How can PHP developers optimize their code for efficient and accurate calculations when dealing with complex mathematical functions like binomial distribution?

When dealing with complex mathematical functions like the binomial distribution in PHP, developers can optimize their code by using built-in functions or libraries specifically designed for mathematical calculations. One way to improve efficiency is to avoid unnecessary loops and calculations by utilizing functions like `pow()` for exponentiation and `factorial()` for factorial calculations. Additionally, developers can consider implementing algorithms that reduce redundant computations and improve overall performance.

// Using the PHP Math library to calculate binomial distribution
function binomialDistribution($n, $k, $p) {
    $coeff = factorial($n) / (factorial($k) * factorial($n - $k));
    $probability = $coeff * pow($p, $k) * pow(1 - $p, $n - $k);
    
    return $probability;
}

// Example of calculating binomial distribution with n=5, k=2, p=0.5
$result = binomialDistribution(5, 2, 0.5);
echo $result;