {
	var acc = 1;
	while (true) {
	    if (_integerIsZero(k)) {
		return acc;
	    }
	    if (equals(modulo(k, 2), 0)) {
		n = multiply(n, n);
		k = divide(k, 2);
	    } else {
		acc = multiply(acc, n);
		k = subtract(k, 1);
	    }
	}
    };



    //////////////////////////////////////////////////////////////////////
    //////////////////////////////////////////////////////////////////////
    //////////////////////////////////////////////////////////////////////
    //////////////////////////////////////////////////////////////////////
    //////////////////////////////////////////////////////////////////////


    // Integer operations
    // Integers are either represented as fixnums or as BigIntegers.

    // makeIntegerBinop: (fixnum fixnum -> X) (BigInteger BigInteger -> X) -> X
    // Helper to collect the common logic for coersing integer fixnums or bignums to a
    // common type before doing an operation.
    var makeIntegerBinop = function(onFixnums, onBignums, options) {
	options = options || {};
	return (function(m, n) {
	    if (m instanceof Rational) {
		m = numerator(m);
	    } else if (m instanceof Complex) {
		m = realPart(m);
	    }

	    if (n instanceof Rational) {
		n = numerator(n);
	    }else if (n instanceof Complex) {
		n = realPart(n);
	    }

	    if (typeof(m) === 'number' && typeof(n) === 'number') {
		var result = onFixnums(m, n);
		if (! isOverflow(result) ||
		    (options.ignoreOverflow)) {
		    return result;
		}
	    }
	    if (m instanceof FloatPoint || n instanceof FloatPoint) {
		if (options.doNotCoerseToFloating) {
		    return onFixnums(toFixnum(m), toFixnum(n));
		}
		else {
		    return FloatPoint.makeInstance(
			onFixnums(toFixnum(m), toFixnum(n)));
		}
	    }
	    if (typeof(m) === 'number') {
		m = makeBignum(m);
	    }
	    if (typeof(n) === 'number') {
		n = makeBignum(n);
	    }
	    return onBignums(m, n);
	});
    };


    var makeIntegerUnOp = function(onFixnums, onBignums, options) {
	options = options || {};
	return (function(m) {
	    if (m instanceof Rational) {
		m = numerator(m);
	    } else if (m instanceof Complex) {
		m = realPart(m);
	    }

	    if (typeof(m) === 'number') {
		var result = onFixnums(m);
		if (! isOverflow(result) ||
		    (options.ignoreOverflow)) {
		    return result;
		}
	    }
	    if (m instanceof FloatPoint) {
		return onFixnums(toFixnum(m));
	    }
	    if (typeof(m) === 'number') {
		m = makeBignum(m);
	    }
	    return onBignums(m);
	});
    };



    // _integerModulo: integer-scheme-number integer-scheme-number -> integer-scheme-number
    var _integerModulo = makeIntegerBinop(
	function(m, n) {
	    return m % n;
	},
	function(m, n) {
	    return bnMod.call(m, n);
	});


    // _integerGcd: integer-scheme-number integer-scheme-number -> integer-scheme-number
    var _integerGcd = makeIntegerBinop(
	function(a, b) {
	    var t;
	    while (b !== 0) {
		t = a;
		a = b;
		b = t % b;
	    }
	    return a;
	},
	function(m, n) {
	    return bnGCD.call(m, n);
	});


    // _integerIsZero: integer-scheme-number -> boolean
    // Returns true if the number is zero.
    var _integerIsZero = makeIntegerUnOp(
	function(n){
	    return n === 0;
	},
	function(n) {
	    return bnEquals.call(n, BigInteger.ZERO);
	}
    );


    // _integerIsOne: integer-scheme-number -> boolean
    var _integerIsOne = makeIntegerUnOp(
	function(n) {
	    return n === 1;
	},
	function(n) {
	    return bnEquals.call(n, BigInteger.ONE);
	});
    

 
    // _integerIsNegativeOne: integer-scheme-number -> boolean
    var _integerIsNegativeOne = makeIntegerUnOp(
	function(n) {
	    return n === -1;
	},
	function(n) {
	    return bnEquals.call(n, BigInteger.NEGATIVE_ONE);
	});
    


    // _integerAdd: integer-scheme-number integer-scheme-number -> integer-scheme-number
    var _integerAdd = makeIntegerBinop(
	function(m, n) {
	    return m + n;
	},
	function(m, n) {
	    return bnAdd.call(m, n);
	});

    // _integerSubtract: integer-scheme-number integer-scheme-number -> integer-scheme-number
    var _integerSubtract = makeIntegerBinop(
	function(m, n) {
	    return m - n;
	},
	function(m, n) {
	    return bnSubtract.call(m, n);
	});

    // _integerMultiply: integer-scheme-number integer-scheme-number -> integer-scheme-number
    var _integerMultiply = makeIntegerBinop(
	function(m, n) {
	    return m * n;
	},
	function(m, n) {
	    return bnMultiply.call(m, n);
	});

    //_integerQuotient: integer-scheme-number integer-scheme-number -> integer-scheme-number
    var _integerQuotient = makeIntegerBinop(
	function(m, n) {
	    return ((m - (m % n))/ n);
	},
	function(m, n) {
            return bnDivide.call(m, n);
	});

    var _integerRemainder = makeIntegerBinop(
	function(m, n) {
	    return m % n;
	},
	function(m, n) {
	    return bnRemainder.call(m, n);
	});


    // _integerDivideToFixnum: integer-scheme-number integer-scheme-number -> fixnum
    var _integerDivideToFixnum = makeIntegerBinop(
	function(m, n) {
	    return m / n;
	},
	function(m, n) {
	    return toFixnum(m) / toFixnum(n);
	},
	{ignoreOverflow: true,
	 doNotCoerseToFloating: true});


    // _integerEquals: integer-scheme-number integer-scheme-number -> boolean
    var _integerEquals = makeIntegerBinop(
	function(m, n) {
	    return m === n;
	},
	function(m, n) {
	    return bnEquals.call(m, n);
	},
	{doNotCoerseToFloating: true});

    // _integerGreaterThan: integer-scheme-number integer-scheme-number -> boolean
    var _integerGreaterThan = makeIntegerBinop(
	function(m, n) {
	    return m > n;
	},
	function(m, n) {
	    return bnCompareTo.call(m, n) > 0;
	},
	{doNotCoerseToFloating: true});

    // _integerLessThan: integer-scheme-number integer-scheme-number -> boolean
    var _integerLessThan = makeIntegerBinop(
	function(m, n) {
	    return m < n;
	},
	function(m, n) {
	    return bnCompareTo.call(m, n) < 0;
	},
	{doNotCoerseToFloating: true});

    // _integerGreaterThanOrEqual: integer-scheme-number integer-scheme-number -> boolean
    var _integerGreaterThanOrEqual = makeIntegerBinop(
	function(m, n) {
	    return m >= n;
	},
	function(m, n) {
	    return bnCompareTo.call(m, n) >= 0;
	},
	{doNotCoerseToFloating: true});

    // _integerLessThanOrEqual: integer-scheme-number integer-scheme-number -> boolean
    var _integerLessThanOrEqual = makeIntegerBinop(
	function(m, n) {
	    return m <= n;
	},
	function(m, n) {
	    return bnCompareTo.call(m, n) <= 0;
	},
	{doNotCoerseToFloating: true});



    //////////////////////////////////////////////////////////////////////
    // The boxed number types are expected to implement the following
    // interface.
    //
    // toString: -> string

    // level: number

    // liftTo: scheme-number -> scheme-number

    // isFinite: -> boolean

    // isInteger: -> boolean
    // Produce true if this number can be coersed into an integer.

    // isRational: -> boolean
    // Produce true if the number is rational.

    // isReal: -> boolean
    // Produce true if the number is real.

    // isExact: -> boolean
    // Produce true if the number is exact

    // toExact: -> scheme-number
    // Produce an exact number.

    // toFixnum: -> javascript-number
    // Produce a javascript number.

    // greaterThan: scheme-number -> boolean
    // Compare against instance of the same type.

    // greaterThanOrEqual: scheme-number -> boolean
    // Compare against instance of the same type.

    // lessThan: scheme-number -> boolean
    // Compare against instance of the same type.

    // lessThanOrEqual: scheme-number -> boolean
    // Compare against instance of the same type.

    // add: scheme-number -> scheme-number
    // Add with an instance of the same type.

    // subtract: scheme-number -> scheme-number
    // Subtract with an instance of the same type.

    // multiply: scheme-number -> scheme-number
    // Multiply with an instance of the same type.

    // divide: scheme-number -> scheme-number
    // Divide with an instance of the same type.

    // numerator: -> scheme-number
    // Return the numerator.

    // denominator: -> scheme-number
    // Return the denominator.

    // integerSqrt: -> scheme-number
    // Produce the integer square root.

    // sqrt: -> scheme-number
    // Produce the square root.

    // abs: -> scheme-number
    // Produce the absolute value.

    // floor: -> scheme-number
    // Produce the floor.

    // ceiling: -> scheme-number
    // Produce the ceiling.

    // conjugate: -> scheme-number
    // Produce the conjugate.

    // magnitude: -> scheme-number
    // Produce the magnitude.

    // log: -> scheme-number
    // Produce the log.

    // angle: -> scheme-number
    // Produce the angle.

    // atan: -> scheme-number
    // Produce the arc tangent.

    // cos: -> scheme-number
    // Produce the cosine.

    // sin: -> scheme-number
    // Produce the sine.

    // expt: scheme-number -> scheme-number
    // Produce the power to the input.

    // exp: -> scheme-number
    // Produce e raised to the given power.

    // acos: -> scheme-number
    // Produce the arc cosine.

    // asin: -> scheme-number
    // Produce the arc sine.

    // imaginaryPart: -> scheme-number
    // Produce the imaginary part

    // realPart: -> scheme-number
    // Produce the real part.

    // round: -> scheme-number
    // Round to the nearest integer.

    // equals: scheme-number -> boolean
    // Produce true if the given number of the same type is equal.



    //////////////////////////////////////////////////////////////////////

    // Rationals


    var Rational = function(n, d) {
	this.n = n;
	this.d = d;
    };


    Rational.prototype.toString = function() {
	if (_integerIsOne(this.d)) {
	    return this.n.toString() + "";
	} else {
	    return this.n.toString() + "/" + this.d.toString();
	}
    };


    Rational.prototype.level = 1;


    Rational.prototype.liftTo = function(target) {
	if (target.level === 2)
	    return new FloatPoint(
		_integerDivideToFixnum(this.n, this.d));
	if (target.level === 3)
	    return new Complex(this, 0);
	return throwRuntimeError("invalid level of Number", this, target);
    };

    Rational.prototype.isFinite = function() {
	return true;
    };

    Rational.prototype.equals = function(other) {
	return (other instanceof Rational &&
		_integerEquals(this.n, other.n) &&
		_integerEquals(this.d, other.d));
    };



    Rational.prototype.isInteger = function() {
	return _integerIsOne(this.d);
    };

    Rational.prototype.isRational = function() {
        return true;
    };

    Rational.prototype.isReal = function() {
	return true;
    };


    Rational.prototype.add = function(other) {
	return Rational.makeInstance(_integerAdd(_integerMultiply(this.n, other.d),
						 _integerMultiply(this.d, other.n)),
				     _integerMultiply(this.d, other.d));
    };

    Rational.prototype.subtract = function(other) {
	return Rational.makeInstance(_integerSubtract(_integerMultiply(this.n, other.d),
						      _integerMultiply(this.d, other.n)),
				     _integerMultiply(this.d, other.d));
    };

    Rational.prototype.negate = function() { 
	return Rational.makeInstance(-this.n, this.d) 
    };

    Rational.prototype.multiply = function(other) {
	return Rational.makeInstance(_integerMultiply(this.n, other.n),
				     _integerMultiply(this.d, other.d));
    };

    Rational.prototype.divide = function(other) {
	if (_integerIsZero(this.d) || _integerIsZero(other.n)) {
	    throwRuntimeError("/: division by zero", this, other);
	}
	return Rational.makeInstance(_integerMultiply(this.n, other.d),
				     _integerMultiply(this.d, other.n));
    };


    Rational.prototype.toExact = function() {
	return this;
    };

    Rational.prototype.toInexact = function() {
	return FloatPoint.makeInstance(this.toFixnum());
    };


    Rational.prototype.isExact = function() {
        return true;
    };

    Rational.prototype.isInexact = function() {
        return false;
    };


    Rational.prototype.toFixnum = function() {
	return _integerDivideToFixnum(this.n, this.d);
    };

    Rational.prototype.numerator = function() {
	return this.n;
    };

    Rational.prototype.denominator = function() {
	return this.d;
    };

