Fastest way to convert a number to radix 64 in JavaScript?

Here is a sketch for a solution for NUMBERS (not arrays of bytes :)

only for positive numbers, ignores fractional parts, and not really tested -- just a sketch!

Base64 = {

    _Rixits :
//   0       8       16      24      32      40      48      56     63
//   v       v       v       v       v       v       v       v      v
    "0123456789ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyz+/",
    // You have the freedom, here, to choose the glyphs you want for 
    // representing your base-64 numbers. The ASCII encoding guys usually
    // choose a set of glyphs beginning with ABCD..., but, looking at
    // your update #2, I deduce that you want glyphs beginning with 
    // 0123..., which is a fine choice and aligns the first ten numbers
    // in base 64 with the first ten numbers in decimal.

    // This cannot handle negative numbers and only works on the 
    //     integer part, discarding the fractional part.
    // Doing better means deciding on whether you're just representing
    // the subset of javascript numbers of twos-complement 32-bit integers 
    // or going with base-64 representations for the bit pattern of the
    // underlying IEEE floating-point number, or representing the mantissae
    // and exponents separately, or some other possibility. For now, bail
    fromNumber : function(number) {
        if (isNaN(Number(number)) || number === null ||
            number === Number.POSITIVE_INFINITY)
            throw "The input is not valid";
        if (number < 0)
            throw "Can't represent negative numbers now";

        var rixit; // like 'digit', only in some non-decimal radix 
        var residual = Math.floor(number);
        var result = '';
        while (true) {
            rixit = residual % 64
            // console.log("rixit : " + rixit);
            // console.log("result before : " + result);
            result = this._Rixits.charAt(rixit) + result;
            // console.log("result after : " + result);
            // console.log("residual before : " + residual);
            residual = Math.floor(residual / 64);
            // console.log("residual after : " + residual);

            if (residual == 0)
                break;
            }
        return result;
    },

    toNumber : function(rixits) {
        var result = 0;
        // console.log("rixits : " + rixits);
        // console.log("rixits.split('') : " + rixits.split(''));
        rixits = rixits.split('');
        for (var e = 0; e < rixits.length; e++) {
            // console.log("_Rixits.indexOf(" + rixits[e] + ") : " + 
                // this._Rixits.indexOf(rixits[e]));
            // console.log("result before : " + result);
            result = (result * 64) + this._Rixits.indexOf(rixits[e]);
            // console.log("result after : " + result);
        }
        return result;
    }
}

UPDATE: Here's some (very lightweight) testing of the above, for running in NodeJs where you have console.log.

function testBase64(x) {
    console.log("My number is " + x);
    var g = Base64.fromNumber(x);
    console.log("My base-64 representation is " + g);
    var h = Base64.toNumber(g);
    console.log("Returning from base-64, I get " + h);
    if (h !== Math.floor(x))
        throw "TEST FAILED";
}

testBase64(0);
try {
    testBase64(-1);
    }
catch (err) {
    console.log("caught >>>>>>  " + err);
    }
try {
    testBase64(undefined);
    }
catch (err) {
    console.log("caught >>>>>>  " + err);
    }
try {
    testBase64(null);
    }
catch (err) {
    console.log("caught >>>>>>  " + err);
    }
try {
    testBase64(Number.NaN);
    }
catch (err) {
    console.log("caught >>>>>>  " + err);
    }
try {
    testBase64(Number.POSITIVE_INFINITY);
    }
catch (err) {
    console.log("caught >>>>>>  " + err);
    }
try {
    testBase64(Number.NEGATIVE_INFINITY);
    }
catch (err) {
    console.log("caught >>>>>>  " + err);
    }

for(i=0; i<100; i++)
    testBase64(Math.random()*1e14);

Here's a version just for 32 bit ints, that is, any number between -2147483648 and 2147483647 (inclusive).

I modified the version in the top answer by Reb Cabin. This should be quite a bit faster because it uses bit operations and lookup tables.

Base64 = (function () {
    var digitsStr = 
    //   0       8       16      24      32      40      48      56     63
    //   v       v       v       v       v       v       v       v      v
        "0123456789ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyz+-";
    var digits = digitsStr.split('');
    var digitsMap = {};
    for (var i = 0; i < digits.length; i++) {
        digitsMap[digits[i]] = i;
    }
    return {
        fromInt: function(int32) {
            var result = '';
            while (true) {
                result = digits[int32 & 0x3f] + result;
                int32 >>>= 6;
                if (int32 === 0)
                    break;
            }
            return result;
        },
        toInt: function(digitsStr) {
            var result = 0;
            var digits = digitsStr.split('');
            for (var i = 0; i < digits.length; i++) {
                result = (result << 6) + digitsMap[digits[i]];
            }
            return result;
        }
    };
})();

For example,

Base64.fromInt(-2147483648); // gives "200000"
Base64.toInt("200000"); // gives -2147483648