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Tangent and Hyperbolic Tangent Reverse Mode Theory

Notation
We use the reverse theory standard math function definition for the functions $H$ and $G$. In addition, we use the forward mode notation in tan_forward for $X(t)$, $Y(t)$ and $Z(t)$.

Eliminating Y(t)
For $j > 0$, the forward mode coefficients are given by $$y^{(j-1)} = \sum_{k=0}^{j-1} z^{(k)} z^{(j-k-1)}$$ Fix $j > 0$ and suppose that $H$ is the same as $G$ except that $y^{(j-1)}$ is replaced as a function of the Taylor coefficients for $Z(t)$. To be specific, for $k = 0 , \ldots , j-1$, $$\begin{array}{rcl} \D{H}{ z^{(k)} } & = & \D{G}{ z^{(k)} } + \D{G}{ y^{(j-1)} } \D{ y^{(j-1)} }{ z^{(k)} } \\ & = & \D{G}{ z^{(k)} } + \D{G}{ y^{(j-1)} } 2 z^{(j-k-1)} \end{array}$$

Positive Orders Z(t)
For order $j > 0$, suppose that $H$ is the same as $G$ except that $z^{(j)}$ is expressed as a function of the coefficients for $X(t)$, and the lower order Taylor coefficients for $Y(t)$, $Z(t)$. $$z^{(j)} = x^{(j)} \pm \frac{1}{j} \sum_{k=1}^j k x^{(k)} y^{(j-k)}$$ For $k = 1 , \ldots , j$, the partial of $H$ with respect to $x^{(k)}$ is given by $$\begin{array}{rcl} \D{H}{ x^{(k)} } & = & \D{G}{ x^{(k)} } + \D{G}{ z^{(j)} } \D{ z^{(j)} }{ x^{(k)} } \\ & = & \D{G}{ x^{(k)} } + \D{G}{ z^{(j)} } \left[ \delta ( j - k ) \pm \frac{k}{j} y^{(j-k)} \right] \end{array}$$ where $\delta ( j - k )$ is one if $j = k$ and zero otherwise. For $k = 1 , \ldots , j$ The partial of $H$ with respect to $y^{j-k}$, is given by $$\begin{array}{rcl} \D{H}{ y^{(j-k)} } & = & \D{G}{ y^{(j-k)} } + \D{G}{ z^{(j)} } \D{ z^{(j)} }{ y^{(j-k)} } \\ & = & \D{G}{ y^{(j-k)} } \pm \D{G}{ z^{(j)} }\frac{k}{j} x^{k} \end{array}$$

Order Zero Z(t)
The order zero coefficients for the tangent and hyperbolic tangent are $$\begin{array}{rcl} z^{(0)} & = & \left\{ \begin{array}{c} \tan ( x^{(0)} ) \\ \tanh ( x^{(0)} ) \end{array} \right. \end{array}$$ Suppose that $H$ is the same as $G$ except that $z^{(0)}$ is expressed as a function of the Taylor coefficients for $X(t)$. In this case, $$\begin{array}{rcl} \D{H}{ x^{(0)} } & = & \D{G}{ x^{(0)} } + \D{G}{ z^{(0)} } \D{ z^{(0)} }{ x^{(0)} } \\ & = & \D{G}{ x^{(0)} } + \D{G}{ z^{(0)} } ( 1 \pm y^{(0)} ) \end{array}$$
Input File: omh/appendix/theory/tan_reverse.omh