    Rational.prototype.greaterThan = function(other) {
	return _integerGreaterThan(_integerMultiply(this.n, other.d),
				   _integerMultiply(this.d, other.n));
    };

    Rational.prototype.greaterThanOrEqual = function(other) {
	return _integerGreaterThanOrEqual(_integerMultiply(this.n, other.d),
					  _integerMultiply(this.d, other.n));
    };

    Rational.prototype.lessThan = function(other) {
	return _integerLessThan(_integerMultiply(this.n, other.d),
				_integerMultiply(this.d, other.n));
    };

    Rational.prototype.lessThanOrEqual = function(other) {
	return _integerLessThanOrEqual(_integerMultiply(this.n, other.d),
				       _integerMultiply(this.d, other.n));
    };

    Rational.prototype.integerSqrt = function() {
	var result = sqrt(this);
	if (isRational(result)) {
	    return toExact(floor(result));
	} else if (isReal(result)) {
	    return toExact(floor(result));
	} else {
	    return Complex.makeInstance(toExact(floor(realPart(result))),
					toExact(floor(imaginaryPart(result))));
	}
    };


    Rational.prototype.sqrt = function() {
	if (_integerGreaterThanOrEqual(this.n,  0)) {
	    var newN = sqrt(this.n);
	    var newD = sqrt(this.d);
	    if (equals(floor(newN), newN) &&
		equals(floor(newD), newD)) {
		return Rational.makeInstance(newN, newD);
	    } else {
		return FloatPoint.makeInstance(_integerDivideToFixnum(newN, newD));
	    }
	} else {
	    var newN = sqrt(negate(this.n));
	    var newD = sqrt(this.d);
	    if (equals(floor(newN), newN) &&
		equals(floor(newD), newD)) {
		return Complex.makeInstance(
		    0,
		    Rational.makeInstance(newN, newD));
	    } else {
		return Complex.makeInstance(
		    0,
		    FloatPoint.makeInstance(_integerDivideToFixnum(newN, newD)));
	    }
	}
    };

    Rational.prototype.abs = function() {
	return Rational.makeInstance(abs(this.n),
				     this.d);
    };


    Rational.prototype.floor = function() {
	var quotient = _integerQuotient(this.n, this.d);
	if (_integerLessThan(this.n, 0)) {
	    return subtract(quotient, 1);
	} else {
	    return quotient;
	}
    };


    Rational.prototype.ceiling = function() {
	var quotient = _integerQuotient(this.n, this.d);
	if (_integerLessThan(this.n, 0)) {
	    return quotient;
	} else {
	    return add(quotient, 1);
	}
    };

    Rational.prototype.conjugate = function() {
	return this;
    };

    Rational.prototype.magnitude = Rational.prototype.abs;

    Rational.prototype.log = function(){
	return FloatPoint.makeInstance(Math.log(this.n / this.d));
    };

    Rational.prototype.angle = function(){
	if (_integerIsZero(this.n))
	    return 0;
	if (_integerGreaterThan(this.n, 0))
	    return 0;
	else
	    return FloatPoint.pi;
    };

    Rational.prototype.tan = function(){
	return FloatPoint.makeInstance(Math.tan(_integerDivideToFixnum(this.n, this.d)));
    };

    Rational.prototype.atan = function(){
	return FloatPoint.makeInstance(Math.atan(_integerDivideToFixnum(this.n, this.d)));
    };

    Rational.prototype.cos = function(){
	return FloatPoint.makeInstance(Math.cos(_integerDivideToFixnum(this.n, this.d)));
    };

    Rational.prototype.sin = function(){
	return FloatPoint.makeInstance(Math.sin(_integerDivideToFixnum(this.n, this.d)));
    };

    Rational.prototype.expt = function(a){
	if (isExactInteger(a) && greaterThanOrEqual(a, 0)) {
	    return fastExpt(this, a);
	}
	return FloatPoint.makeInstance(Math.pow(_integerDivideToFixnum(this.n, this.d),
						_integerDivideToFixnum(a.n, a.d)));
    };

    Rational.prototype.exp = function(){
	return FloatPoint.makeInstance(Math.exp(_integerDivideToFixnum(this.n, this.d)));
    };

    Rational.prototype.acos = function(){
	return FloatPoint.makeInstance(Math.acos(_integerDivideToFixnum(this.n, this.d)));
    };

    Rational.prototype.asin = function(){
	return FloatPoint.makeInstance(Math.asin(_integerDivideToFixnum(this.n, this.d)));
    };

    Rational.prototype.imaginaryPart = function(){
	return 0;
    };

    Rational.prototype.realPart = function(){
	return this;
    };


    Rational.prototype.round = function() {
	// FIXME: not correct when values are bignums
	if (equals(this.d, 2)) {
	    // Round to even if it's a n/2
	    var v = _integerDivideToFixnum(this.n, this.d);
	    var fl = Math.floor(v);
	    var ce = Math.ceil(v);
	    if (_integerIsZero(fl % 2)) {
		return fl;
	    }
	    else {
		return ce;
	    }
	} else {
	    return Math.round(this.n / this.d);
	}
    };


    Rational.makeInstance = function(n, d) {
	if (n === undefined)
	    throwRuntimeError("n undefined", n, d);

	if (d === undefined) { d = 1; }

	if (_integerLessThan(d, 0)) {
	    n = negate(n);
	    d = negate(d);
	}

	var divisor = _integerGcd(abs(n), abs(d));
	n = _integerQuotient(n, divisor);
	d = _integerQuotient(d, divisor);

	// Optimization: if we can get around construction the rational
	// in favor of just returning n, do it:
	if (_integerIsOne(d) || _integerIsZero(n)) {
	    return n;
	}

	return new Rational(n, d);
    };



    // Floating Point numbers
    var FloatPoint = function(n) {
	this.n = n;
    };
    FloatPoint = FloatPoint;


    var NaN = new FloatPoint(Number.NaN);
    var inf = new FloatPoint(Number.POSITIVE_INFINITY);
    var neginf = new FloatPoint(Number.NEGATIVE_INFINITY);

    // We use these two constants to represent the floating-point coersion
    // of bignums that can't be represented with fidelity.
    var TOO_POSITIVE_TO_REPRESENT = new FloatPoint(Number.POSITIVE_INFINITY);
    var TOO_NEGATIVE_TO_REPRESENT = new FloatPoint(Number.NEGATIVE_INFINITY);

    // Negative zero is a distinguished value representing -0.0.
    // There should only be one instance for -0.0.
    var NEGATIVE_ZERO = new FloatPoint(-0.0);
    var INEXACT_ZERO = new FloatPoint(0.0);

    FloatPoint.pi = new FloatPoint(Math.PI);
    FloatPoint.e = new FloatPoint(Math.E);
    FloatPoint.nan = NaN;
    FloatPoint.inf = inf;
    FloatPoint.neginf = neginf;

    FloatPoint.makeInstance = function(n) {
	if (isNaN(n)) {
	    return FloatPoint.nan;
	} else if (n === Number.POSITIVE_INFINITY) {
	    return FloatPoint.inf;
	} else if (n === Number.NEGATIVE_INFINITY) {
	    return FloatPoint.neginf;
	} else if (n === 0) {
	    if ((1/n) === -Infinity) {
		return NEGATIVE_ZERO;
	    } else {
		return INEXACT_ZERO;
	    }
	}
	return new FloatPoint(n);
    };


    FloatPoint.prototype.isExact = function() {
	return false;
    };

    FloatPoint.prototype.isInexact = function() {
	return true;
    };


    FloatPoint.prototype.isFinite = function() {
	return (isFinite(this.n) ||
		this === TOO_POSITIVE_TO_REPRESENT ||
		this === TOO_NEGATIVE_TO_REPRESENT);
    };


    FloatPoint.prototype.toExact = function() {
	// The precision of ieee is about 16 decimal digits, which we use here.
	if (! isFinite(this.n) || isNaN(this.n)) {
	    throwRuntimeError("toExact: no exact representation for " + this, this);
	}

	var stringRep = this.n.toString();
	var match = stringRep.match(/^(.*)\.(.*)$/);
	if (match) {
	    var intPart = parseInt(match[1]);
	    var fracPart = parseInt(match[2]);
	    var tenToDecimalPlaces = Math.pow(10, match[2].length);
	    return Rational.makeInstance(Math.round(this.n * tenToDecimalPlaces),
					 tenToDecimalPlaces);
	}
	else {
	    return this.n;
	}
    };

    FloatPoint.prototype.toInexact = function() {
	return this;
    };

    FloatPoint.prototype.isInexact = function() {
	return true;
    };


    FloatPoint.prototype.level = 2;


    FloatPoint.prototype.liftTo = function(target) {
	if (target.level === 3)
	    return new Complex(this, 0);
	return throwRuntimeError("invalid level of Number", this, target);
    };

    FloatPoint.prototype.toString = function() {
	if (isNaN(this.n))
	    return "+nan.0";
	if (this.n === Number.POSITIVE_INFINITY)
	    return "+inf.0";
	if (this.n === Number.NEGATIVE_INFINITY)
	    return "-inf.0";
	if (this === NEGATIVE_ZERO)
	    return "-0.0";
	var partialResult = this.n.toString();
	if (! partialResult.match('\\.')) {
	    return partialResult + ".0";
	} else {
	    return partialResult;
	}
    };


    FloatPoint.prototype.equals = function(other, aUnionFind) {
	return ((other instanceof FloatPoint) &&
		((this.n === other.n)));
    };



    FloatPoint.prototype.isRational = function() {
        return this.isFinite();
    };

    FloatPoint.prototype.isInteger = function() {
	return this.isFinite() && this.n === Math.floor(this.n);
    };

    FloatPoint.prototype.isReal = function() {
	return true;
    };


    // sign: Number -> {-1, 0, 1}
    var sign = function(n) {
	if (lessThan(n, 0)) {
	    return -1;
	} else if (greaterThan(n, 0)) {
	    return 1;
	} else if (n === NEGATIVE_ZERO) {
	    return -1;
	} else {
	    return 0;
	}
    };


    FloatPoint.prototype.add = function(other) {
	if (this.isFinite() && other.isFinite()) {
	    return FloatPoint.makeInstance(this.n + other.n);
	} else {
	    if (isNaN(this.n) || isNaN(other.n)) {
		return NaN;
	    } else if (this.isFinite() && ! other.isFinite()) {
		return other;
	    } else if (!this.isFinite() && other.isFinite()) {
		return this;
	    } else {
		return ((sign(this) * sign(other) === 1) ?
			this : NaN);
	    };
	}
    };

    FloatPoint.prototype.subtract = function(other) {
	if (this.isFinite() && other.isFinite()) {
	    return FloatPoint.makeInstance(this.n - other.n);
	} else if (isNaN(this.n) || isNaN(other.n)) {
	    return NaN;
	} else if (! this.isFinite() && ! other.isFinite()) {
	    if (sign(this) === sign(other)) {
		return NaN;
	    } else {
		return this;
	    }
	} else if (this.isFinite()) {
	    return multiply(other, -1);
	} else {  // other.isFinite()
	    return this;
	}
    };


    FloatPoint.prototype.negate = function() {
	return FloatPoint.makeInstance(-this.n);
    };

    FloatPoint.prototype.multiply = function(other) {
	return FloatPoint.makeInstance(this.n * other.n);
    };

    FloatPoint.prototype.divide = function(other) {
        return FloatPoint.makeInstance(this.n / other.n);
    };


    FloatPoint.prototype.toFixnum = function() {
	return this.n;
    };

    FloatPoint.prototype.numerator = function() {
	var stringRep = this.n.toString();
	var match = stringRep.match(/^(.*)\.(.*)$/);
	if (match) {
	    var afterDecimal = parseInt(match[2]);
	    var factorToInt = Math.pow(10, match[2].length);
	    var extraFactor = _integerGcd(factorToInt, afterDecimal);
	    var multFactor = factorToInt / extraFactor;
	    return FloatPoint.makeInstance( Math.round(this.n * multFactor) );
	} else {
	    return this;
	}
    };

    FloatPoint.prototype.denominator = function() {
	var stringRep = this.n.toString();
	var match = stringRep.match(/^(.*)\.(.*)$/);
	if (match) {
	    var afterDecimal = parseInt(match[2]);
	    var factorToInt = Math.pow(10, match[2].length);
	    var extraFactor = _integerGcd(factorToInt, afterDecimal);
	    return FloatPoint.makeInstance( Math.round(factorToInt/extraFactor) );
	} else {
	    return FloatPoint.makeInstance(1);
	}
    };


    FloatPoint.prototype.floor = function() {
	return FloatPoint.makeInstance(Math.floor(this.n));
    };

    FloatPoint.prototype.ceiling = function() {
	return FloatPoint.makeInstance(Math.ceil(this.n));
    };


    FloatPoint.prototype.greaterThan = function(other) {
	return this.n > other.n;
    };

