We can make this example more formal. Automatic differentiation is a formalization of the previous example (see graph).
We define \( \boldsymbol{x}\in x_1,\dots, x_l \) input variables to a given function \( f(\boldsymbol{x}) \) and \( x_{l+1},\dots, x_L \) intermediate variables.
In the above example we have only one input variable, \( l=1 \) and four intermediate variables, that is
$$ \begin{bmatrix} x_1=x & x_2 = x^2=a & x_3 =\exp{a}= b & x_4=c=a+b & x_5 = \sqrt{c}=d \end{bmatrix}. $$Furthemore, for \( i=l+1, \dots, L \) (here \( i=2,3,4,5 \) and \( f=x_L=d \)), we define the elementary functions \( g_i(x_{Pa(x_i)}) \) where \( x_{Pa(x_i)} \) are the parent nodes of the variable \( x_i \).
In our case, we have for example for \( x_3=g_3(x_{Pa(x_i)})=\exp{a} \), that \( g_3=\exp{()} \) and \( x_{Pa(x_3)}=a \).