Initial program 13.6
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
- Using strategy
rm Applied distribute-rgt1-in13.6
\[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{\left(wj + 1\right) \cdot e^{wj}}}\]
Applied *-un-lft-identity13.6
\[\leadsto wj - \frac{\color{blue}{1 \cdot \left(wj \cdot e^{wj} - x\right)}}{\left(wj + 1\right) \cdot e^{wj}}\]
Applied times-frac13.6
\[\leadsto wj - \color{blue}{\frac{1}{wj + 1} \cdot \frac{wj \cdot e^{wj} - x}{e^{wj}}}\]
Simplified13.0
\[\leadsto wj - \frac{1}{wj + 1} \cdot \color{blue}{\left(wj - \frac{x}{e^{wj}}\right)}\]
- Using strategy
rm Applied sub-neg13.0
\[\leadsto wj - \frac{1}{wj + 1} \cdot \color{blue}{\left(wj + \left(-\frac{x}{e^{wj}}\right)\right)}\]
Applied distribute-lft-in13.0
\[\leadsto wj - \color{blue}{\left(\frac{1}{wj + 1} \cdot wj + \frac{1}{wj + 1} \cdot \left(-\frac{x}{e^{wj}}\right)\right)}\]
Applied associate--r+6.7
\[\leadsto \color{blue}{\left(wj - \frac{1}{wj + 1} \cdot wj\right) - \frac{1}{wj + 1} \cdot \left(-\frac{x}{e^{wj}}\right)}\]
Simplified6.7
\[\leadsto \left(wj - \frac{1}{wj + 1} \cdot wj\right) - \color{blue}{\frac{-x}{(\left(e^{wj}\right) \cdot wj + \left(e^{wj}\right))_*}}\]
Taylor expanded around 0 1.0
\[\leadsto \color{blue}{\left(\left({wj}^{2} + {wj}^{4}\right) - {wj}^{3}\right)} - \frac{-x}{(\left(e^{wj}\right) \cdot wj + \left(e^{wj}\right))_*}\]
Simplified1.0
\[\leadsto \color{blue}{(\left(1 - wj\right) \cdot \left(wj \cdot wj\right) + \left({wj}^{4}\right))_*} - \frac{-x}{(\left(e^{wj}\right) \cdot wj + \left(e^{wj}\right))_*}\]
Final simplification1.0
\[\leadsto (\left(1 - wj\right) \cdot \left(wj \cdot wj\right) + \left({wj}^{4}\right))_* - \frac{-x}{(\left(e^{wj}\right) \cdot wj + \left(e^{wj}\right))_*}\]
