This procedure converts a given polynomial r into Horner form, also known as nested multiplication form. This is a form which minimizes the number of arithmetic operations required to evaluate the polynomial.

If r is a rational function (i.e. a quotient of polynomials) then the numerator and denominator are each converted into Horner form.

If the second argument x is present then the first argument must be a polynomial (or rational expression) in the variable x. If the second argument is omitted then either r is an operator such that  yields a polynomial (or rational expression) in y, or else r is an expression with exactly one indeterminate (determined via indets).

Note that for the purpose of evaluating a polynomial efficiently, the Horner form minimizes the number of arithmetic operations for a general polynomial. Specifically, the cost of evaluating a polynomial of degree n in Horner form is: n multiplications and n additions.

The command with(numapprox,hornerform) allows the use of the abbreviated form of this command.