Approximates the square root of any number. Note that the approximation will be an approximation for numbers that contain a large number of digits. Example use sqrt(16) will return 4.

sqrt.coffee

sqrt = (() ->
    sqrtIter = (guess, x) ->
        if goodGuess(guess, x) then guess
        else sqrtIter((guess + x / guess) / 2, x)
    goodGuess = (guess, x) -> Math.abs(guess * guess - x) / x < 0.0000001
 
    (x) -> sqrtIter(1.0, x)
)()

sqrt.js

var sqrt = function (value) {
    // Returns -1 if value is negative or not a number.
    if (!(value >= 0)) {return undefined; }
    
    // Otherwise it approximates the square root.
    var guess = value / 2.0;
    var diff = (guess * guess) - value;
    var n = 1;

    while ((Math.abs(diff) > 0.00000000000000001) && (n < 1000000)) {
        guess -= diff / (2.0 * guess);
        diff = (guess * guess) - value;
        n += 1;
    }

    return guess;
};

// Testing code.
var testSqrt = function () {
    var testSqrtValue = function (arg, value, id) {
        console.log(sqrt(arg) === value ? "Test " + id + " passed." : "Test " + id + " failed.");
    };

    testSqrtValue(16, 4, 1);
    testSqrtValue(-1, -1, 2);
    testSqrtValue(0, 0, 3);
    testSqrtValue(121, 11, 4);
    testSqrtValue(1.5, 1.224744871391589, 5);
    testSqrtValue("hello", -1, 6);
};

sqrt.py

# Uses the Isaac Newton/Joseph Raphson method to calculate the
# square root of a wide range of numbers in reasonable time only.
def sqrt(value):
    assert value > 0, "Value must be more than 0."
    value = float(value)
    guess = value / 2.0
    diff = guess ** 2 - value
    n = 1
    acc = 0.00000000000000001

    while abs(diff) > acc and n < 1000000:
        guess = guess - diff / (2.0 * guess)
        diff = guess ** 2 - value
        n += 1

    return "guess = {0} iterations = {1} accuracy = {2}".format(guess, n, acc)