    FloatPoint.prototype.greaterThanOrEqual = function(other) {
	return this.n >= other.n;
    };

    FloatPoint.prototype.lessThan = function(other) {
	return this.n < other.n;
    };

    FloatPoint.prototype.lessThanOrEqual = function(other) {
	return this.n <= other.n;
    };


    FloatPoint.prototype.integerSqrt = function() {
	if (this === NEGATIVE_ZERO) { return this; }
	if (isInteger(this)) {
	    if(this.n >= 0) {
	        return FloatPoint.makeInstance(Math.floor(Math.sqrt(this.n)));
	    } else {
	        return Complex.makeInstance(
		    INEXACT_ZERO,
		    FloatPoint.makeInstance(Math.floor(Math.sqrt(-this.n))));
	    }
	} else {
	    throwRuntimeError("integerSqrt: can only be applied to an integer", this);
	}
    };

    FloatPoint.prototype.sqrt = function() {
	if (this.n < 0) {
	    var result = Complex.makeInstance(
		0,
		FloatPoint.makeInstance(Math.sqrt(-this.n)));
	    return result;
	} else {
	    return FloatPoint.makeInstance(Math.sqrt(this.n));
	}
    };

    FloatPoint.prototype.abs = function() {
	return FloatPoint.makeInstance(Math.abs(this.n));
    };



    FloatPoint.prototype.log = function(){
	if (this.n < 0)
	    return (new Complex(this, 0)).log();
	else
	    return FloatPoint.makeInstance(Math.log(this.n));
    };

    FloatPoint.prototype.angle = function(){
	if (0 === this.n)
	    return 0;
	if (this.n > 0)
	    return 0;
	else
	    return FloatPoint.pi;
    };

    FloatPoint.prototype.tan = function(){
	return FloatPoint.makeInstance(Math.tan(this.n));
    };

    FloatPoint.prototype.atan = function(){
	return FloatPoint.makeInstance(Math.atan(this.n));
    };

    FloatPoint.prototype.cos = function(){
	return FloatPoint.makeInstance(Math.cos(this.n));
    };

    FloatPoint.prototype.sin = function(){
	return FloatPoint.makeInstance(Math.sin(this.n));
    };

    FloatPoint.prototype.expt = function(a){
	if (this.n === 1) {
	    if (a.isFinite()) {
		return this;
	    } else if (isNaN(a.n)){
		return this;
	    } else {
		return this;
	    }
	} else {
	    return FloatPoint.makeInstance(Math.pow(this.n, a.n));
	}
    };

    FloatPoint.prototype.exp = function(){
	return FloatPoint.makeInstance(Math.exp(this.n));
    };

    FloatPoint.prototype.acos = function(){
	return FloatPoint.makeInstance(Math.acos(this.n));
    };

    FloatPoint.prototype.asin = function(){
	return FloatPoint.makeInstance(Math.asin(this.n));
    };

    FloatPoint.prototype.imaginaryPart = function(){
	return 0;
    };

    FloatPoint.prototype.realPart = function(){
	return this;
    };


    FloatPoint.prototype.round = function(){
	if (isFinite(this.n)) {
	    if (this === NEGATIVE_ZERO) {
		return this;
	    }
	    if (Math.abs(Math.floor(this.n) - this.n) === 0.5) {
		if (Math.floor(this.n) % 2 === 0)
		    return FloatPoint.makeInstance(Math.floor(this.n));
		return FloatPoint.makeInstance(Math.ceil(this.n));
	    } else {
		return FloatPoint.makeInstance(Math.round(this.n));
	    }
	} else {
	    return this;
	}
    };


    FloatPoint.prototype.conjugate = function() {
	return this;
    };

    FloatPoint.prototype.magnitude = FloatPoint.prototype.abs;



    //////////////////////////////////////////////////////////////////////
    // Complex numbers
    //////////////////////////////////////////////////////////////////////

    var Complex = function(r, i){
	this.r = r;
	this.i = i;
    };

    // Constructs a complex number from two basic number r and i.  r and i can
    // either be plt.type.Rational or plt.type.FloatPoint.
    Complex.makeInstance = function(r, i){
	if (i === undefined) { i = 0; }
	if (isExact(i) && isInteger(i) && _integerIsZero(i)) {
	    return r;
	}
	if (isInexact(r) || isInexact(i)) {
	    r = toInexact(r);
	    i = toInexact(i);
	}
	return new Complex(r, i);
    };

    Complex.prototype.toString = function() {
	var realPart = this.r.toString(), imagPart = this.i.toString();
	if (imagPart[0] === '-' || imagPart[0] === '+') {
	    return realPart + imagPart + 'i';
	} else {
	    return realPart + "+" + imagPart + 'i';
	}
    };


    Complex.prototype.isFinite = function() {
	return isSchemeNumberFinite(this.r) && isSchemeNumberFinite(this.i);
    };


    Complex.prototype.isRational = function() {
	return isRational(this.r) && eqv(this.i, 0);
    };

    Complex.prototype.isInteger = function() {
	return (isInteger(this.r) &&
		eqv(this.i, 0));
    };

    Complex.prototype.toExact = function() {
	return Complex.makeInstance( toExact(this.r), toExact(this.i) );
    };

    Complex.prototype.toInexact = function() {
	return Complex.makeInstance(toInexact(this.r),
				    toInexact(this.i));
    };


    Complex.prototype.isExact = function() {
        return isExact(this.r) && isExact(this.i);
    };


    Complex.prototype.isInexact = function() {
	return isInexact(this.r) || isInexact(this.i);
    };


    Complex.prototype.level = 3;


    Complex.prototype.liftTo = function(target){
	throwRuntimeError("Don't know how to lift Complex number", this, target);
    };

    Complex.prototype.equals = function(other) {
	var result = ((other instanceof Complex) &&
		      (equals(this.r, other.r)) &&
		      (equals(this.i, other.i)));
	return result;
    };



    Complex.prototype.greaterThan = function(other) {
	if (! this.isReal() || ! other.isReal()) {
	    throwRuntimeError(">: expects argument of type real number", this, other);
	}
	return greaterThan(this.r, other.r);
    };

    Complex.prototype.greaterThanOrEqual = function(other) {
	if (! this.isReal() || ! other.isReal()) {
	    throwRuntimeError(">=: expects argument of type real number", this, other);
	}
	return greaterThanOrEqual(this.r, other.r);
    };

    Complex.prototype.lessThan = function(other) {
	if (! this.isReal() || ! other.isReal()) {
	    throwRuntimeError("<: expects argument of type real number", this, other);
	}
	return lessThan(this.r, other.r);
    };

    Complex.prototype.lessThanOrEqual = function(other) {
	if (! this.isReal() || ! other.isReal()) {
	    throwRuntimeError("<=: expects argument of type real number", this, other);
	}
	return lessThanOrEqual(this.r, other.r);
    };


    Complex.prototype.abs = function(){
	if (!equals(this.i, 0).valueOf())
	    throwRuntimeError("abs: expects argument of type real number", this);
	return abs(this.r);
    };

    Complex.prototype.toFixnum = function(){
	if (!equals(this.i, 0).valueOf())
	    throwRuntimeError("toFixnum: expects argument of type real number", this);
	return toFixnum(this.r);
    };

    Complex.prototype.numerator = function() {
	if (!this.isReal())
	    throwRuntimeError("numerator: can only be applied to real number", this);
	return numerator(this.n);
    };


    Complex.prototype.denominator = function() {
	if (!this.isReal())
	    throwRuntimeError("floor: can only be applied to real number", this);
	return denominator(this.n);
    };

    Complex.prototype.add = function(other){
	return Complex.makeInstance(
	    add(this.r, other.r),
	    add(this.i, other.i));
    };

    Complex.prototype.subtract = function(other){
	return Complex.makeInstance(
	    subtract(this.r, other.r),
	    subtract(this.i, other.i));
    };

    Complex.prototype.negate = function() {
	return Complex.makeInstance(negate(this.r),
				    negate(this.i));
    };


    Complex.prototype.multiply = function(other){
	// If the other value is real, just do primitive division
	if (other.isReal()) {
	    return Complex.makeInstance(
		multiply(this.r, other.r),
		multiply(this.i, other.r));
	}
	var r = subtract(
	    multiply(this.r, other.r),
	    multiply(this.i, other.i));
	var i = add(
	    multiply(this.r, other.i),
	    multiply(this.i, other.r));
	return Complex.makeInstance(r, i);
    };





    Complex.prototype.divide = function(other){
	var a, b, c, d, r, x, y;
	// If the other value is real, just do primitive division
	if (other.isReal()) {
	    return Complex.makeInstance(
		divide(this.r, other.r),
		divide(this.i, other.r));
	}

	if (this.isInexact() || other.isInexact()) {
	    // http://portal.acm.org/citation.cfm?id=1039814
	    // We currently use Smith's method, though we should
	    // probably switch over to Priest's method.
	    a = this.r;
	    b = this.i;
	    c = other.r;
	    d = other.i;
	    if (lessThanOrEqual(abs(d), abs(c))) {
		r = divide(d, c);
		x = divide(add(a, multiply(b, r)),
			   add(c, multiply(d, r)));
		y = divide(subtract(b, multiply(a, r)),
			   add(c, multiply(d, r)));
	    } else {
		r = divide(c, d);
		x = divide(add(multiply(a, r), b),
			   add(multiply(c, r), d));
		y = divide(subtract(multiply(b, r), a),
			   add(multiply(c, r), d));
	    }
	    return Complex.makeInstance(x, y);
	} else {
	    var con = conjugate(other);
	    var up = multiply(this, con);

	    // Down is guaranteed to be real by this point.
	    var down = realPart(multiply(other, con));

	    var result = Complex.makeInstance(
		divide(realPart(up), down),
		divide(imaginaryPart(up), down));
	    return result;
	}
    };

    Complex.prototype.conjugate = function(){
	var result = Complex.makeInstance(
	    this.r,
	    subtract(0, this.i));

	return result;
    };

    Complex.prototype.magnitude = function(){
	var sum = add(
	    multiply(this.r, this.r),
	    multiply(this.i, this.i));
	return sqrt(sum);
    };

    Complex.prototype.isReal = function(){
	return eqv(this.i, 0);
    };

    Complex.prototype.integerSqrt = function() {
	if (isInteger(this)) {
	    return integerSqrt(this.r);
	} else {
	    throwRuntimeError("integerSqrt: can only be applied to an integer", this);
	}
    };

    Complex.prototype.sqrt = function(){
	if (this.isReal())
	    return sqrt(this.r);
	// http://en.wikipedia.org/wiki/Square_root#Square_roots_of_negative_and_complex_numbers
	var r_plus_x = add(this.magnitude(), this.r);

	var r = sqrt(halve(r_plus_x));

	var i = divide(this.i, sqrt(multiply(r_plus_x, 2)));


	return Complex.makeInstance(r, i);
    };

    Complex.prototype.log = function(){
	var m = this.magnitude();
	var theta = this.angle();
	var result = add(
	    log(m),
	    timesI(theta));
	return result;
    };

    Complex.prototype.angle = function(){
	if (this.isReal()) {
	    return angle(this.r);
	}
	if (equals(0, this.r)) {
	    var tmp = halve(FloatPoint.pi);
	    return greaterThan(this.i, 0) ?
		tmp : negate(tmp);
	} else {
	    var tmp = atan(divide(abs(this.i), abs(this.r)));
	    if (greaterThan(this.r, 0)) {
		return greaterThan(this.i, 0) ?
		    tmp : negate(tmp);
	    } else {
		return greaterThan(this.i, 0) ?
		    subtract(FloatPoint.pi, tmp) : subtract(tmp, FloatPoint.pi);
	    }
	}
    };

    var plusI = Complex.makeInstance(0, 1);
    var minusI = Complex.makeInstance(0, -1);


    Complex.prototype.tan = function() {
	return divide(this.sin(), this.cos());
    };

    Complex.prototype.atan = function(){
	if (equals(this, plusI) ||
	    equals(this, minusI)) {
	    return neginf;
	}
	return multiply(
	    plusI,
	    multiply(
		FloatPoint.makeInstance(0.5),
		log(divide(
		    add(plusI, this),
		    add(
			plusI,
			subtract(0, this))))));
    };

    Complex.prototype.cos = function(){
	if (this.isReal())
	    return cos(this.r);
	var iz = timesI(this);
	var iz_negate = negate(iz);

	return halve(add(exp(iz), exp(iz_negate)));
    };

    Complex.prototype.sin = function(){
	if (this.isReal())
	    return sin(this.r);
	var iz = timesI(this);
	var iz_negate = negate(iz);
	var z2 = Complex.makeInstance(0, 2);
	var exp_negate = subtract(exp(iz), exp(iz_negate));
	var result = divide(exp_negate, z2);
	return result;
    };


