The convert(f, 'horner', x) command converts the input to horner form with respect to x, or the names in x if this argument is a list or set.

If x is omitted, it is taken as indets(f,name).

If x is given as a list or set of names, the conversion is applied recursively to the coefficients of f in the first variable given.  A list allows for control over the order of the conversions.

If f is a set, list or relation, the result is map(convert, f, 'horner', x).

Any subexpressions of f which are not either polynomials or rational functions in x are frozen via a call to frontend prior to conversion.

$\mathrm{convert}\left(x^2+3x+4\,\mathrm{horner}\,x\right)$

$4+\left(3+x\right)x$

$\mathrm{poly}≔y^2{x}^2+2x{y}^2+2x^{2}y+x^2+2x$

$\mathrm{poly}≔y^2{x}^2+2x^{2}y+2x{y}^2+x^2+2x$

$\mathrm{convert}\left(\mathrm{poly}\,\mathrm{horner}\,x\right)$

$\left(2y^2+2+\left(y^2+2y+1\right)x\right)x$

$\mathrm{convert}\left(\mathrm{poly}\,\mathrm{horner}\,\left[x\,y\right]\right)$

$\left(2y^2+2+\left(1+\left(2+y\right)y\right)x\right)x$

$\mathrm{convert}\left(\mathrm{poly}\,\mathrm{horner}\,\left[y\,x\right]\right)$

$\left(2+x\right)x+\left(2x^2+\left(2+x\right)xy\right)y$

$\mathrm{convert}\left(\frac{{x\left[i\right]}^2-3x\left[i\right]+2}{4x\left[i\right]-2{x\left[i\right]}^3-2+{x\left[i\right]}^2}\,\mathrm{horner}\right)$

$\frac{2+\left(-3+x_i\right)x_i}{-2+\left(4+\left(1-2x_i\right)x_i\right)x_i}$

$\mathrm{convert}\left(a^3+3a^2\sin \left(t\right)-a\cos \left(t\right)+1+\mathrm{sqrt}\left(a\right)=1-a\tan \left(t\right)+a^2\,\mathrm{horner}\,a\right)$

$\sqrt{a}+1+\left(-\cos \left(t\right)+\left(3\sin \left(t\right)+a\right)a\right)a=1+\left(-\tan \left(t\right)+a\right)a$

$\mathrm{convert}\left(\left[3b^3-2b^2+b-1\,\frac{2-b}{3b^2-5b+6}\right]\,\mathrm{horner}\right)$

$\left[-1+\left(1+\left(-2+3b\right)b\right)b\,\frac{2-b}{6+\left(-5+3b\right)b}\right]$

$\mathrm{convert}\left(\left\{3b^3-2b^2+b-1\,\frac{2-b}{3b^2-5b+6}\right\}\,\mathrm{horner}\right)$

$\left\{\frac{2-b}{6+\left(-5+3b\right)b}\,-1+\left(1+\left(-2+3b\right)b\right)b\right\}$