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 $r\left(y\right)$ 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.

$\mathrm{with}\left(\mathrm{numapprox}\right)\:$

$f≔t\mapsto a\cdot t^4+b\cdot t^3+c\cdot t^2+d\cdot t+e$

$f≔t\mapsto a\cdot t^4+b\cdot t^3+c\cdot t^2+d\cdot t+e$

$\mathrm{hornerform}\left(f\right)$

$t\mapsto e+\left(d+\left(c+\left(a\cdot t+b\right)\cdot t\right)\cdot t\right)\cdot t$

$s≔\mathrm{taylor}\left(\exp \left(x\right)\,x\right)$

$s≔1+x+\frac{1}{2}{x}^2+\frac{1}{6}{x}^3+\frac{1}{24}{x}^4+\frac{1}{120}{x}^5+O\left(x^6\right)$

$\mathrm{hornerform}\left(s\right)$

$1+\left(1+\left(\frac{1}{2}+\left(\frac{1}{6}+\left(\frac{1}{24}+\frac{x}{120}\right)x\right)x\right)x\right)x$

$r≔\mathrm{pade}\left(\exp \left(ax\right)\,x\,\left[3\,3\right]\right)$

$r≔\frac{a^3{x}^3+12a^2{x}^2+60ax+120}{-a^3{x}^3+12a^2{x}^2-60ax+120}$

$\mathrm{hornerform}\left(r\,x\right)$

$\frac{120+\left(60a+\left(a^{3}x+12a^2\right)x\right)x}{120+\left(-60a+\left(-a^{3}x+12a^2\right)x\right)x}$