Powers

The power function is a function with a variable raised to a fixed real number. Usually these functions are already implemented in the C language. However, for numerical reasons and efficiency PINS redefines them

Power 1 regular

\[x^+ \approx \pow_{1,h}(x)\DEF \begin{cases} \dfrac{\epsilon}{1-z} & x<0 \\[1em] \dfrac{a_0+z(a_1+z(a_2+z(a_3+a_4 z)))}{(1+z)^3} & \textrm{otherwise} \end{cases} \qquad z = \dfrac{x}{h}\]

\[ a_0 = \epsilon,\qquad a_1 = 4\epsilon,\qquad a_2 = 7\epsilon,\qquad a_3 = 8\epsilon,\qquad a_4 = 1-20\epsilon\]

which is continous up to the third derivative included, \(\epsilon<1/20\).

Power2 regular

\[ (x^+)^2\approx \pow_{2,h}(x)\DEF \begin{cases} \dfrac{\epsilon}{1-z} & x<0 \\[1em] \dfrac{\epsilon(1+z(3+4z(1+z))) + (1-12\epsilon)z^4}{(1+z)^2} & \textrm{otherwise} \end{cases}\]

which is continous up to the third derivative included, , \(\epsilon<1/12\).

Power4 regular

\[(x^+)^4 \approx \pow_{4,h}(x)\DEF \begin{cases} \epsilon\exp(z) & x<0 \\\\[0.5em] \epsilon+z(a+z(b+z(c+zd))) & \textrm{otherwise} \end{cases}\]

\[ a = \epsilon,\qquad b = (1/2)\epsilon,\qquad c = (1/6)\epsilon,\qquad d = 1-(5/3)\epsilon\]

\(\epsilon\) max \(3/10\) to hase asymthotically \(x^4/2\)