This procedure converts a given rational function r into the continued-fraction form which minimizes the number of arithmetic operations required for evaluation.

If the second argument x is present then the first argument must be a 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 rational expression in y, or else r is a rational expression with exactly one indeterminate (determined via indets).

Note that for the purpose of evaluating a rational function efficiently (i.e. minimizing the number of arithmetic operations), the rational function should be converted to continued-fraction form. In general, the cost of evaluating a rational function of degree $\left(m\,n\right)$ when each of numerator and denominator is expressed in Horner (nested multiplication) form, with the denominator made monic, is

whereas the same rational function can be evaluated in continued-fraction form with a cost not exceeding

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

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

$f≔t\mapsto \frac{1.1\cdot t^2-20.5\cdot t+5.3}{t^2+7.6\cdot t+0.1}$

$f≔t\mapsto \frac{1.1\cdot t^2-20.5\cdot t+5.3}{t^2+7.6\cdot t+0.1}$

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

$t\mapsto \frac{5.3+\left(-20.5+1.1\cdot t\right)\cdot t}{0.1+\left(7.6+t\right)\cdot t}$

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

$y\mapsto 1.100000000-\frac{28.86000000}{y+7.779833680+\frac{1.499076119}{y-0.1798336798}}$

$e≔\mathrm{pade}\left(\exp \left(x\right)\,x\,\left[2\,2\right]\right)$

$e≔\frac{\frac{1}{12}{x}^2+\frac{1}{2}x+1}{\frac{1}{12}{x}^2-\frac{1}{2}x+1}$

$\mathrm{confracform}\left(e\,x\right)$

$1+\frac{12}{x-6+\frac{12}{x}}$

$r\left[2,3\right]≔\mathrm{minimax}\left(\frac{\tan \left(x\right)}{x}\,x=0..\frac{\mathrm{\pi }}{4}\,\left[2\,3\right]\right)$

$r_{2,3}≔\frac{1.130422926+\left(-0.07842798254-0.07066118710x\right)x}{1.130423032+\left(-0.07843711579+\left(-0.4473405792+0.02547897687x\right)x\right)x}$

$\mathrm{confracform}\left(r\left[2,3\right]\right)$

$-\frac{2.773313366}{x-18.66715865+\frac{33.63826614}{x+8.668760300+\frac{49.52801799}{x-7.558844295}}}$