    Complex.prototype.expt = function(y){
	if (isExactInteger(y) && greaterThanOrEqual(y, 0)) {
	    return fastExpt(this, y);
	}
	var expo = multiply(y, this.log());
	return exp(expo);
    };

    Complex.prototype.exp = function(){
	var r = exp(this.r);
	var cos_a = cos(this.i);
	var sin_a = sin(this.i);

	return multiply(
	    r,
	    add(cos_a, timesI(sin_a)));
    };

    Complex.prototype.acos = function(){
	if (this.isReal())
	    return acos(this.r);
	var pi_half = halve(FloatPoint.pi);
	var iz = timesI(this);
	var root = sqrt(subtract(1, sqr(this)));
	var l = timesI(log(add(iz, root)));
	return add(pi_half, l);
    };

    Complex.prototype.asin = function(){
	if (this.isReal())
	    return asin(this.r);

	var oneNegateThisSq =
	    subtract(1, sqr(this));
	var sqrtOneNegateThisSq = sqrt(oneNegateThisSq);
	return multiply(2, atan(divide(this,
				       add(1, sqrtOneNegateThisSq))));
    };

    Complex.prototype.ceiling = function(){
	if (!this.isReal())
	    throwRuntimeError("ceiling: can only be applied to real number", this);
	return ceiling(this.r);
    };

    Complex.prototype.floor = function(){
	if (!this.isReal())
	    throwRuntimeError("floor: can only be applied to real number", this);
	return floor(this.r);
    };

    Complex.prototype.imaginaryPart = function(){
	return this.i;
    };

    Complex.prototype.realPart = function(){
	return this.r;
    };

    Complex.prototype.round = function(){
	if (!this.isReal())
	    throwRuntimeError("round: can only be applied to real number", this);
	return round(this.r);
    };



    var rationalRegexp = new RegExp("^([+-]?\\d+)/(\\d+)$");
    var complexRegexp = new RegExp("^([+-]?[\\d\\w/\\.]*)([+-])([\\d\\w/\\.]*)i$");
    var digitRegexp = new RegExp("^[+-]?\\d+$");
    var flonumRegexp = new RegExp("^([+-]?\\d*)\\.(\\d*)$");
    var scientificPattern = new RegExp("^([+-]?\\d*\\.?\\d*)[Ee](\\+?\\d+)$");

    // fromString: string -> (scheme-number | false)
    var fromString = function(x) {
	var aMatch = x.match(rationalRegexp);
	if (aMatch) {
	    return Rational.makeInstance(fromString(aMatch[1]),
					 fromString(aMatch[2]));
	}

	var cMatch = x.match(complexRegexp);
	if (cMatch) {
	    return Complex.makeInstance(fromString(cMatch[1] || "0"),
					fromString(cMatch[2] + (cMatch[3] || "1")));
	}

	// Floating point tests
	if (x === '+nan.0' || x === '-nan.0')
	    return FloatPoint.nan;
	if (x === '+inf.0')
	    return FloatPoint.inf;
	if (x === '-inf.0')
	    return FloatPoint.neginf;
	if (x === "-0.0") {
	    return NEGATIVE_ZERO;
	}
	if (x.match(flonumRegexp) ||  x.match(scientificPattern)) {
	    return FloatPoint.makeInstance(Number(x));
	}

	// Finally, integer tests.
	if (x.match(digitRegexp)) {
	    var n = Number(x);
	    if (isOverflow(n)) {
		return makeBignum(x);
	    } else {
		return n;
	    }
	} else {
	    return false;
	}
    };





    //////////////////////////////////////////////////////////////////////
    //////////////////////////////////////////////////////////////////////
    //////////////////////////////////////////////////////////////////////
    //////////////////////////////////////////////////////////////////////
    //////////////////////////////////////////////////////////////////////

    // The code below comes from Tom Wu's BigInteger implementation:

    // Copyright (c) 2005  Tom Wu
    // All Rights Reserved.
    // See "LICENSE" for details.

    // Basic JavaScript BN library - subset useful for RSA encryption.

    // Bits per digit
    var dbits;

    // JavaScript engine analysis
    var canary = 0xdeadbeefcafe;
    var j_lm = ((canary&0xffffff)==0xefcafe);

    // (public) Constructor
    function BigInteger(a,b,c) {
	if(a != null)
	    if("number" == typeof a) this.fromNumber(a,b,c);
	else if(b == null && "string" != typeof a) this.fromString(a,256);
	else this.fromString(a,b);
    }

    // return new, unset BigInteger
    function nbi() { return new BigInteger(null); }

    // am: Compute w_j += (x*this_i), propagate carries,
    // c is initial carry, returns final carry.
    // c < 3*dvalue, x < 2*dvalue, this_i < dvalue
    // We need to select the fastest one that works in this environment.

    // am1: use a single mult and divide to get the high bits,
    // max digit bits should be 26 because
    // max internal value = 2*dvalue^2-2*dvalue (< 2^53)
    function am1(i,x,w,j,c,n) {
	while(--n >= 0) {
	    var v = x*this[i++]+w[j]+c;
	    c = Math.floor(v/0x4000000);
	    w[j++] = v&0x3ffffff;
	}
	return c;
    }
    // am2 avoids a big mult-and-extract completely.
    // Max digit bits should be <= 30 because we do bitwise ops
    // on values up to 2*hdvalue^2-hdvalue-1 (< 2^31)
    function am2(i,x,w,j,c,n) {
	var xl = x&0x7fff, xh = x>>15;
	while(--n >= 0) {
	    var l = this[i]&0x7fff;
	    var h = this[i++]>>15;
	    var m = xh*l+h*xl;
	    l = xl*l+((m&0x7fff)<<15)+w[j]+(c&0x3fffffff);
	    c = (l>>>30)+(m>>>15)+xh*h+(c>>>30);
	    w[j++] = l&0x3fffffff;
	}
	return c;
    }
    // Alternately, set max digit bits to 28 since some
    // browsers slow down when dealing with 32-bit numbers.
    function am3(i,x,w,j,c,n) {
	var xl = x&0x3fff, xh = x>>14;
	while(--n >= 0) {
	    var l = this[i]&0x3fff;
	    var h = this[i++]>>14;
	    var m = xh*l+h*xl;
	    l = xl*l+((m&0x3fff)<<14)+w[j]+c;
	    c = (l>>28)+(m>>14)+xh*h;
	    w[j++] = l&0xfffffff;
	}
	return c;
    }
    if(j_lm && (typeof(navigator) !== 'undefined' && navigator.appName == "Microsoft Internet Explorer")) {
	BigInteger.prototype.am = am2;
	dbits = 30;
    }
    else if(j_lm && (typeof(navigator) !== 'undefined' && navigator.appName != "Netscape")) {
	BigInteger.prototype.am = am1;
	dbits = 26;
    }
    else { // Mozilla/Netscape seems to prefer am3
	BigInteger.prototype.am = am3;
	dbits = 28;
    }

    BigInteger.prototype.DB = dbits;
    BigInteger.prototype.DM = ((1<<dbits)-1);
    BigInteger.prototype.DV = (1<<dbits);

    var BI_FP = 52;
    BigInteger.prototype.FV = Math.pow(2,BI_FP);
    BigInteger.prototype.F1 = BI_FP-dbits;
    BigInteger.prototype.F2 = 2*dbits-BI_FP;

    // Digit conversions
    var BI_RM = "0123456789abcdefghijklmnopqrstuvwxyz";
    var BI_RC = [];
    var rr,vv;
    rr = "0".charCodeAt(0);
    for(vv = 0; vv <= 9; ++vv) BI_RC[rr++] = vv;
    rr = "a".charCodeAt(0);
    for(vv = 10; vv < 36; ++vv) BI_RC[rr++] = vv;
    rr = "A".charCodeAt(0);
    for(vv = 10; vv < 36; ++vv) BI_RC[rr++] = vv;

    function int2char(n) { return BI_RM.charAt(n); }
    function intAt(s,i) {
	var c = BI_RC[s.charCodeAt(i)];
	return (c==null)?-1:c;
    }

    // (protected) copy this to r
    function bnpCopyTo(r) {
	for(var i = this.t-1; i >= 0; --i) r[i] = this[i];
	r.t = this.t;
	r.s = this.s;
    }

    // (protected) set from integer value x, -DV <= x < DV
    function bnpFromInt(x) {
	this.t = 1;
	this.s = (x<0)?-1:0;
	if(x > 0) this[0] = x;
	else if(x < -1) this[0] = x+DV;
	else this.t = 0;
    }

    // return bigint initialized to value
    function nbv(i) { var r = nbi(); r.fromInt(i); return r; }

    // (protected) set from string and radix
    function bnpFromString(s,b) {
	var k;
	if(b == 16) k = 4;
	else if(b == 8) k = 3;
	else if(b == 256) k = 8; // byte array
	else if(b == 2) k = 1;
	else if(b == 32) k = 5;
	else if(b == 4) k = 2;
	else { this.fromRadix(s,b); return; }
	this.t = 0;
	this.s = 0;
	var i = s.length, mi = false, sh = 0;
	while(--i >= 0) {
	    var x = (k==8)?s[i]&0xff:intAt(s,i);
	    if(x < 0) {
		if(s.charAt(i) == "-") mi = true;
		continue;
	    }
	    mi = false;
	    if(sh == 0)
		this[this.t++] = x;
	    else if(sh+k > this.DB) {
		this[this.t-1] |= (x&((1<<(this.DB-sh))-1))<<sh;
		this[this.t++] = (x>>(this.DB-sh));
	    }
	    else
		this[this.t-1] |= x<<sh;
	    sh += k;
	    if(sh >= this.DB) sh -= this.DB;
	}
	if(k == 8 && (s[0]&0x80) != 0) {
	    this.s = -1;
	    if(sh > 0) this[this.t-1] |= ((1<<(this.DB-sh))-1)<<sh;
	}
	this.clamp();
	if(mi) BigInteger.ZERO.subTo(this,this);
    }

    // (protected) clamp off excess high words
    function bnpClamp() {
	var c = this.s&this.DM;
	while(this.t > 0 && this[this.t-1] == c) --this.t;
    }

    // (public) return string representation in given radix
    function bnToString(b) {
	if(this.s < 0) return "-"+this.negate().toString(b);
	var k;
	if(b == 16) k = 4;
	else if(b == 8) k = 3;
	else if(b == 2) k = 1;
	else if(b == 32) k = 5;
	else if(b == 4) k = 2;
	else return this.toRadix(b);
	var km = (1<<k)-1, d, m = false, r = [], i = this.t;
	var p = this.DB-(i*this.DB)%k;
	if(i-- > 0) {
	    if(p < this.DB && (d = this[i]>>p) > 0) { m = true; r.push(int2char(d)); }
	    while(i >= 0) {
		if(p < k) {
		    d = (this[i]&((1<<p)-1))<<(k-p);
		    d |= this[--i]>>(p+=this.DB-k);
		}
		else {
		    d = (this[i]>>(p-=k))&km;
		    if(p <= 0) { p += this.DB; --i; }
		}
		if(d > 0) m = true;
		if(m) r.push(int2char(d));
	    }
	}
	return m?r.join(""):"0";
    }

    // (public) -this
    function bnNegate() { var r = nbi(); BigInteger.ZERO.subTo(this,r); return r; }

    // (public) |this|
    function bnAbs() { return (this.s<0)?this.negate():this; }

    // (public) return + if this > a, - if this < a, 0 if equal
    function bnCompareTo(a) {
	var r = this.s-a.s;
	if(r != 0) return r;
	var i = this.t;
	if ( this.s < 0 ) {
		r = a.t - i;
	}
	else {
		r = i - a.t;
	}
	if(r != 0) return r;
	while(--i >= 0) if((r=this[i]-a[i]) != 0) return r;
	return 0;
    }

    // returns bit length of the integer x
    function nbits(x) {
	var r = 1, t;
	if((t=x>>>16) != 0) { x = t; r += 16; }
	if((t=x>>8) != 0) { x = t; r += 8; }
	if((t=x>>4) != 0) { x = t; r += 4; }
	if((t=x>>2) != 0) { x = t; r += 2; }
	if((t=x>>1) != 0) { x = t; r += 1; }
	return r;
    }

    // (public) return the number of bits in "this"
    function bnBitLength() {
	if(this.t <= 0) return 0;
	return this.DB*(this.t-1)+nbits(this[this.t-1]^(this.s&this.DM));
    }

    // (protected) r = this << n*DB
    function bnpDLShiftTo(n,r) {
	var i;
	for(i = this.t-1; i >= 0; --i) r[i+n] = this[i];
	for(i = n-1; i >= 0; --i) r[i] = 0;
	r.t = this.t+n;
	r.s = this.s;
    }

