Why does Math.round(0.49999999999999994) return 1?

double, floating-point, java, rounding

Solution

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. https://bugs.java.com/bugdatabase/view_bug?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

Problem

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.

Original source

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