public static String add(String bits1, String bits2) {
	int len1 = bits1.length();
	int len2 = bits2.length();
	int max = Math.max(len1, len2);
	StringBuilder result = new StringBuilder(1 + max);
	boolean one = false;
	for (int i = 1; i <= max; i++) {
		final char c1 = len1 - i >= 0 ? bits1.charAt(len1 - i) : '0';
		final char c2 = len2 - i >= 0 ? bits2.charAt(len2 - i) : '0';
		if (('0' ^ c1) > 1 || ('0' ^ c2) > 1)
			throw new IllegalArgumentException(bits1 + " + " + bits2);
		if (c1 + c2 == 2 * '0') {
			result.append(one ? '1' : '0');
			one = false;
		} else if (c1 + c2 == '0' + '1') {
			result.append(one ? '0' : '1');
		} else if (c1 + c2 == 2 * '1') {
			result.append(one ? '1' : '0');
			one = true;
		} else
			throw new IllegalArgumentException(bits1 + " + " + bits2);
	}
	if (one)
		result.append('1');
	return result.reverse().toString();
}