Question

When my Floating-Point Guide was yesterday published on slashdot, I got a lot of flak for my suggested comparison function, which was indeed inadequate. So I finally did the sensible thing and wrote a test suite to see whether I could get them all to pass. Here is my result so far. And I wonder if this is really as good as one can get with a generic (i.e. not application specific) float comparison function, or whether I still missed some edge cases.

(Code updated to fix error)

import static org.junit.Assert.assertFalse;
import static org.junit.Assert.assertTrue;

import org.junit.Test;

/**
 * Test suite to demonstrate a good method for comparing floating-point values using an epsilon. Run via JUnit 4.
 *
 * Note: this function attempts a "one size fits all" solution. There may be some edge cases for which it still
 * produces unexpected results, and some of the tests it was developed to pass probably specify behaviour that is
 * not appropriate for some applications. Before using it, make sure it's appropriate for your application!
 *
 * From http://floating-point-gui.de
 *
 * @author Michael Borgwardt
 */
public class NearlyEqualsTest {
    public static boolean nearlyEqual(float a, float b, float epsilon) {
        final float absA = Math.abs(a);
        final float absB = Math.abs(b);
        final float diff = Math.abs(a - b);

        if (a * b == 0) { // a or b or both are zero
            // relative error is not meaningful here
            return diff < (epsilon * epsilon);
        } else { // use relative error
            return diff / (absA + absB) < epsilon;
        }
    }

    public static boolean nearlyEqual(float a, float b) {
        return nearlyEqual(a, b, 0.000001f);
    }

    /** Regular large numbers - generally not problematic */
    @Test
    public void big() {
        assertTrue(nearlyEqual(1000000f, 1000001f));
        assertTrue(nearlyEqual(1000001f, 1000000f));
        assertFalse(nearlyEqual(10000f, 10001f));
        assertFalse(nearlyEqual(10001f, 10000f));
    }

    /** Negative large numbers */
    @Test
    public void bigNeg() {
        assertTrue(nearlyEqual(-1000000f, -1000001f));
        assertTrue(nearlyEqual(-1000001f, -1000000f));
        assertFalse(nearlyEqual(-10000f, -10001f));
        assertFalse(nearlyEqual(-10001f, -10000f));
    }

    /** Numbers around 1 */
    @Test
    public void mid() {
        assertTrue(nearlyEqual(1.0000001f, 1.0000002f));
        assertTrue(nearlyEqual(1.0000002f, 1.0000001f));
        assertFalse(nearlyEqual(1.0002f, 1.0001f));
        assertFalse(nearlyEqual(1.0001f, 1.0002f));
    }

    /** Numbers around -1 */
    @Test
    public void midNeg() {
        assertTrue(nearlyEqual(-1.000001f, -1.000002f));
        assertTrue(nearlyEqual(-1.000002f, -1.000001f));
        assertFalse(nearlyEqual(-1.0001f, -1.0002f));
        assertFalse(nearlyEqual(-1.0002f, -1.0001f));
    }

    /** Numbers between 1 and 0 */
    @Test
    public void small() {
        assertTrue(nearlyEqual(0.000000001000001f, 0.000000001000002f));
        assertTrue(nearlyEqual(0.000000001000002f, 0.000000001000001f));
        assertFalse(nearlyEqual(0.000000000001002f, 0.000000000001001f));
        assertFalse(nearlyEqual(0.000000000001001f, 0.000000000001002f));
    }

    /** Numbers between -1 and 0 */
    @Test
    public void smallNeg() {
        assertTrue(nearlyEqual(-0.000000001000001f, -0.000000001000002f));
        assertTrue(nearlyEqual(-0.000000001000002f, -0.000000001000001f));
        assertFalse(nearlyEqual(-0.000000000001002f, -0.000000000001001f));
        assertFalse(nearlyEqual(-0.000000000001001f, -0.000000000001002f));
    }

