Machine Epsilon

Question

I was looking at a simple way to approximate Machine Epsilon in Java:

float machEps = 1.0f;

do
    machEps /= 2.0f;
while ((float) (1.0 + (machEps / 2.0)) != 1.0);

System.out.println( machEps);

This returns:

1.1920929E-7

However, when I remove the conversion to float in the while loop:

float machEps = 1.0f;

do
   machEps /= 2.0f;
while ( (1.0 + (machEps / 2.0)) != 1.0);

System.out.println( machEps);

I get:

2.220446E-16

I'm not quite sure why this is....my guess is that in the second case Java attempts to expand machEps from a float to a double. However, I'm not sure if that is an accurate statement or if there is another reason why I get two different answers.

Answer 1

1.0 and 2.0 are doubles.

Arithmetic between doubles and floats will implicitly convert the floats to doubles.

You need to force the entire expression to use floats by adding the f suffix to all of your literals.

Answer 2

Are you looking for java.lang.Math.ulp?

Returns the size of an ulp of the argument. An ulp of a double value is the positive distance between this floating-point value and the double value next larger in magnitude.

Answer 3

You can easily find the smallest float that is strictly greater than 1.0 using Math.nextUp(1.0f).