Question

While doing unsigned int conversions in Java I got some “anomalous” results, which I have cut down to the minimal case shown below. The same code in C generates similar results.

The problem is that when I calculate the theoretical sum of the absolute value of all integer values using the Gauss summation formula n(n+1)/2 (see https://en.wikipedia.org/wiki/Summation) the value I calculate does not match the total if I actually add up all the absolute values one by one.

Note that when I calculate the total using the summation formula the division by 2 “/2” is omitted because I am adding both the negative and positive numbers as absolute values, and at the end I have to add (longIntegerMax + 1) because the negative numbers have one extra number at the end (Integer.MIN) which has the absolute value of Integer.MAX + 1.

public static void main( String[] asArguments ){
    long longAbsoluteTotal = 0;
    long longNumberOfIntegers = 0;
    long longIntegerMax = Integer.MAX_VALUE;
    long longIntegerMin = Integer.MIN_VALUE;
    for( int i = Integer.MIN_VALUE;; i++ ){
        longNumberOfIntegers++;
        if( i < 0 ){
            longAbsoluteTotal += i * -1;
        } else {
            longAbsoluteTotal += i;
        }
        if( i == Integer.MAX_VALUE ) break;
    }
    long longCalculatedTotal = longIntegerMax * (longIntegerMax + 1) + longIntegerMax + 1; 
    System.out.println( "count of all integers: " + longNumberOfIntegers ); 
    System.out.println( "total of absolute value of all integers: " + longAbsoluteTotal );
    System.out.println( "calculated total of absolute value of all integers: " + longCalculatedTotal );
}

output:

 count of all integers: 4294967296
 total of absolute value of all integers: 4611686014132420608
 calculated total of absolute value of all integers: 4611686018427387904

As you can see, the calculated total is close to the real total, but does not match it exactly. Why not?

Answer 1

The problem is here ..

longAbsoluteTotal += i * -1;

The ( i * -1 ) is still integer arithmetic, it produces a number greater than Integer.MAX_VALUE and overflows back to Integer.MIN_VALUE;

You could fix this as @Evgeniy suggested, or you could use

longAbsoluteTotal += i * -1L; 

to force long arithmetic.