Initial program 45.3
\[(x \cdot y + z)_* - \left(1 + \left(x \cdot y + z\right)\right)\]
Initial simplification45.3
\[\leadsto (x \cdot y + z)_* + \left(\left(-1 - z\right) - x \cdot y\right)\]
- Using strategy
rm Applied add-log-exp46.2
\[\leadsto (x \cdot y + z)_* + \left(\left(-1 - z\right) - \color{blue}{\log \left(e^{x \cdot y}\right)}\right)\]
Applied add-log-exp47.1
\[\leadsto (x \cdot y + z)_* + \left(\color{blue}{\log \left(e^{-1 - z}\right)} - \log \left(e^{x \cdot y}\right)\right)\]
Applied diff-log47.1
\[\leadsto (x \cdot y + z)_* + \color{blue}{\log \left(\frac{e^{-1 - z}}{e^{x \cdot y}}\right)}\]
Applied add-log-exp47.5
\[\leadsto \color{blue}{\log \left(e^{(x \cdot y + z)_*}\right)} + \log \left(\frac{e^{-1 - z}}{e^{x \cdot y}}\right)\]
Applied sum-log47.5
\[\leadsto \color{blue}{\log \left(e^{(x \cdot y + z)_*} \cdot \frac{e^{-1 - z}}{e^{x \cdot y}}\right)}\]
Simplified31.0
\[\leadsto \log \color{blue}{\left(e^{\left(-1 - x \cdot y\right) - \left(z - (x \cdot y + z)_*\right)}\right)}\]
- Using strategy
rm Applied associate--l-14.9
\[\leadsto \log \left(e^{\color{blue}{-1 - \left(x \cdot y + \left(z - (x \cdot y + z)_*\right)\right)}}\right)\]
- Using strategy
rm Applied associate-+r-7.8
\[\leadsto \log \left(e^{-1 - \color{blue}{\left(\left(x \cdot y + z\right) - (x \cdot y + z)_*\right)}}\right)\]
Final simplification7.8
\[\leadsto \log \left(e^{-1 - \left(\left(y \cdot x + z\right) - (x \cdot y + z)_*\right)}\right)\]