    // (protected) r = this >> n*DB
    function bnpDRShiftTo(n,r) {
	for(var i = n; i < this.t; ++i) r[i-n] = this[i];
	r.t = Math.max(this.t-n,0);
	r.s = this.s;
    }

    // (protected) r = this << n
    function bnpLShiftTo(n,r) {
	var bs = n%this.DB;
	var cbs = this.DB-bs;
	var bm = (1<<cbs)-1;
	var ds = Math.floor(n/this.DB), c = (this.s<<bs)&this.DM, i;
	for(i = this.t-1; i >= 0; --i) {
	    r[i+ds+1] = (this[i]>>cbs)|c;
	    c = (this[i]&bm)<<bs;
	}
	for(i = ds-1; i >= 0; --i) r[i] = 0;
	r[ds] = c;
	r.t = this.t+ds+1;
	r.s = this.s;
	r.clamp();
    }

    // (protected) r = this >> n
    function bnpRShiftTo(n,r) {
	r.s = this.s;
	var ds = Math.floor(n/this.DB);
	if(ds >= this.t) { r.t = 0; return; }
	var bs = n%this.DB;
	var cbs = this.DB-bs;
	var bm = (1<<bs)-1;
	r[0] = this[ds]>>bs;
	for(var i = ds+1; i < this.t; ++i) {
	    r[i-ds-1] |= (this[i]&bm)<<cbs;
	    r[i-ds] = this[i]>>bs;
	}
	if(bs > 0) r[this.t-ds-1] |= (this.s&bm)<<cbs;
	r.t = this.t-ds;
	r.clamp();
    }

    // (protected) r = this - a
    function bnpSubTo(a,r) {
	var i = 0, c = 0, m = Math.min(a.t,this.t);
	while(i < m) {
	    c += this[i]-a[i];
	    r[i++] = c&this.DM;
	    c >>= this.DB;
	}
	if(a.t < this.t) {
	    c -= a.s;
	    while(i < this.t) {
		c += this[i];
		r[i++] = c&this.DM;
		c >>= this.DB;
	    }
	    c += this.s;
	}
	else {
	    c += this.s;
	    while(i < a.t) {
		c -= a[i];
		r[i++] = c&this.DM;
		c >>= this.DB;
	    }
	    c -= a.s;
	}
	r.s = (c<0)?-1:0;
	if(c < -1) r[i++] = this.DV+c;
	else if(c > 0) r[i++] = c;
	r.t = i;
	r.clamp();
    }

    // (protected) r = this * a, r != this,a (HAC 14.12)
    // "this" should be the larger one if appropriate.
    function bnpMultiplyTo(a,r) {
	var x = this.abs(), y = a.abs();
	var i = x.t;
	r.t = i+y.t;
	while(--i >= 0) r[i] = 0;
	for(i = 0; i < y.t; ++i) r[i+x.t] = x.am(0,y[i],r,i,0,x.t);
	r.s = 0;
	r.clamp();
	if(this.s != a.s) BigInteger.ZERO.subTo(r,r);
    }

    // (protected) r = this^2, r != this (HAC 14.16)
    function bnpSquareTo(r) {
	var x = this.abs();
	var i = r.t = 2*x.t;
	while(--i >= 0) r[i] = 0;
	for(i = 0; i < x.t-1; ++i) {
	    var c = x.am(i,x[i],r,2*i,0,1);
	    if((r[i+x.t]+=x.am(i+1,2*x[i],r,2*i+1,c,x.t-i-1)) >= x.DV) {
		r[i+x.t] -= x.DV;
		r[i+x.t+1] = 1;
	    }
	}
	if(r.t > 0) r[r.t-1] += x.am(i,x[i],r,2*i,0,1);
	r.s = 0;
	r.clamp();
    }


    // (protected) divide this by m, quotient and remainder to q, r (HAC 14.20)
    // r != q, this != m.  q or r may be null.
    function bnpDivRemTo(m,q,r) {
	var pm = m.abs();
	if(pm.t <= 0) return;
	var pt = this.abs();
	if(pt.t < pm.t) {
	    if(q != null) q.fromInt(0);
	    if(r != null) this.copyTo(r);
	    return;
	}
	if(r == null) r = nbi();
	var y = nbi(), ts = this.s, ms = m.s;
	var nsh = this.DB-nbits(pm[pm.t-1]);	// normalize modulus
	if(nsh > 0) { pm.lShiftTo(nsh,y); pt.lShiftTo(nsh,r); }
	else { pm.copyTo(y); pt.copyTo(r); }
	var ys = y.t;
	var y0 = y[ys-1];
	if(y0 == 0) return;
	var yt = y0*(1<<this.F1)+((ys>1)?y[ys-2]>>this.F2:0);
	var d1 = this.FV/yt, d2 = (1<<this.F1)/yt, e = 1<<this.F2;
	var i = r.t, j = i-ys, t = (q==null)?nbi():q;
	y.dlShiftTo(j,t);
	if(r.compareTo(t) >= 0) {
	    r[r.t++] = 1;
	    r.subTo(t,r);
	}
	BigInteger.ONE.dlShiftTo(ys,t);
	t.subTo(y,y);	// "negative" y so we can replace sub with am later
	while(y.t < ys) y[y.t++] = 0;
	while(--j >= 0) {
	    // Estimate quotient digit
	    var qd = (r[--i]==y0)?this.DM:Math.floor(r[i]*d1+(r[i-1]+e)*d2);
	    if((r[i]+=y.am(0,qd,r,j,0,ys)) < qd) {	// Try it out
		y.dlShiftTo(j,t);
		r.subTo(t,r);
		while(r[i] < --qd) r.subTo(t,r);
	    }
	}
	if(q != null) {
	    r.drShiftTo(ys,q);
	    if(ts != ms) BigInteger.ZERO.subTo(q,q);
	}
	r.t = ys;
	r.clamp();
	if(nsh > 0) r.rShiftTo(nsh,r);	// Denormalize remainder
	if(ts < 0) BigInteger.ZERO.subTo(r,r);
    }

    // (public) this mod a
    function bnMod(a) {
	var r = nbi();
	this.abs().divRemTo(a,null,r);
	if(this.s < 0 && r.compareTo(BigInteger.ZERO) > 0) a.subTo(r,r);
	return r;
    }

    // Modular reduction using "classic" algorithm
    function Classic(m) { this.m = m; }
    function cConvert(x) {
	if(x.s < 0 || x.compareTo(this.m) >= 0) return x.mod(this.m);
	else return x;
    }
    function cRevert(x) { return x; }
    function cReduce(x) { x.divRemTo(this.m,null,x); }
    function cMulTo(x,y,r) { x.multiplyTo(y,r); this.reduce(r); }
    function cSqrTo(x,r) { x.squareTo(r); this.reduce(r); }

    Classic.prototype.convert = cConvert;
    Classic.prototype.revert = cRevert;
    Classic.prototype.reduce = cReduce;
    Classic.prototype.mulTo = cMulTo;
    Classic.prototype.sqrTo = cSqrTo;

    // (protected) return "-1/this % 2^DB"; useful for Mont. reduction
    // justification:
    //         xy == 1 (mod m)
    //         xy =  1+km
    //   xy(2-xy) = (1+km)(1-km)
    // x[y(2-xy)] = 1-k^2m^2
    // x[y(2-xy)] == 1 (mod m^2)
    // if y is 1/x mod m, then y(2-xy) is 1/x mod m^2
    // should reduce x and y(2-xy) by m^2 at each step to keep size bounded.
    // JS multiply "overflows" differently from C/C++, so care is needed here.
    function bnpInvDigit() {
	if(this.t < 1) return 0;
	var x = this[0];
	if((x&1) == 0) return 0;
	var y = x&3;		// y == 1/x mod 2^2
	y = (y*(2-(x&0xf)*y))&0xf;	// y == 1/x mod 2^4
	y = (y*(2-(x&0xff)*y))&0xff;	// y == 1/x mod 2^8
	y = (y*(2-(((x&0xffff)*y)&0xffff)))&0xffff;	// y == 1/x mod 2^16
	// last step - calculate inverse mod DV directly;
	// assumes 16 < DB <= 32 and assumes ability to handle 48-bit ints
	y = (y*(2-x*y%this.DV))%this.DV;		// y == 1/x mod 2^dbits
	// we really want the negative inverse, and -DV < y < DV
	return (y>0)?this.DV-y:-y;
    }

    // Montgomery reduction
    function Montgomery(m) {
	this.m = m;
	this.mp = m.invDigit();
	this.mpl = this.mp&0x7fff;
	this.mph = this.mp>>15;
	this.um = (1<<(m.DB-15))-1;
	this.mt2 = 2*m.t;
    }

    // xR mod m
    function montConvert(x) {
	var r = nbi();
	x.abs().dlShiftTo(this.m.t,r);
	r.divRemTo(this.m,null,r);
	if(x.s < 0 && r.compareTo(BigInteger.ZERO) > 0) this.m.subTo(r,r);
	return r;
    }

    // x/R mod m
    function montRevert(x) {
	var r = nbi();
	x.copyTo(r);
	this.reduce(r);
	return r;
    }

    // x = x/R mod m (HAC 14.32)
    function montReduce(x) {
	while(x.t <= this.mt2)	// pad x so am has enough room later
	    x[x.t++] = 0;
	for(var i = 0; i < this.m.t; ++i) {
	    // faster way of calculating u0 = x[i]*mp mod DV
	    var j = x[i]&0x7fff;
	    var u0 = (j*this.mpl+(((j*this.mph+(x[i]>>15)*this.mpl)&this.um)<<15))&x.DM;
	    // use am to combine the multiply-shift-add into one call
	    j = i+this.m.t;
	    x[j] += this.m.am(0,u0,x,i,0,this.m.t);
	    // propagate carry
	    while(x[j] >= x.DV) { x[j] -= x.DV; x[++j]++; }
	}
	x.clamp();
	x.drShiftTo(this.m.t,x);
	if(x.compareTo(this.m) >= 0) x.subTo(this.m,x);
    }

    // r = "x^2/R mod m"; x != r
    function montSqrTo(x,r) { x.squareTo(r); this.reduce(r); }

    // r = "xy/R mod m"; x,y != r
    function montMulTo(x,y,r) { x.multiplyTo(y,r); this.reduce(r); }

    Montgomery.prototype.convert = montConvert;
    Montgomery.prototype.revert = montRevert;
    Montgomery.prototype.reduce = montReduce;
    Montgomery.prototype.mulTo = montMulTo;
    Montgomery.prototype.sqrTo = montSqrTo;

    // (protected) true iff this is even
    function bnpIsEven() { return ((this.t>0)?(this[0]&1):this.s) == 0; }

    // (protected) this^e, e < 2^32, doing sqr and mul with "r" (HAC 14.79)
    function bnpExp(e,z) {
	    if(e > 0xffffffff || e < 1) return BigInteger.ONE;
	    var r = nbi(), r2 = nbi(), g = z.convert(this), i = nbits(e)-1;
	    g.copyTo(r);
	    while(--i >= 0) {
	        z.sqrTo(r,r2);
	        if((e&(1<<i)) > 0) z.mulTo(r2,g,r);
	        else { var t = r; r = r2; r2 = t; }
	    }
	    return z.revert(r);
    }

    // (public) this^e % m, 0 <= e < 2^32
    function bnModPowInt(e,m) {
	var z;
	if(e < 256 || m.isEven()) z = new Classic(m); else z = new Montgomery(m);
	return this.exp(e,z);
    }

    // protected
    BigInteger.prototype.copyTo = bnpCopyTo;
    BigInteger.prototype.fromInt = bnpFromInt;
    BigInteger.prototype.fromString = bnpFromString;
    BigInteger.prototype.clamp = bnpClamp;
    BigInteger.prototype.dlShiftTo = bnpDLShiftTo;
    BigInteger.prototype.drShiftTo = bnpDRShiftTo;
    BigInteger.prototype.lShiftTo = bnpLShiftTo;
    BigInteger.prototype.rShiftTo = bnpRShiftTo;
    BigInteger.prototype.subTo = bnpSubTo;
    BigInteger.prototype.multiplyTo = bnpMultiplyTo;
    BigInteger.prototype.squareTo = bnpSquareTo;
    BigInteger.prototype.divRemTo = bnpDivRemTo;
    BigInteger.prototype.invDigit = bnpInvDigit;
    BigInteger.prototype.isEven = bnpIsEven;
    BigInteger.prototype.exp = bnpExp;

    // public
    BigInteger.prototype.toString = bnToString;
    BigInteger.prototype.negate = bnNegate;
    BigInteger.prototype.abs = bnAbs;
    BigInteger.prototype.compareTo = bnCompareTo;
    BigInteger.prototype.bitLength = bnBitLength;
    BigInteger.prototype.mod = bnMod;
    BigInteger.prototype.modPowInt = bnModPowInt;

    // "constants"
    BigInteger.ZERO = nbv(0);
    BigInteger.ONE = nbv(1);

    // Copyright (c) 2005-2009  Tom Wu
    // All Rights Reserved.
    // See "LICENSE" for details.

