private function doAdd(string $a, string $b) : string
    {
        $length = $this->pad($a, $b);

        $carry = 0;
        $result = '';

        for ($i = $length - $this->maxDigits;; $i -= $this->maxDigits) {
            $blockLength = $this->maxDigits;

            if ($i < 0) {
                $blockLength += $i;
                $i = 0;
            }

            $blockA = \substr($a, $i, $blockLength);
            $blockB = \substr($b, $i, $blockLength);

            $sum = (string) ($blockA + $blockB + $carry);
            $sumLength = \strlen($sum);

            if ($sumLength > $blockLength) {
                $sum = \substr($sum, 1);
                $carry = 1;
            } else {
                if ($sumLength < $blockLength) {
                    $sum = \str_repeat('0', $blockLength - $sumLength) . $sum;
                }
                $carry = 0;
            }

            $result = $sum . $result;

            if ($i === 0) {
                break;
            }
        }

        if ($carry === 1) {
            $result = '1' . $result;
        }

        return $result;
    }