    /** Comparisons involving zero */
    @Test
    public void zero() {
        assertTrue(nearlyEqual(0.0f, 0.0f));
        assertTrue(nearlyEqual(0.0f, -0.0f));
        assertTrue(nearlyEqual(-0.0f, -0.0f));
        assertFalse(nearlyEqual(0.00000001f, 0.0f));
        assertFalse(nearlyEqual(0.0f, 0.00000001f));
        assertFalse(nearlyEqual(-0.00000001f, 0.0f));
        assertFalse(nearlyEqual(0.0f, -0.00000001f));

        assertTrue(nearlyEqual(0.0f, 0.00000001f, 0.01f));
        assertTrue(nearlyEqual(0.00000001f, 0.0f, 0.01f));
        assertFalse(nearlyEqual(0.00000001f, 0.0f, 0.000001f));
        assertFalse(nearlyEqual(0.0f, 0.00000001f, 0.000001f));

        assertTrue(nearlyEqual(0.0f, -0.00000001f, 0.1f));
        assertTrue(nearlyEqual(-0.00000001f, 0.0f, 0.1f));
        assertFalse(nearlyEqual(-0.00000001f, 0.0f, 0.00000001f));
        assertFalse(nearlyEqual(0.0f, -0.00000001f, 0.00000001f));
    }

    /** Comparisons of numbers on opposite sides of 0 */
    @Test
    public void opposite() {
        assertFalse(nearlyEqual(1.000000001f, -1.0f));
        assertFalse(nearlyEqual(-1.0f, 1.000000001f));
        assertFalse(nearlyEqual(-1.000000001f, 1.0f));
        assertFalse(nearlyEqual(1.0f, -1.000000001f));
        assertTrue(nearlyEqual(1e10f * Float.MIN_VALUE, -1e10f * Float.MIN_VALUE));
    }

    /**
     * The really tricky part - comparisons of numbers very close to zero.
     */
    @Test
    public void ulp() {
        assertTrue(nearlyEqual(Float.MIN_VALUE, -Float.MIN_VALUE));
        assertTrue(nearlyEqual(-Float.MIN_VALUE, Float.MIN_VALUE));
        assertTrue(nearlyEqual(Float.MIN_VALUE, 0));
        assertTrue(nearlyEqual(0, Float.MIN_VALUE));
        assertTrue(nearlyEqual(-Float.MIN_VALUE, 0));
        assertTrue(nearlyEqual(0, -Float.MIN_VALUE));

        assertFalse(nearlyEqual(0.000000001f, -Float.MIN_VALUE));
        assertFalse(nearlyEqual(0.000000001f, Float.MIN_VALUE));
        assertFalse(nearlyEqual(Float.MIN_VALUE, 0.000000001f));
        assertFalse(nearlyEqual(-Float.MIN_VALUE, 0.000000001f));

        assertFalse(nearlyEqual(1e25f * Float.MIN_VALUE, 0.0f, 1e-12f));
        assertFalse(nearlyEqual(0.0f, 1e25f * Float.MIN_VALUE, 1e-12f));
        assertFalse(nearlyEqual(1e25f * Float.MIN_VALUE, -1e25f * Float.MIN_VALUE, 1e-12f));

        assertTrue(nearlyEqual(1e25f * Float.MIN_VALUE, 0.0f, 1e-5f));
        assertTrue(nearlyEqual(0.0f, 1e25f * Float.MIN_VALUE, 1e-5f));
        assertTrue(nearlyEqual(1e20f * Float.MIN_VALUE, -1e20f * Float.MIN_VALUE, 1e-5f));
    }

}

Answer 1

After sleeping over it, I’ve realized that this part was rubbish:

    if (a*b==0) {
        return diff < Float.MIN_VALUE / epsilon;

This becomes less strict as epsilon gets smaller! A more sensible version:

    if (a * b == 0) {
        return diff < (epsilon * epsilon);

Still, the two branches of the if are not very consistent with each other. It’s much stricter when a or b is very small than when one of them is zero. I’m really starting to think that using integer comparison is an overall better method.