- Split input into 2 regimes
if wj < 1.8976396421588022e-08
Initial program 13.3
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
- Using strategy
rm Applied div-sub13.3
\[\leadsto wj - \color{blue}{\left(\frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)}\]
Applied associate--r-6.7
\[\leadsto \color{blue}{\left(wj - \frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}}\right) + \frac{x}{e^{wj} + wj \cdot e^{wj}}}\]
Taylor expanded around 0 0.3
\[\leadsto \color{blue}{\left(\left({wj}^{2} + {wj}^{4}\right) - {wj}^{3}\right)} + \frac{x}{e^{wj} + wj \cdot e^{wj}}\]
Simplified0.3
\[\leadsto \color{blue}{\left({wj}^{4} - \left(wj \cdot wj - wj\right) \cdot wj\right)} + \frac{x}{e^{wj} + wj \cdot e^{wj}}\]
if 1.8976396421588022e-08 < wj
Initial program 22.8
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
- Using strategy
rm Applied distribute-rgt1-in22.9
\[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{\left(wj + 1\right) \cdot e^{wj}}}\]
Applied *-un-lft-identity22.9
\[\leadsto wj - \frac{\color{blue}{1 \cdot \left(wj \cdot e^{wj} - x\right)}}{\left(wj + 1\right) \cdot e^{wj}}\]
Applied times-frac22.8
\[\leadsto wj - \color{blue}{\frac{1}{wj + 1} \cdot \frac{wj \cdot e^{wj} - x}{e^{wj}}}\]
Simplified2.4
\[\leadsto wj - \frac{1}{wj + 1} \cdot \color{blue}{\left(wj - \frac{x}{e^{wj}}\right)}\]
- Recombined 2 regimes into one program.
Final simplification0.4
\[\leadsto \begin{array}{l}
\mathbf{if}\;wj \le 1.8976396421588022 \cdot 10^{-08}:\\
\;\;\;\;\frac{x}{wj \cdot e^{wj} + e^{wj}} + \left({wj}^{4} - wj \cdot \left(wj \cdot wj - wj\right)\right)\\
\mathbf{else}:\\
\;\;\;\;wj - \frac{1}{1 + wj} \cdot \left(wj - \frac{x}{e^{wj}}\right)\\
\end{array}\]
