- Split input into 2 regimes
if wj < 4.608621108479766e-09
Initial program 13.4
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
Taylor expanded around 0 0.9
\[\leadsto \color{blue}{\left({wj}^{2} + x\right) - 2 \cdot \left(x \cdot wj\right)}\]
Simplified0.9
\[\leadsto \color{blue}{x + \left(wj + -2 \cdot x\right) \cdot wj}\]
if 4.608621108479766e-09 < wj
Initial program 25.3
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
- Using strategy
rm Applied distribute-rgt1-in25.4
\[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{\left(wj + 1\right) \cdot e^{wj}}}\]
Applied *-un-lft-identity25.4
\[\leadsto wj - \frac{\color{blue}{1 \cdot \left(wj \cdot e^{wj} - x\right)}}{\left(wj + 1\right) \cdot e^{wj}}\]
Applied times-frac25.3
\[\leadsto wj - \color{blue}{\frac{1}{wj + 1} \cdot \frac{wj \cdot e^{wj} - x}{e^{wj}}}\]
Simplified2.6
\[\leadsto wj - \frac{1}{wj + 1} \cdot \color{blue}{\left(wj - \frac{x}{e^{wj}}\right)}\]
- Using strategy
rm Applied flip3-+2.8
\[\leadsto wj - \frac{1}{\color{blue}{\frac{{wj}^{3} + {1}^{3}}{wj \cdot wj + \left(1 \cdot 1 - wj \cdot 1\right)}}} \cdot \left(wj - \frac{x}{e^{wj}}\right)\]
Applied associate-/r/2.5
\[\leadsto wj - \color{blue}{\left(\frac{1}{{wj}^{3} + {1}^{3}} \cdot \left(wj \cdot wj + \left(1 \cdot 1 - wj \cdot 1\right)\right)\right)} \cdot \left(wj - \frac{x}{e^{wj}}\right)\]
Simplified2.5
\[\leadsto wj - \left(\frac{1}{{wj}^{3} + {1}^{3}} \cdot \color{blue}{\left(\left(1 - wj\right) + wj \cdot wj\right)}\right) \cdot \left(wj - \frac{x}{e^{wj}}\right)\]
- Recombined 2 regimes into one program.
Final simplification0.9
\[\leadsto \begin{array}{l}
\mathbf{if}\;wj \le 4.608621108479766 \cdot 10^{-09}:\\
\;\;\;\;x + wj \cdot \left(x \cdot -2 + wj\right)\\
\mathbf{else}:\\
\;\;\;\;wj - \left(\frac{1}{1 + {wj}^{3}} \cdot \left(\left(1 - wj\right) + wj \cdot wj\right)\right) \cdot \left(wj - \frac{x}{e^{wj}}\right)\\
\end{array}\]