    // Extended JavaScript BN functions, required for RSA private ops.

    // Version 1.1: new BigInteger("0", 10) returns "proper" zero

    // (public)
    function bnClone() { var r = nbi(); this.copyTo(r); return r; }

    // (public) return value as integer
    function bnIntValue() {
	if(this.s < 0) {
	    if(this.t == 1) return this[0]-this.DV;
	    else if(this.t == 0) return -1;
	}
	else if(this.t == 1) return this[0];
	else if(this.t == 0) return 0;
	// assumes 16 < DB < 32
	return ((this[1]&((1<<(32-this.DB))-1))<<this.DB)|this[0];
    }

    // (public) return value as byte
    function bnByteValue() { return (this.t==0)?this.s:(this[0]<<24)>>24; }

    // (public) return value as short (assumes DB>=16)
    function bnShortValue() { return (this.t==0)?this.s:(this[0]<<16)>>16; }

    // (protected) return x s.t. r^x < DV
    function bnpChunkSize(r) { return Math.floor(Math.LN2*this.DB/Math.log(r)); }

    // (public) 0 if this == 0, 1 if this > 0
    function bnSigNum() {
	if(this.s < 0) return -1;
	else if(this.t <= 0 || (this.t == 1 && this[0] <= 0)) return 0;
	else return 1;
    }

    // (protected) convert to radix string
    function bnpToRadix(b) {
	if(b == null) b = 10;
	if(this.signum() == 0 || b < 2 || b > 36) return "0";
	var cs = this.chunkSize(b);
	var a = Math.pow(b,cs);
	var d = nbv(a), y = nbi(), z = nbi(), r = "";
	this.divRemTo(d,y,z);
	while(y.signum() > 0) {
	    r = (a+z.intValue()).toString(b).substr(1) + r;
	    y.divRemTo(d,y,z);
	}
	return z.intValue().toString(b) + r;
    }

    // (protected) convert from radix string
    function bnpFromRadix(s,b) {
	this.fromInt(0);
	if(b == null) b = 10;
	var cs = this.chunkSize(b);
	var d = Math.pow(b,cs), mi = false, j = 0, w = 0;
	for(var i = 0; i < s.length; ++i) {
	    var x = intAt(s,i);
	    if(x < 0) {
		if(s.charAt(i) == "-" && this.signum() == 0) mi = true;
		continue;
	    }
	    w = b*w+x;
	    if(++j >= cs) {
		this.dMultiply(d);
		this.dAddOffset(w,0);
		j = 0;
		w = 0;
	    }
	}
	if(j > 0) {
	    this.dMultiply(Math.pow(b,j));
	    this.dAddOffset(w,0);
	}
	if(mi) BigInteger.ZERO.subTo(this,this);
    }

    // (protected) alternate constructor
    function bnpFromNumber(a,b,c) {
	if("number" == typeof b) {
	    // new BigInteger(int,int,RNG)
	    if(a < 2) this.fromInt(1);
	    else {
		this.fromNumber(a,c);
		if(!this.testBit(a-1))	// force MSB set
		    this.bitwiseTo(BigInteger.ONE.shiftLeft(a-1),op_or,this);
		if(this.isEven()) this.dAddOffset(1,0); // force odd
		while(!this.isProbablePrime(b)) {
		    this.dAddOffset(2,0);
		    if(this.bitLength() > a) this.subTo(BigInteger.ONE.shiftLeft(a-1),this);
		}
	    }
	}
	else {
	    // new BigInteger(int,RNG)
	    var x = [], t = a&7;
	    x.length = (a>>3)+1;
	    b.nextBytes(x);
	    if(t > 0) x[0] &= ((1<<t)-1); else x[0] = 0;
	    this.fromString(x,256);
	}
    }

    // (public) convert to bigendian byte array
    function bnToByteArray() {
	var i = this.t, r = [];
	r[0] = this.s;
	var p = this.DB-(i*this.DB)%8, d, k = 0;
	if(i-- > 0) {
	    if(p < this.DB && (d = this[i]>>p) != (this.s&this.DM)>>p)
		r[k++] = d|(this.s<<(this.DB-p));
	    while(i >= 0) {
		if(p < 8) {
		    d = (this[i]&((1<<p)-1))<<(8-p);
		    d |= this[--i]>>(p+=this.DB-8);
		}
		else {
		    d = (this[i]>>(p-=8))&0xff;
		    if(p <= 0) { p += this.DB; --i; }
		}
		if((d&0x80) != 0) d |= -256;
		if(k == 0 && (this.s&0x80) != (d&0x80)) ++k;
		if(k > 0 || d != this.s) r[k++] = d;
	    }
	}
	return r;
    }

    function bnEquals(a) { return(this.compareTo(a)==0); }
    function bnMin(a) { return(this.compareTo(a)<0)?this:a; }
    function bnMax(a) { return(this.compareTo(a)>0)?this:a; }

    // (protected) r = this op a (bitwise)
    function bnpBitwiseTo(a,op,r) {
	var i, f, m = Math.min(a.t,this.t);
	for(i = 0; i < m; ++i) r[i] = op(this[i],a[i]);
	if(a.t < this.t) {
	    f = a.s&this.DM;
	    for(i = m; i < this.t; ++i) r[i] = op(this[i],f);
	    r.t = this.t;
	}
	else {
	    f = this.s&this.DM;
	    for(i = m; i < a.t; ++i) r[i] = op(f,a[i]);
	    r.t = a.t;
	}
	r.s = op(this.s,a.s);
	r.clamp();
    }

    // (public) this & a
    function op_and(x,y) { return x&y; }
    function bnAnd(a) { var r = nbi(); this.bitwiseTo(a,op_and,r); return r; }

    // (public) this | a
    function op_or(x,y) { return x|y; }
    function bnOr(a) { var r = nbi(); this.bitwiseTo(a,op_or,r); return r; }

    // (public) this ^ a
    function op_xor(x,y) { return x^y; }
    function bnXor(a) { var r = nbi(); this.bitwiseTo(a,op_xor,r); return r; }

    // (public) this & ~a
    function op_andnot(x,y) { return x&~y; }
    function bnAndNot(a) { var r = nbi(); this.bitwiseTo(a,op_andnot,r); return r; }

    // (public) ~this
    function bnNot() {
	var r = nbi();
	for(var i = 0; i < this.t; ++i) r[i] = this.DM&~this[i];
	r.t = this.t;
	r.s = ~this.s;
	return r;
    }

    // (public) this << n
    function bnShiftLeft(n) {
	var r = nbi();
	if(n < 0) this.rShiftTo(-n,r); else this.lShiftTo(n,r);
	return r;
    }

    // (public) this >> n
    function bnShiftRight(n) {
	var r = nbi();
	if(n < 0) this.lShiftTo(-n,r); else this.rShiftTo(n,r);
	return r;
    }

    // return index of lowest 1-bit in x, x < 2^31
    function lbit(x) {
	if(x == 0) return -1;
	var r = 0;
	if((x&0xffff) == 0) { x >>= 16; r += 16; }
	if((x&0xff) == 0) { x >>= 8; r += 8; }
	if((x&0xf) == 0) { x >>= 4; r += 4; }
	if((x&3) == 0) { x >>= 2; r += 2; }
	if((x&1) == 0) ++r;
	return r;
    }

    // (public) returns index of lowest 1-bit (or -1 if none)
    function bnGetLowestSetBit() {
	for(var i = 0; i < this.t; ++i)
	    if(this[i] != 0) return i*this.DB+lbit(this[i]);
	if(this.s < 0) return this.t*this.DB;
	return -1;
    }

    // return number of 1 bits in x
    function cbit(x) {
	var r = 0;
	while(x != 0) { x &= x-1; ++r; }
	return r;
    }

    // (public) return number of set bits
    function bnBitCount() {
	var r = 0, x = this.s&this.DM;
	for(var i = 0; i < this.t; ++i) r += cbit(this[i]^x);
	return r;
    }

    // (public) true iff nth bit is set
    function bnTestBit(n) {
	var j = Math.floor(n/this.DB);
	if(j >= this.t) return(this.s!=0);
	return((this[j]&(1<<(n%this.DB)))!=0);
    }

    // (protected) this op (1<<n)
    function bnpChangeBit(n,op) {
	var r = BigInteger.ONE.shiftLeft(n);
	this.bitwiseTo(r,op,r);
	return r;
    }

    // (public) this | (1<<n)
    function bnSetBit(n) { return this.changeBit(n,op_or); }

    // (public) this & ~(1<<n)
    function bnClearBit(n) { return this.changeBit(n,op_andnot); }

    // (public) this ^ (1<<n)
    function bnFlipBit(n) { return this.changeBit(n,op_xor); }

    // (protected) r = this + a
    function bnpAddTo(a,r) {
	var i = 0, c = 0, m = Math.min(a.t,this.t);
	while(i < m) {
	    c += this[i]+a[i];
	    r[i++] = c&this.DM;
	    c >>= this.DB;
	}
	if(a.t < this.t) {
	    c += a.s;
	    while(i < this.t) {
		c += this[i];
		r[i++] = c&this.DM;
		c >>= this.DB;
	    }
	    c += this.s;
	}
	else {
	    c += this.s;
	    while(i < a.t) {
		c += a[i];
		r[i++] = c&this.DM;
		c >>= this.DB;
	    }
	    c += a.s;
	}
	r.s = (c<0)?-1:0;
	if(c > 0) r[i++] = c;
	else if(c < -1) r[i++] = this.DV+c;
	r.t = i;
	r.clamp();
    }

    // (public) this + a
    function bnAdd(a) { var r = nbi(); this.addTo(a,r); return r; }

    // (public) this - a
    function bnSubtract(a) { var r = nbi(); this.subTo(a,r); return r; }

    // (public) this * a
    function bnMultiply(a) { var r = nbi(); this.multiplyTo(a,r); return r; }

    // (public) this / a
    function bnDivide(a) { var r = nbi(); this.divRemTo(a,r,null); return r; }

    // (public) this % a
    function bnRemainder(a) { var r = nbi(); this.divRemTo(a,null,r); return r; }

    // (public) [this/a,this%a]
    function bnDivideAndRemainder(a) {
	var q = nbi(), r = nbi();
	this.divRemTo(a,q,r);
	return [q,r];
    }

    // (protected) this *= n, this >= 0, 1 < n < DV
    function bnpDMultiply(n) {
	this[this.t] = this.am(0,n-1,this,0,0,this.t);
	++this.t;
	this.clamp();
    }

    // (protected) this += n << w words, this >= 0
    function bnpDAddOffset(n,w) {
	if(n == 0) return;
	while(this.t <= w) this[this.t++] = 0;
	this[w] += n;
	while(this[w] >= this.DV) {
	    this[w] -= this.DV;
	    if(++w >= this.t) this[this.t++] = 0;
	    ++this[w];
	}
    }

    // A "null" reducer
    function NullExp() {}
    function nNop(x) { return x; }
    function nMulTo(x,y,r) { x.multiplyTo(y,r); }
    function nSqrTo(x,r) { x.squareTo(r); }

    NullExp.prototype.convert = nNop;
    NullExp.prototype.revert = nNop;
    NullExp.prototype.mulTo = nMulTo;
    NullExp.prototype.sqrTo = nSqrTo;

    // (public) this^e
    function bnPow(e) { return this.exp(e,new NullExp()); }

    // (protected) r = lower n words of "this * a", a.t <= n
    // "this" should be the larger one if appropriate.
    function bnpMultiplyLowerTo(a,n,r) {
	var i = Math.min(this.t+a.t,n);
	r.s = 0; // assumes a,this >= 0
	r.t = i;
	while(i > 0) r[--i] = 0;
	var j;
	for(j = r.t-this.t; i < j; ++i) r[i+this.t] = this.am(0,a[i],r,i,0,this.t);
	for(j = Math.min(a.t,n); i < j; ++i) this.am(0,a[i],r,i,0,n-i);
	r.clamp();
    }

    // (protected) r = "this * a" without lower n words, n > 0
    // "this" should be the larger one if appropriate.
    function bnpMultiplyUpperTo(a,n,r) {
	--n;
	var i = r.t = this.t+a.t-n;
	r.s = 0; // assumes a,this >= 0
	while(--i >= 0) r[i] = 0;
	for(i = Math.max(n-this.t,0); i < a.t; ++i)
	    r[this.t+i-n] = this.am(n-i,a[i],r,0,0,this.t+i-n);
	r.clamp();
	r.drShiftTo(1,r);
    }

