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Exponential Function Reverse Mode Theory
We use the reverse theory standard math function definition for the functions  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  \[
\begin{array}{rcl}
\D{H}{ x^{(k)} } & = & 
\D{G}{ x^{(k)} }  + \D{G}{ z^{(j)} } \frac{1}{j}  k z^{(j-k)}
\\
\D{H}{ z^{(j-k)} } & = & 
\D{G}{ z^{(j-k)} }  + \D{G}{ z^{(j)} } \frac{1}{j}  k x^{(k)}
\end{array}
\] 

Input File: omh/exp_reverse.omh