Why does Math.round(0.49999999999999994) return 1?

Question

In the following program you can see that each value slightly less than .5 is rounded down, except for 0.5.

for (int i = 10; i >= 0; i--) {     long l = Double.doubleToLongBits(i + 0.5);     double x;     do {         x = Double.longBitsToDouble(l);         System.out.println(x + " rounded is " + Math.round(x));         l--;     } while (Math.round(x) > i); } 

prints

10.5 rounded is 11 10.499999999999998 rounded is 10 9.5 rounded is 10 9.499999999999998 rounded is 9 8.5 rounded is 9 8.499999999999998 rounded is 8 7.5 rounded is 8 7.499999999999999 rounded is 7 6.5 rounded is 7 6.499999999999999 rounded is 6 5.5 rounded is 6 5.499999999999999 rounded is 5 4.5 rounded is 5 4.499999999999999 rounded is 4 3.5 rounded is 4 3.4999999999999996 rounded is 3 2.5 rounded is 3 2.4999999999999996 rounded is 2 1.5 rounded is 2 1.4999999999999998 rounded is 1 0.5 rounded is 1 0.49999999999999994 rounded is 1 0.4999999999999999 rounded is 0 

I am using Java 6 update 31.

Answer 1

Summary

In Java 6 (and presumably earlier), round(x) is implemented as floor(x+0.5).¹ This is a specification bug, for precisely this one pathological case.² Java 7 no longer mandates this broken implementation.³

The problem

0.5+0.49999999999999994 is exactly 1 in double precision:

static void print(double d) {     System.out.printf("%016x\n", Double.doubleToLongBits(d)); }  public static void main(String args[]) {     double a = 0.5;     double b = 0.49999999999999994;      print(a);      // 3fe0000000000000     print(b);      // 3fdfffffffffffff     print(a+b);    // 3ff0000000000000     print(1.0);    // 3ff0000000000000 } 

This is because 0.49999999999999994 has a smaller exponent than 0.5, so when they're added, its mantissa is shifted, and the ULP gets bigger.

The solution

Since Java 7, OpenJDK (for example) implements it thus:⁴

public static long round(double a) {     if (a != 0x1.fffffffffffffp-2) // greatest double value less than 0.5         return (long)floor(a + 0.5d);     else         return 0; } 

---

_{1. http://docs.oracle.com/javase/6/docs/api/java/lang/Math.html#round%28double%29}

_{2. http://bugs.java.com/bugdatabase/view_bug.do?bug_id=6430675 (credits to @SimonNickerson for finding this)}

_{3. http://docs.oracle.com/javase/7/docs/api/java/lang/Math.html#round%28double%29}

_{4. http://grepcode.com/file/repository.grepcode.com/java/root/jdk/openjdk/7u40-b43/java/lang/Math.java#Math.round%28double%29}