Note on Benford’s Law

While dealing with Benford’s Law i noticed, that the algorithm, which used the product formula, works more than 2 times faster, then the algorithm with the summation formula. Below you can see my two implementations of Benford’s Law in C. Thanks to Prof. Dr. Klaus Neumann for any suggestions.

Params

int n := position of the numeral in the number

int d := numeral

return := p(d) at position n

Example

1337 -> 1. benford(1,1) -> 2. benford(2,3) -> benford(3,3) -> benford(4,7)

Benford’s Law with summation formula

float benford10(int n, int d){

float j = pow(10, n – 1), s = 0, i;

for(i = floor(pow(10, (n – 2))); i <= j – 1; i++)

s += log10(1 + (1 / ((i * 10) + d)));

return s;

}

Benford’s Law with product formula

float benford10(int n, int d){

float j = pow(10, n – 1) – 1, p = 1, i;

for(i = floor(pow(10, (n – 2))); i <= j; i++)

p *= (1 + (1/((10 * i) + d)));

return log10(p);

}