How Math in JDK8 Detect Overflow

JDk8 add methods such as add/subtract/multiply/increment/decrement/Exact etc to throw ArithmeticException if it the result overflows.

The code:

public static long addExact(long x, long y) {
  long r = x + y;
  // HD 2-12 Overflow iff both arguments have the opposite sign of the result
  if (((x ^ r) & (y ^ r)) < 0) {
    throw new ArithmeticException("long overflow");
  }
  return r;
}

public static long subtractExact(long x, long y) {
  long r = x - y;
  // HD 2-12 Overflow iff the arguments have different signs and
  // the sign of the result is different than the sign of x
  if (((x ^ y) & (x ^ r)) < 0) {
    throw new ArithmeticException("long overflow");
  }
  return r;
}

public static int multiplyExact(int x, int y) {
  long r = (long)x * (long)y;
  if ((int)r != r) {
    throw new ArithmeticException("integer overflow");
  }
  return (int)r;
}
public static long multiplyExact(long x, long y) {
  long r = x * y;
  long ax = Math.abs(x);
  long ay = Math.abs(y);
  if (((ax | ay) >>> 31 != 0)) {
    // Some bits greater than 2^31 that might cause overflow
    // Check the result using the divide operator
    // and check for the special case of Long.MIN_VALUE * -1
    if (((y != 0) && (r / y != x)) ||
      (x == Long.MIN_VALUE && y == -1)) {
      throw new ArithmeticException("long overflow");
    }
  }
  return r;
}
public static long incrementExact(long a) {
  if (a == Long.MAX_VALUE) {
    throw new ArithmeticException("long overflow");
  }

  return a + 1L;
}
public static long negateExact(long a) {
  if (a == Long.MIN_VALUE) {
    throw new ArithmeticException("long overflow");
  }

  return -a;
}  
public static int floorDiv(int x, int y) {
  int r = x / y;
  // if the signs are different and modulo not zero, round down
  if ((x ^ y) < 0 && (r * y != x)) {
    r--;
  }
  return r;
}

Resources
Hacker's Delight
Hacker's Delight: The Basics
The Aggregate Magic Algorithms, 26 bit twiddling algorithms from Hank Dietz and others
Bit Twiddling Hacks, by Sean Anderson