- Split input into 2 regimes
if wj < 6.35250989482101e-09
Initial program 14.1
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
Initial simplification14.1
\[\leadsto \frac{(\left(-wj\right) \cdot \left(e^{wj}\right) + x)_*}{(\left(e^{wj}\right) \cdot wj + \left(e^{wj}\right))_*} + wj\]
Taylor expanded around 0 0.8
\[\leadsto \color{blue}{\left({wj}^{2} + x\right) - 2 \cdot \left(x \cdot wj\right)}\]
Simplified0.8
\[\leadsto \color{blue}{(wj \cdot \left((x \cdot -2 + wj)_*\right) + x)_*}\]
if 6.35250989482101e-09 < wj
Initial program 23.5
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
Initial simplification23.5
\[\leadsto \frac{(\left(-wj\right) \cdot \left(e^{wj}\right) + x)_*}{(\left(e^{wj}\right) \cdot wj + \left(e^{wj}\right))_*} + wj\]
Taylor expanded around inf 23.5
\[\leadsto \frac{\color{blue}{x - e^{wj} \cdot wj}}{(\left(e^{wj}\right) \cdot wj + \left(e^{wj}\right))_*} + wj\]
- Using strategy
rm Applied add-sqr-sqrt23.7
\[\leadsto \frac{x - e^{wj} \cdot wj}{\color{blue}{\sqrt{(\left(e^{wj}\right) \cdot wj + \left(e^{wj}\right))_*} \cdot \sqrt{(\left(e^{wj}\right) \cdot wj + \left(e^{wj}\right))_*}}} + wj\]
Applied associate-/r*23.7
\[\leadsto \color{blue}{\frac{\frac{x - e^{wj} \cdot wj}{\sqrt{(\left(e^{wj}\right) \cdot wj + \left(e^{wj}\right))_*}}}{\sqrt{(\left(e^{wj}\right) \cdot wj + \left(e^{wj}\right))_*}}} + wj\]
- Using strategy
rm Applied *-un-lft-identity23.7
\[\leadsto \frac{\frac{x - e^{wj} \cdot wj}{\sqrt{(\left(e^{wj}\right) \cdot wj + \left(e^{wj}\right))_*}}}{\sqrt{\color{blue}{1 \cdot (\left(e^{wj}\right) \cdot wj + \left(e^{wj}\right))_*}}} + wj\]
Applied sqrt-prod23.7
\[\leadsto \frac{\frac{x - e^{wj} \cdot wj}{\sqrt{(\left(e^{wj}\right) \cdot wj + \left(e^{wj}\right))_*}}}{\color{blue}{\sqrt{1} \cdot \sqrt{(\left(e^{wj}\right) \cdot wj + \left(e^{wj}\right))_*}}} + wj\]
Applied div-inv23.7
\[\leadsto \frac{\color{blue}{\left(x - e^{wj} \cdot wj\right) \cdot \frac{1}{\sqrt{(\left(e^{wj}\right) \cdot wj + \left(e^{wj}\right))_*}}}}{\sqrt{1} \cdot \sqrt{(\left(e^{wj}\right) \cdot wj + \left(e^{wj}\right))_*}} + wj\]
Applied times-frac23.7
\[\leadsto \color{blue}{\frac{x - e^{wj} \cdot wj}{\sqrt{1}} \cdot \frac{\frac{1}{\sqrt{(\left(e^{wj}\right) \cdot wj + \left(e^{wj}\right))_*}}}{\sqrt{(\left(e^{wj}\right) \cdot wj + \left(e^{wj}\right))_*}}} + wj\]
Applied fma-def23.6
\[\leadsto \color{blue}{(\left(\frac{x - e^{wj} \cdot wj}{\sqrt{1}}\right) \cdot \left(\frac{\frac{1}{\sqrt{(\left(e^{wj}\right) \cdot wj + \left(e^{wj}\right))_*}}}{\sqrt{(\left(e^{wj}\right) \cdot wj + \left(e^{wj}\right))_*}}\right) + wj)_*}\]
Simplified23.6
\[\leadsto (\color{blue}{\left((\left(e^{wj}\right) \cdot \left(-wj\right) + x)_*\right)} \cdot \left(\frac{\frac{1}{\sqrt{(\left(e^{wj}\right) \cdot wj + \left(e^{wj}\right))_*}}}{\sqrt{(\left(e^{wj}\right) \cdot wj + \left(e^{wj}\right))_*}}\right) + wj)_*\]
- Recombined 2 regimes into one program.
Final simplification1.5
\[\leadsto \begin{array}{l}
\mathbf{if}\;wj \le 6.35250989482101 \cdot 10^{-09}:\\
\;\;\;\;(wj \cdot \left((x \cdot -2 + wj)_*\right) + x)_*\\
\mathbf{else}:\\
\;\;\;\;(\left((\left(e^{wj}\right) \cdot \left(-wj\right) + x)_*\right) \cdot \left(\frac{\frac{1}{\sqrt{(\left(e^{wj}\right) \cdot wj + \left(e^{wj}\right))_*}}}{\sqrt{(\left(e^{wj}\right) \cdot wj + \left(e^{wj}\right))_*}}\right) + wj)_*\\
\end{array}\]
