Question

Why is a mod (%) operation more expensive than a multiplication (*) by a bit more than a factor of 2?

Please be more specific about how CPU performs division operation and returns the result for MOD operation.

In the following example the threads each run for a second. The test was performed on a SPARC processor.

// multiplication
void someThread() {

    int a = 10234;
    while (true) {
        opers++;
        a = a * a;
        a++;
    }

    // opers ~ 26 * 10^6 in a sec.
}

// MOD
void someThread() {

    int a = 10234;
    while (true) {
        opers++;
        a = a % 10000007;
        a++;
    }

    // opers ~ 12 * 10^6 in a sec.
}

Answer 1

Algorithms (processors execute the division and the multiplication by algorithms implemented in gates) for division are more costly than for multiplication. As a matter of fact, some algorithms for division which have a good complexity are using the multiplication as a basic step.

Even if you use the naive algorithms that are learned in school. They both have the same asymptotic complexity, but the constant for the division is greater (you have to find out the digit and that is not trivial, so you can mess up and have to fix the mess).