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H
and
G
.
The forward mode formulas for the
exponential
function are
\[
z^{(j)} = \exp ( x^{(0)} )
\]
if
j = 0
, and
\[
z^{(j)} = \frac{1}{j}
\sum_{k=1}^{j} k x^{(k)} z^{(j-k)}
\]
for the case
j = 0
, and for
j > 0
,
\[
\begin{array}{rcl}
\D{H}{ x^{(j)} } & = &
\D{G}{ x^{(j)} } + \D{G}{ z^{(j)} } z^{(j)}
\end{array}
\]
If
j > 0
, then for
k = 1 , \ldots , j
\[
\begin{array}{rcl}
\D{H}{ x^{(k)} } & = &
\D{G}{ x^{(k)} } + \D{G}{ z^{(j)} } \frac{1}{j} k z^{(j-k)}
\\
\D{H}{ z^{(j-k)} } & = &
\D{G}{ z^{(j-k)} } + \D{G}{ z^{(j)} } \frac{1}{j} k x^{(k)}
\end{array}
\]