compareTo with primitives -> Integer / int

Wrapping int primitive into Integer object will cost you some memory, but the difference will be only significant in very rare(memory demand) cases (array with 1000+ elements). I will not recommend using new Integer(int a) constructor this way. This will suffice :

Integer a = 3; 

About comparision there is Math.signum(double d).

compare= (int) Math.signum(a-b); 

For performance, it usually best to make the code as simple and clear as possible and this will often perform well (as the JIT will optimise this code best). In your case, the simplest examples are also likely to be the fastest.

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I would do either

int cmp = a > b ? +1 : a < b ? -1 : 0;

or a longer version

int cmp;
if (a > b)
   cmp = +1;
else if (a < b)
   cmp = -1;
else
   cmp = 0;

or

int cmp = Integer.compare(a, b); // in Java 7
int cmp = Double.compare(a, b); // before Java 7

It's best not to create an object if you don't need to.

Performance wise, the first is best.

If you know for sure that you won't get an overflow you can use

int cmp = a - b; // if you know there wont be an overflow.

you won't get faster than this.

Use Integer.compare(int, int). And don'try to micro-optimize your code unless you can prove that you have a performance issue.

May I propose a third

((Integer) a).compareTo(b)