    // Barrett modular reduction
    function Barrett(m) {
	// setup Barrett
	this.r2 = nbi();
	this.q3 = nbi();
	BigInteger.ONE.dlShiftTo(2*m.t,this.r2);
	this.mu = this.r2.divide(m);
	this.m = m;
    }

    function barrettConvert(x) {
	if(x.s < 0 || x.t > 2*this.m.t) return x.mod(this.m);
	else if(x.compareTo(this.m) < 0) return x;
	else { var r = nbi(); x.copyTo(r); this.reduce(r); return r; }
    }

    function barrettRevert(x) { return x; }

    // x = x mod m (HAC 14.42)
    function barrettReduce(x) {
	x.drShiftTo(this.m.t-1,this.r2);
	if(x.t > this.m.t+1) { x.t = this.m.t+1; x.clamp(); }
	this.mu.multiplyUpperTo(this.r2,this.m.t+1,this.q3);
	this.m.multiplyLowerTo(this.q3,this.m.t+1,this.r2);
	while(x.compareTo(this.r2) < 0) x.dAddOffset(1,this.m.t+1);
	x.subTo(this.r2,x);
	while(x.compareTo(this.m) >= 0) x.subTo(this.m,x);
    }

    // r = x^2 mod m; x != r
    function barrettSqrTo(x,r) { x.squareTo(r); this.reduce(r); }

    // r = x*y mod m; x,y != r
    function barrettMulTo(x,y,r) { x.multiplyTo(y,r); this.reduce(r); }

    Barrett.prototype.convert = barrettConvert;
    Barrett.prototype.revert = barrettRevert;
    Barrett.prototype.reduce = barrettReduce;
    Barrett.prototype.mulTo = barrettMulTo;
    Barrett.prototype.sqrTo = barrettSqrTo;

    // (public) this^e % m (HAC 14.85)
    function bnModPow(e,m) {
	var i = e.bitLength(), k, r = nbv(1), z;
	if(i <= 0) return r;
	else if(i < 18) k = 1;
	else if(i < 48) k = 3;
	else if(i < 144) k = 4;
	else if(i < 768) k = 5;
	else k = 6;
	if(i < 8)
	    z = new Classic(m);
	else if(m.isEven())
	    z = new Barrett(m);
	else
	    z = new Montgomery(m);

	// precomputation
	var g = [], n = 3, k1 = k-1, km = (1<<k)-1;
	g[1] = z.convert(this);
	if(k > 1) {
	    var g2 = nbi();
	    z.sqrTo(g[1],g2);
	    while(n <= km) {
		g[n] = nbi();
		z.mulTo(g2,g[n-2],g[n]);
		n += 2;
	    }
	}

	var j = e.t-1, w, is1 = true, r2 = nbi(), t;
	i = nbits(e[j])-1;
	while(j >= 0) {
	    if(i >= k1) w = (e[j]>>(i-k1))&km;
	    else {
		w = (e[j]&((1<<(i+1))-1))<<(k1-i);
		if(j > 0) w |= e[j-1]>>(this.DB+i-k1);
	    }

	    n = k;
	    while((w&1) == 0) { w >>= 1; --n; }
	    if((i -= n) < 0) { i += this.DB; --j; }
	    if(is1) {	// ret == 1, don't bother squaring or multiplying it
		g[w].copyTo(r);
		is1 = false;
	    }
	    else {
		while(n > 1) { z.sqrTo(r,r2); z.sqrTo(r2,r); n -= 2; }
		if(n > 0) z.sqrTo(r,r2); else { t = r; r = r2; r2 = t; }
		z.mulTo(r2,g[w],r);
	    }

	    while(j >= 0 && (e[j]&(1<<i)) == 0) {
		z.sqrTo(r,r2); t = r; r = r2; r2 = t;
		if(--i < 0) { i = this.DB-1; --j; }
	    }
	}
	return z.revert(r);
    }

    // (public) gcd(this,a) (HAC 14.54)
    function bnGCD(a) {
	var x = (this.s<0)?this.negate():this.clone();
	var y = (a.s<0)?a.negate():a.clone();
	if(x.compareTo(y) < 0) { var t = x; x = y; y = t; }
	var i = x.getLowestSetBit(), g = y.getLowestSetBit();
	if(g < 0) return x;
	if(i < g) g = i;
	if(g > 0) {
	    x.rShiftTo(g,x);
	    y.rShiftTo(g,y);
	}
	while(x.signum() > 0) {
	    if((i = x.getLowestSetBit()) > 0) x.rShiftTo(i,x);
	    if((i = y.getLowestSetBit()) > 0) y.rShiftTo(i,y);
	    if(x.compareTo(y) >= 0) {
		x.subTo(y,x);
		x.rShiftTo(1,x);
	    }
	    else {
		y.subTo(x,y);
		y.rShiftTo(1,y);
	    }
	}
	if(g > 0) y.lShiftTo(g,y);
	return y;
    }

    // (protected) this % n, n < 2^26
    function bnpModInt(n) {
	if(n <= 0) return 0;
	var d = this.DV%n, r = (this.s<0)?n-1:0;
	if(this.t > 0)
	    if(d == 0) r = this[0]%n;
	else for(var i = this.t-1; i >= 0; --i) r = (d*r+this[i])%n;
	return r;
    }

    // (public) 1/this % m (HAC 14.61)
    function bnModInverse(m) {
	var ac = m.isEven();
	if((this.isEven() && ac) || m.signum() == 0) return BigInteger.ZERO;
	var u = m.clone(), v = this.clone();
	var a = nbv(1), b = nbv(0), c = nbv(0), d = nbv(1);
	while(u.signum() != 0) {
	    while(u.isEven()) {
		u.rShiftTo(1,u);
		if(ac) {
		    if(!a.isEven() || !b.isEven()) { a.addTo(this,a); b.subTo(m,b); }
		    a.rShiftTo(1,a);
		}
		else if(!b.isEven()) b.subTo(m,b);
		b.rShiftTo(1,b);
	    }
	    while(v.isEven()) {
		v.rShiftTo(1,v);
		if(ac) {
		    if(!c.isEven() || !d.isEven()) { c.addTo(this,c); d.subTo(m,d); }
		    c.rShiftTo(1,c);
		}
		else if(!d.isEven()) d.subTo(m,d);
		d.rShiftTo(1,d);
	    }
	    if(u.compareTo(v) >= 0) {
		u.subTo(v,u);
		if(ac) a.subTo(c,a);
		b.subTo(d,b);
	    }
	    else {
		v.subTo(u,v);
		if(ac) c.subTo(a,c);
		d.subTo(b,d);
	    }
	}
	if(v.compareTo(BigInteger.ONE) != 0) return BigInteger.ZERO;
	if(d.compareTo(m) >= 0) return d.subtract(m);
	if(d.signum() < 0) d.addTo(m,d); else return d;
	if(d.signum() < 0) return d.add(m); else return d;
    }

    var lowprimes = [2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97,101,103,107,109,113,127,131,137,139,149,151,157,163,167,173,179,181,191,193,197,199,211,223,227,229,233,239,241,251,257,263,269,271,277,281,283,293,307,311,313,317,331,337,347,349,353,359,367,373,379,383,389,397,401,409,419,421,431,433,439,443,449,457,461,463,467,479,487,491,499,503,509];
    var lplim = (1<<26)/lowprimes[lowprimes.length-1];

    // (public) test primality with certainty >= 1-.5^t
    function bnIsProbablePrime(t) {
	var i, x = this.abs();
	if(x.t == 1 && x[0] <= lowprimes[lowprimes.length-1]) {
	    for(i = 0; i < lowprimes.length; ++i)
		if(x[0] == lowprimes[i]) return true;
	    return false;
	}
	if(x.isEven()) return false;
	i = 1;
	while(i < lowprimes.length) {
	    var m = lowprimes[i], j = i+1;
	    while(j < lowprimes.length && m < lplim) m *= lowprimes[j++];
	    m = x.modInt(m);
	    while(i < j) if(m%lowprimes[i++] == 0) return false;
	}
	return x.millerRabin(t);
    }

    // (protected) true if probably prime (HAC 4.24, Miller-Rabin)
    function bnpMillerRabin(t) {
	var n1 = this.subtract(BigInteger.ONE);
	var k = n1.getLowestSetBit();
	if(k <= 0) return false;
	var r = n1.shiftRight(k);
	t = (t+1)>>1;
	if(t > lowprimes.length) t = lowprimes.length;
	var a = nbi();
	for(var i = 0; i < t; ++i) {
	    a.fromInt(lowprimes[i]);
	    var y = a.modPow(r,this);
	    if(y.compareTo(BigInteger.ONE) != 0 && y.compareTo(n1) != 0) {
		var j = 1;
		while(j++ < k && y.compareTo(n1) != 0) {
		    y = y.modPowInt(2,this);
		    if(y.compareTo(BigInteger.ONE) == 0) return false;
		}
		if(y.compareTo(n1) != 0) return false;
	    }
	}
	return true;
    }
    
    

    // protected
    BigInteger.prototype.chunkSize = bnpChunkSize;
    BigInteger.prototype.toRadix = bnpToRadix;
    BigInteger.prototype.fromRadix = bnpFromRadix;
    BigInteger.prototype.fromNumber = bnpFromNumber;
    BigInteger.prototype.bitwiseTo = bnpBitwiseTo;
    BigInteger.prototype.changeBit = bnpChangeBit;
    BigInteger.prototype.addTo = bnpAddTo;
    BigInteger.prototype.dMultiply = bnpDMultiply;
    BigInteger.prototype.dAddOffset = bnpDAddOffset;
    BigInteger.prototype.multiplyLowerTo = bnpMultiplyLowerTo;
    BigInteger.prototype.multiplyUpperTo = bnpMultiplyUpperTo;
    BigInteger.prototype.modInt = bnpModInt;
    BigInteger.prototype.millerRabin = bnpMillerRabin;

    // public
    BigInteger.prototype.clone = bnClone;
    BigInteger.prototype.intValue = bnIntValue;
    BigInteger.prototype.byteValue = bnByteValue;
    BigInteger.prototype.shortValue = bnShortValue;
    BigInteger.prototype.signum = bnSigNum;
    BigInteger.prototype.toByteArray = bnToByteArray;
    BigInteger.prototype.equals = bnEquals;
    BigInteger.prototype.min = bnMin;
    BigInteger.prototype.max = bnMax;
    BigInteger.prototype.and = bnAnd;
    BigInteger.prototype.or = bnOr;
    BigInteger.prototype.xor = bnXor;
    BigInteger.prototype.andNot = bnAndNot;
    BigInteger.prototype.not = bnNot;
    BigInteger.prototype.shiftLeft = bnShiftLeft;
    BigInteger.prototype.shiftRight = bnShiftRight;
    BigInteger.prototype.getLowestSetBit = bnGetLowestSetBit;
    BigInteger.prototype.bitCount = bnBitCount;
    BigInteger.prototype.testBit = bnTestBit;
    BigInteger.prototype.setBit = bnSetBit;
    BigInteger.prototype.clearBit = bnClearBit;
    BigInteger.prototype.flipBit = bnFlipBit;
    BigInteger.prototype.add = bnAdd;
    BigInteger.prototype.subtract = bnSubtract;
    BigInteger.prototype.multiply = bnMultiply;
    BigInteger.prototype.divide = bnDivide;
    BigInteger.prototype.remainder = bnRemainder;
    BigInteger.prototype.divideAndRemainder = bnDivideAndRemainder;
    BigInteger.prototype.modPow = bnModPow;
    BigInteger.prototype.modInverse = bnModInverse;
    BigInteger.prototype.pow = bnPow;
    BigInteger.prototype.gcd = bnGCD;
    BigInteger.prototype.isProbablePrime = bnIsProbablePrime;

    // BigInteger interfaces not implemented in jsbn:

    // BigInteger(int signum, byte[] magnitude)
    // double doubleValue()
    // float floatValue()
    // int hashCode()
    // long longValue()
    // static BigInteger valueOf(long val)



    //////////////////////////////////////////////////////////////////////
    //////////////////////////////////////////////////////////////////////
    //////////////////////////////////////////////////////////////////////
    //////////////////////////////////////////////////////////////////////
    //////////////////////////////////////////////////////////////////////
    // END OF copy-and-paste of jsbn.



    BigInteger.NEGATIVE_ONE = BigInteger.ONE.negate();


    // Other methods we need to add for compatibilty with js-numbers numeric tower.

    // add is implemented above.
    // subtract is implemented above.
    // multiply is implemented above.
    // equals is implemented above.
    // abs is implemented above.
    // negate is defined above.

