In mathematics, Horner's method (or Horner's scheme) is an algorithm for polynomial evaluation. With this method, it is possible to evaluate a polynomial with only n additions and n multiplications. Hence, its storage requirements are n times the number of bits of x.
Horner's method can be based on the following identity:
This identity is called Horner's rule.
To solve the right part of the identity above, for a given x, we start by iterating through the polynomial from the inside out, accumulating each iteration result. After n iterations, with n being the order of the polynomial, the accumulated result gives us the polynomial evaluation.
Using the polynomial:
4 * x^4 + 2 * x^3 + 3 * x^2 + x^1 + 3, a traditional approach to evaluate it at x = 2, could be representing it as an array [3, 1, 3, 2, 4] and iterate over it saving each iteration value at an accumulator, such as acc += pow(x=2, index) * array[index]. In essence, each power of a number (pow) operation is n-1 multiplications. So, in this scenario, a total of 14 operations would have happened, composed of 4 additions, 5 multiplications, and 5 pows (we're assuming that each power is calculated by repeated multiplication).
Now, using the same scenario but with Horner's rule, the polynomial can be re-written as x * (x * (x * (4 * x + 2) + 3) + 1) + 3, representing it as [4, 2, 3, 1, 3] it is possible to save the first iteration as acc = arr[0] * (x=2) + arr[1], and then finish iterations for acc *= (x=2) + arr[index]. In the same scenario but using Horner's rule, a total of 10 operations would have happened, composed of only 4 additions and 4 multiplications.