"Inside out" approach to sqrt with convergence as described in https://news.ycombinator.com/item?id=7043943

gistfile1.php

<?php

function _sqrt($n) {
    $x = 1.0;
    $y = 0.5 * (1.0 + 1.0/$n);
    $z = 1e-10;

    while (abs($x/$y - 1) > $z) {
        $x = $y;
        $y = (0.5 * ($y + $n/$y));
    }

    return $y;
}

function _sqrt2($n) {
    $pair =  head(filter('converged', sliding(sqrtGuesses($n))));

    return $pair[0];
}

function sqrtGuesses($n) {
    $y = 1.0;
    while (true) {
        $y = (0.5 * ($y + $n/$y));
        yield $y;
    }
}

function converged(array $pair) {
    list($x, $y) = $pair;

    return abs($x/$y - 1) < 1e-10;
}

$x = _sqrt(3);
var_dump($x);

$y = _sqrt2(3);
var_dump($y);

/*
 * "library functions"
 */
function sliding($xs, $n = 2) {
    $window = [];

    $i = 1;
    foreach ($xs as $x) {
        array_push($window, $x);

        if ($i < $n) {
            $i++;
            continue;
        }

        yield $window;
        array_shift($window);
    }
}

function head($xs) {
    foreach ($xs as $x) {
        return $x;
    }

    throw new RuntimeException('No head in $xs, is it empty?');
}

function filter($predicate, $xs) {
    foreach ($xs as $x) {
        if ($predicate($x)) {
            yield $x;
        }
    }
}