Initial program 44.8
\[(x \cdot y + z)_* - \left(1 + \left(x \cdot y + z\right)\right)\]
- Using strategy
rm Applied add-log-exp45.9
\[\leadsto (x \cdot y + z)_* - \left(1 + \left(x \cdot y + \color{blue}{\log \left(e^{z}\right)}\right)\right)\]
Applied add-log-exp46.8
\[\leadsto (x \cdot y + z)_* - \left(1 + \left(\color{blue}{\log \left(e^{x \cdot y}\right)} + \log \left(e^{z}\right)\right)\right)\]
Applied sum-log46.8
\[\leadsto (x \cdot y + z)_* - \left(1 + \color{blue}{\log \left(e^{x \cdot y} \cdot e^{z}\right)}\right)\]
Applied add-log-exp46.8
\[\leadsto (x \cdot y + z)_* - \left(\color{blue}{\log \left(e^{1}\right)} + \log \left(e^{x \cdot y} \cdot e^{z}\right)\right)\]
Applied sum-log46.8
\[\leadsto (x \cdot y + z)_* - \color{blue}{\log \left(e^{1} \cdot \left(e^{x \cdot y} \cdot e^{z}\right)\right)}\]
Applied add-log-exp47.2
\[\leadsto \color{blue}{\log \left(e^{(x \cdot y + z)_*}\right)} - \log \left(e^{1} \cdot \left(e^{x \cdot y} \cdot e^{z}\right)\right)\]
Applied diff-log47.2
\[\leadsto \color{blue}{\log \left(\frac{e^{(x \cdot y + z)_*}}{e^{1} \cdot \left(e^{x \cdot y} \cdot e^{z}\right)}\right)}\]
Simplified13.1
\[\leadsto \log \color{blue}{\left(\frac{e^{\left((x \cdot y + z)_* - x \cdot y\right) - z}}{e}\right)}\]
- Using strategy
rm Applied log-div13.1
\[\leadsto \color{blue}{\log \left(e^{\left((x \cdot y + z)_* - x \cdot y\right) - z}\right) - \log e}\]
Simplified8.1
\[\leadsto \color{blue}{\left((x \cdot y + z)_* - \left(x \cdot y + z\right)\right)} - \log e\]
Final simplification8.1
\[\leadsto \left((x \cdot y + z)_* - \left(z + x \cdot y\right)\right) - 1\]
