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atan_forward

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Inverse Tangent and Hyperbolic Tangent Forward Mode Theory

Derivatives

\begin{eqnarray} \R{atan}^{(1)} (x) & = & 1 / ( 1 + x * x ) \\ \R{atanh}^{(1)} (x) & = & 1 / ( 1 - x * x ) \end{eqnarray}

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{eqnarray} 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{eqnarray}