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$\newcommand{\W}[1]{ \; #1 \; } \newcommand{\R}[1]{ {\rm #1} } \newcommand{\B}[1]{ {\bf #1} } \newcommand{\D}[2]{ \frac{\partial #1}{\partial #2} } \newcommand{\DD}[3]{ \frac{\partial^2 #1}{\partial #2 \partial #3} } \newcommand{\Dpow}[2]{ \frac{\partial^{#1}}{\partial {#2}^{#1}} } \newcommand{\dpow}[2]{ \frac{ {\rm d}^{#1}}{{\rm d}\, {#2}^{#1}} }$
Inverse Tangent and Hyperbolic Tangent Forward Mode Theory

Derivatives
$$\begin{array}{rcl} \R{atan}^{(1)} (x) & = & 1 / ( 1 + x * x ) \\ \R{atanh}^{(1)} (x) & = & 1 / ( 1 - x * x ) \end{array}$$If $F(x)$ is $\R{atan} (x)$ or $\R{atanh} (x)$, the corresponding derivative satisfies the equation $$(1 \pm x * x ) * F^{(1)} (x) - 0 * F (x) = 1$$ and in the standard math function differential equation , $A(x) = 0$, $B(x) = 1 \pm x * x$, and $D(x) = 1$. We use $a$, $b$, $d$ and $z$ to denote the Taylor coefficients for $A [ X (t) ]$, $B [ X (t) ]$, $D [ X (t) ]$, and $F [ X(t) ]$ respectively.

Taylor Coefficients Recursion
For $j = 0 , 1, \ldots$, $$\begin{array}{rcl} z^{(0)} & = & F( x^{(0)} ) \\ b^{(j)} & = & \left\{ \begin{array}{ll} 1 \pm x^{(0)} * x^{(0)} & {\rm if} \; j = 0 \\ \pm \sum_{k=0}^j x^{(k)} x^{(j-k)} & {\rm otherwise} \end{array} \right. \\ e^{(j)} & = & d^{(j)} + \sum_{k=0}^{j} a^{(j-k)} * z^{(k)} \\ & = & \left\{ \begin{array}{ll} 1 & {\rm if} \; j = 0 \\ 0 & {\rm otherwise} \end{array} \right. \\ z^{(j+1)} & = & \frac{1}{j+1} \frac{1}{ b^{(0)} } \left( \sum_{k=0}^j e^{(k)} (j+1-k) x^{(j+1-k)} - \sum_{k=1}^j b^{(k)} (j+1-k) z^{(j+1-k)} \right) \\ z^{(j+1)} & = & \frac{1}{j+1} \frac{1}{ b^{(0)} } \left( (j+1) x^{(j+1)} - \sum_{k=1}^j k z^{(k)} b^{(j+1-k)} \right) \end{array}$$
Input File: omh/appendix/theory/atan_forward.omh