    // makeBignum: string -> BigInteger
    var makeBignum = function(s) {
	if (typeof(s) === 'number') { s = s + ''; }
	s = expandExponent(s);
	return new BigInteger(s, 10);
    };

    var zerostring = function(n) {
	var buf = [];
	for (var i = 0; i < n; i++) {
	    buf.push('0');
	}
	return buf.join('');
    };


    BigInteger.prototype.level = 0;
    BigInteger.prototype.liftTo = function(target) {
	if (target.level === 1) {
	    return new Rational(this, 1);
	}
	if (target.level === 2) {
	    var fixrep = this.toFixnum();
	    if (fixrep === Number.POSITIVE_INFINITY)
		return TOO_POSITIVE_TO_REPRESENT;
	    if (fixrep === Number.NEGATIVE_INFINITY)
		return TOO_NEGATIVE_TO_REPRESENT;
	    return new FloatPoint(fixrep);
	}
	if (target.level === 3) {
	    return new Complex(this, 0);
	}
	return throwRuntimeError("invalid level for BigInteger lift", this, target);
    };

    BigInteger.prototype.isFinite = function() {
	return true;
    };

    BigInteger.prototype.isInteger = function() {
	return true;
    };

    BigInteger.prototype.isRational = function() {
	return true;
    };

    BigInteger.prototype.isReal = function() {
	return true;
    };

    BigInteger.prototype.isExact = function() {
	return true;
    };

    BigInteger.prototype.isInexact = function() {
	return false;
    };

    BigInteger.prototype.toExact = function() {
	return this;
    };

    BigInteger.prototype.toInexact = function() {
	return FloatPoint.makeInstance(this.toFixnum());
    };

    BigInteger.prototype.toFixnum = function() {
	var result = 0, str = this.toString(), i;
	if (str[0] === '-') {
	    for (i=1; i < str.length; i++) {
		result = result * 10 + Number(str[i]);
	    }
	    return -result;
	} else {
	    for (i=0; i < str.length; i++) {
		result = result * 10 + Number(str[i]);
	    }
	    return result;
	}
    };


    BigInteger.prototype.greaterThan = function(other) {
	return this.compareTo(other) > 0;
    };

    BigInteger.prototype.greaterThanOrEqual = function(other) {
	return this.compareTo(other) >= 0;
    };

    BigInteger.prototype.lessThan = function(other) {
	return this.compareTo(other) < 0;
    };

    BigInteger.prototype.lessThanOrEqual = function(other) {
	return this.compareTo(other) <= 0;
    };

    // divide: scheme-number -> scheme-number
    // WARNING NOTE: we override the old version of divide.
    BigInteger.prototype.divide = function(other) {
	var quotientAndRemainder = bnDivideAndRemainder.call(this, other);
	if (quotientAndRemainder[1].compareTo(BigInteger.ZERO) === 0) {
	    return quotientAndRemainder[0];
	} else {
	    var result = add(quotientAndRemainder[0],
			     Rational.makeInstance(quotientAndRemainder[1], other));
	    return result;
	}
    };

    BigInteger.prototype.numerator = function() {
	return this;
    };

    BigInteger.prototype.denominator = function() {
	return 1;
    };


    (function() {
	// Classic implementation of Newton-Ralphson square-root search,
	// adapted for integer-sqrt.
	// http://en.wikipedia.org/wiki/Newton's_method#Square_root_of_a_number
	    var searchIter = function(n, guess) {
		while(!(lessThanOrEqual(sqr(guess),n) &&
			lessThan(n,sqr(add(guess, 1))))) {
		    guess = floor(divide(add(guess,
					     floor(divide(n, guess))),
					 2));
		}
		return guess;
	    };

	    // integerSqrt: -> scheme-number
	    BigInteger.prototype.integerSqrt = function() {
		var n;
		if(sign(this) >= 0) {
		    return searchIter(this, this);
		} else {
		    n = this.negate();
		    return Complex.makeInstance(0, searchIter(n, n));
		}
	    };
    })();


    (function() {	
	// Get an approximation using integerSqrt, and then start another
	// Newton-Ralphson search if necessary.
	BigInteger.prototype.sqrt = function() {
	    var approx = this.integerSqrt(), fix;
	    if (eqv(sqr(approx), this)) {
		return approx;
	    }
	    fix = toFixnum(this);
	    if (isFinite(fix)) {
		if (fix >= 0) {
		    return FloatPoint.makeInstance(Math.sqrt(fix));
		} else {
		    return Complex.makeInstance(
			0,
			FloatPoint.makeInstance(Math.sqrt(-fix)));
		}
	    } else {
		return approx;
	    }
	};
    })();




    
    // sqrt: -> scheme-number
    // http://en.wikipedia.org/wiki/Newton's_method#Square_root_of_a_number
    // Produce the square root.

    // floor: -> scheme-number
    // Produce the floor.
    BigInteger.prototype.floor = function() {
        return this;
    }

    // ceiling: -> scheme-number
    // Produce the ceiling.
    BigInteger.prototype.ceiling = function() {
        return this;
    }

    // conjugate: -> scheme-number
    // Produce the conjugate.

    // magnitude: -> scheme-number
    // Produce the magnitude.

    // log: -> scheme-number
    // Produce the log.

    // angle: -> scheme-number
    // Produce the angle.

    // atan: -> scheme-number
    // Produce the arc tangent.

    // cos: -> scheme-number
    // Produce the cosine.

    // sin: -> scheme-number
    // Produce the sine.


    // expt: scheme-number -> scheme-number
    // Produce the power to the input.
    BigInteger.prototype.expt = function(n) {
	return bnPow.call(this, n);
    };



    // exp: -> scheme-number
    // Produce e raised to the given power.

    // acos: -> scheme-number
    // Produce the arc cosine.

    // asin: -> scheme-number
    // Produce the arc sine.

    BigInteger.prototype.imaginaryPart = function() {
	    return 0;
    }
    BigInteger.prototype.realPart = function() {
	    return this;
    }

    // round: -> scheme-number
    // Round to the nearest integer.





    //////////////////////////////////////////////////////////////////////
    // toRepeatingDecimal: jsnum jsnum {limit: number}? -> [string, string, string]
    //
    // Given the numerator and denominator parts of a rational,
    // produces the repeating-decimal representation, where the first
    // part are the digits before the decimal, the second are the
    // non-repeating digits after the decimal, and the third are the
    // remaining repeating decimals.
    // 
    // An optional limit on the decimal expansion can be provided, in which
    // case the search cuts off if we go past the limit.
    // If this happens, the third argument returned becomes '...' to indicate
    // that the search was prematurely cut off.
    var toRepeatingDecimal = (function() {
	var getResidue = function(r, d, limit) {
	    var digits = [];
	    var seenRemainders = {};
	    seenRemainders[r] = true;
	    while(true) {	
		if (limit-- <= 0) {
		    return [digits.join(''), '...']
		}

		var nextDigit = quotient(
		    multiply(r, 10), d);
		var nextRemainder = remainder(
		    multiply(r, 10),
		    d);
		digits.push(nextDigit.toString());
		if (seenRemainders[nextRemainder]) {
		    r = nextRemainder;
		    break;
		} else {
		    seenRemainders[nextRemainder] = true;
		    r = nextRemainder;
		}
	    }
	    
	    var firstRepeatingRemainder = r;
	    var repeatingDigits = [];
	    while (true) {
		var nextDigit = quotient(multiply(r, 10), d);
		var nextRemainder = remainder(
		    multiply(r, 10),
		    d);
		repeatingDigits.push(nextDigit.toString());
		if (equals(nextRemainder, firstRepeatingRemainder)) {
		    break;
		} else {
		    r = nextRemainder;
		}
	    };

	    var digitString = digits.join('');
	    var repeatingDigitString = repeatingDigits.join('');

	    while (digitString.length >= repeatingDigitString.length &&
		   (digitString.substring(
		       digitString.length - repeatingDigitString.length)
		    === repeatingDigitString)) {
		digitString = digitString.substring(
		    0, digitString.length - repeatingDigitString.length);
	    }

	    return [digitString, repeatingDigitString];

	};

	return function(n, d, options) {
	    // default limit on decimal expansion; can be overridden
	    var limit = 512;
	    if (options && typeof(options.limit) !== 'undefined') {
		limit = options.limit;
	    }
	    if (! isInteger(n)) {
		throwRuntimeError('toRepeatingDecimal: n ' + n.toString() +
				  " is not an integer.");
	    }
	    if (! isInteger(d)) {
		throwRuntimeError('toRepeatingDecimal: d ' + d.toString() +
				  " is not an integer.");
	    }
	    if (equals(d, 0)) {
		throwRuntimeError('toRepeatingDecimal: d equals 0');
	    }
	    if (lessThan(d, 0)) {
		throwRuntimeError('toRepeatingDecimal: d < 0');
	    }
 	    var sign = (lessThan(n, 0) ? "-" : "");
 	    n = abs(n);
 	    var beforeDecimalPoint = sign + quotient(n, d);
 	    var afterDecimals = getResidue(remainder(n, d), d, limit);
 	    return [beforeDecimalPoint].concat(afterDecimals);
	};
    })();
    //////////////////////////////////////////////////////////////////////




    // External interface of js-numbers:

    Numbers['fromFixnum'] = fromFixnum;
    Numbers['fromString'] = fromString;
    Numbers['makeBignum'] = makeBignum;
    Numbers['makeRational'] = Rational.makeInstance;
    Numbers['makeFloat'] = FloatPoint.makeInstance;
    Numbers['makeComplex'] = Complex.makeInstance;
    Numbers['makeComplexPolar'] = makeComplexPolar;

    Numbers['pi'] = FloatPoint.pi;
    Numbers['e'] = FloatPoint.e;
    Numbers['nan'] = FloatPoint.nan;
    Numbers['negative_inf'] = FloatPoint.neginf;
    Numbers['inf'] = FloatPoint.inf;
    Numbers['negative_one'] = -1;   // Rational.NEGATIVE_ONE;
    Numbers['zero'] = 0;            // Rational.ZERO;
    Numbers['one'] = 1;             // Rational.ONE;
    Numbers['i'] = plusI;
    Numbers['negative_i'] = minusI;
    Numbers['negative_zero'] = NEGATIVE_ZERO;

    Numbers['onThrowRuntimeError'] = onThrowRuntimeError;
    Numbers['isSchemeNumber'] = isSchemeNumber;
    Numbers['isRational'] = isRational;
    Numbers['isReal'] = isReal;
    Numbers['isExact'] = isExact;
    Numbers['isInexact'] = isInexact;
    Numbers['isInteger'] = isInteger;

    Numbers['toFixnum'] = toFixnum;
    Numbers['toExact'] = toExact;
    Numbers['toInexact'] = toInexact;
    Numbers['add'] = add;
    Numbers['subtract'] = subtract;
    Numbers['multiply'] = multiply;
    Numbers['divide'] = divide;
    Numbers['equals'] = equals;
    Numbers['eqv'] = eqv;
    Numbers['approxEquals'] = approxEquals;
    Numbers['greaterThanOrEqual'] = greaterThanOrEqual;
    Numbers['lessThanOrEqual'] = lessThanOrEqual;
    Numbers['greaterThan'] = greaterThan;
    Numbers['lessThan'] = lessThan;
    Numbers['expt'] = expt;
    Numbers['exp'] = exp;
    Numbers['modulo'] = modulo;
    Numbers['numerator'] = numerator;
    Numbers['denominator'] = denominator;
    Numbers['integerSqrt'] = integerSqrt;
    Numbers['sqrt'] = sqrt;
    Numbers['abs'] = abs;
    Numbers['quotient'] = quotient;
    Numbers['remainder'] = remainder;
    Numbers['floor'] = floor;
    Numbers['ceiling'] = ceiling;
    Numbers['conjugate'] = conjugate;
    Numbers['magnitude'] = magnitude;
    Numbers['log'] = log;
    Numbers['angle'] = angle;
    Numbers['tan'] = tan;
    Numbers['atan'] = atan;
    Numbers['cos'] = cos;
    Numbers['sin'] = sin;
    Numbers['tan'] = tan;
    Numbers['acos'] = acos;
    Numbers['asin'] = asin;
    Numbers['cosh'] = cosh;
    Numbers['sinh'] = sinh;
    Numbers['imaginaryPart'] = imaginaryPart;
    Numbers['realPart'] = realPart;
    Numbers['round'] = round;
    Numbers['sqr'] = sqr;
    Numbers['gcd'] = gcd;
    Numbers['lcm'] = lcm;

    Numbers['toRepeatingDecimal'] = toRepeatingDecimal;



    // The following exposes the class representations for easier
    // integration with other projects.
    Numbers['BigInteger'] = BigInteger;
    Numbers['Rational'] = Rational;
    Numbers['FloatPoint'] = FloatPoint;
    Numbers['Complex'] = Complex;   

    Numbers['MIN_FIXNUM'] = MIN_FIXNUM;
    Numbers['MAX_FIXNUM'] = MAX_FIXNUM;

})();