Why wont my for loop add fractions past 1.0/10,000,000.0?

Question

I need to add a series of fractions. I looked through other for loops on adding fractions, but did not come across the problem I am having. The fractions denominator increases by 1.0 each time until it adds up 1.0/1.0 + 1.0/2.0 + .... + 1.0/ 150,000,000.0. My code has worked for me by adding the fractions until I go a little over 10,000,000 in my for loop. Once I go over say 15,000,000 it just doesn't print anything and the program continues to run. What error did I make? Why won't it print the answer when it's above a certain number?

Here is the code:

float sum = 0;
float numerator = 1;
float denominator = 1;

for(float i = 1; i <= 10000000; i++)
{
    sum = ((float)numerator/denominator) + sum;
    denominator++;
}
System.out.println("The sum is " + sum);

Answer 1

Adding 1 to a sufficiently large float won't have any effect any more. The problem is the limited precision of a float -- 23 bits. Once you get to about 16 million (about 2²⁴), the difference between consecutive float values is greater than 1. You can see this by printing out the result of Math.ulp (unit in last place) value.

if (denominator % 1000 == 0)
    System.out.println("Denominator is " + denominator + ", ulp is " + Math.ulp(denominator));

At the point where the output stops, the ulp is 1.0.

...
Denominator is 1.6775E7, ulp is 1.0
Denominator is 1.6776E7, ulp is 1.0
Denominator is 1.6777E7, ulp is 1.0

At 2²⁴, the ulp jumps to 2.0, and adding 1.0 follows IEEE's round towards even rule, and the sum is unchanged.

Change the datatypes to double for a much larger precision (53 bits). Just beware that this change would only prolong the problem until you reach 2⁵³. That would also make your program run much longer. And if you really want to go beyond that amount, use BigDecimals.

Answer 2

Instead of writing more comments here are my thoughts:

Floats have limited precision. It means that if you add very, very small numbers, the outcome might not be what you'd expect. It also works the other way - adding 1 to a very large number won't change it's value.

In your program to avoid this problem you should increment integer values and cast them to float before summation. Like this:

public class Sum {

public static void main(String[] args) {
    // TODO Auto-generated method stub


    float sum = 0;
    float numerator = 1;
    int denominator = 1; //changed to int

    for(int i = 1; i <= 10000000; i++)
    {
        sum = (numerator/(float)denominator) + sum; //this casts denominator to float
        denominator++; //this will now increment integer

    }
    System.out.println("The sum is " + sum);




} 
}

Further reading: Floating point on wikipedia