Initial program 13.7
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
Initial simplification6.7
\[\leadsto \left(wj - \frac{wj}{wj + 1}\right) + \frac{\frac{x}{e^{wj}}}{wj + 1}\]
Taylor expanded around 0 1.2
\[\leadsto \color{blue}{\left(\left({wj}^{2} + {wj}^{4}\right) - {wj}^{3}\right)} + \frac{\frac{x}{e^{wj}}}{wj + 1}\]
- Using strategy
rm Applied *-un-lft-identity1.2
\[\leadsto \left(\left({wj}^{2} + {wj}^{4}\right) - {wj}^{3}\right) + \frac{\frac{x}{e^{wj}}}{\color{blue}{1 \cdot \left(wj + 1\right)}}\]
Applied div-inv1.2
\[\leadsto \left(\left({wj}^{2} + {wj}^{4}\right) - {wj}^{3}\right) + \frac{\color{blue}{x \cdot \frac{1}{e^{wj}}}}{1 \cdot \left(wj + 1\right)}\]
Applied times-frac1.2
\[\leadsto \left(\left({wj}^{2} + {wj}^{4}\right) - {wj}^{3}\right) + \color{blue}{\frac{x}{1} \cdot \frac{\frac{1}{e^{wj}}}{wj + 1}}\]
Simplified1.2
\[\leadsto \left(\left({wj}^{2} + {wj}^{4}\right) - {wj}^{3}\right) + \color{blue}{x} \cdot \frac{\frac{1}{e^{wj}}}{wj + 1}\]
- Using strategy
rm Applied flip3-+1.2
\[\leadsto \left(\left({wj}^{2} + {wj}^{4}\right) - {wj}^{3}\right) + x \cdot \frac{\frac{1}{e^{wj}}}{\color{blue}{\frac{{wj}^{3} + {1}^{3}}{wj \cdot wj + \left(1 \cdot 1 - wj \cdot 1\right)}}}\]
Applied associate-/r/1.2
\[\leadsto \left(\left({wj}^{2} + {wj}^{4}\right) - {wj}^{3}\right) + x \cdot \color{blue}{\left(\frac{\frac{1}{e^{wj}}}{{wj}^{3} + {1}^{3}} \cdot \left(wj \cdot wj + \left(1 \cdot 1 - wj \cdot 1\right)\right)\right)}\]
Simplified1.2
\[\leadsto \left(\left({wj}^{2} + {wj}^{4}\right) - {wj}^{3}\right) + x \cdot \left(\frac{\frac{1}{e^{wj}}}{{wj}^{3} + {1}^{3}} \cdot \color{blue}{\left(\left(1 - wj\right) + wj \cdot wj\right)}\right)\]
Final simplification1.2
\[\leadsto \left(\left(wj \cdot wj + \left(1 - wj\right)\right) \cdot \frac{\frac{1}{e^{wj}}}{1 + {wj}^{3}}\right) \cdot x + \left(\left({wj}^{2} + {wj}^{4}\right) - {wj}^{3}\right)\]
