MathPHP\Functions\Special::regularizedIncompleteBeta PHP Method

    public static function regularizedIncompleteBeta($x, $a, $b)
    {
        $limits = ['x' => '[0, 1]', 'a' => '(0,∞)', 'b' => '(0,∞)'];
        Support::checkLimits($limits, ['x' => $x, 'a' => $a, 'b' => $b]);
        if ($x == 1 || $x == 0) {
            return $x;
        }
        if ($a == 1) {
            return 1 - (1 - $x) ** $b;
        }
        if ($b == 1) {
            return $x ** $a;
        }
        if ($x > 0.9 || $b > $a && $x > 0.5) {
            $y = 1 - $x;
            return 1 - self::regularizedIncompleteBeta($y, $b, $a);
        }
        if ($a > 1 && $b > 1) {
            // Tolerance on evaluating the continued fraction.
            $tol = 1.0E-15;
            $dif = $tol + 1;
            // Initialize
            // We will calculate the continuous fraction with a minimum depth of 10.
            $m = 10;
            // Counter
            do {
                $I_new = self::iBetaCF($m, $x, $a, $b);
                if ($m > 10) {
                    $dif = abs(($I - $I_new) / $I_new);
                }
                $I = $I_new;
                $m++;
            } while ($dif > $tol);
            return $I;
        } else {
            if ($a <= 1) {
                // We shift a up by one, to the region that the continuous fraction works best.
                $offset = $x ** $a * (1 - $x) ** $b / $a / self::beta($a, $b);
                return self::regularizedIncompleteBeta($x, $a + 1, $b) + $offset;
            } else {
                // $b <= 1
                // We shift a up by one, to the region that the continuous fraction works best.
                $offset = $x ** $a * (1 - $x) ** $b / $b / self::beta($a, $b);
                return self::regularizedIncompleteBeta($x, $a, $b + 1) - $offset;
            }
        }
    }

 /**
  * Cumulative distribution function
  * Calculate the cumulative t value up to a point, left tail.
  *
  * cdf = 1 - ½Iₓ₍t₎(ν/2, ½)
  *
  *                 ν
  *  where x(t) = ------
  *               t² + ν
  *
  *        Iₓ₍t₎(ν/2, ½) is the regularized incomplete beta function
  *
  * @param number $t t score
  * @param int    $ν degrees of freedom > 0
  */
 public static function CDF($t, int $ν)
 {
     Support::checkLimits(self::LIMITS, ['t' => $t, 'ν' => $ν]);
     if ($t == 0) {
         return 0.5;
     }
     $x⟮t⟯ = $ν / ($t ** 2 + $ν);
     $ν／2 = $ν / 2;
     $½ = 0.5;
     $Iₓ = Special::regularizedIncompleteBeta($x⟮t⟯, $ν／2, $½);
     if ($t < 0) {
         return $½ * $Iₓ;
     }
     // $t ≥ 0
     return 1 - $½ * $Iₓ;
 }