Initial program 0.4
\[\log \left(1 + e^{x}\right) - x \cdot y\]
- Using strategy
rm Applied flip3-+0.5
\[\leadsto \log \color{blue}{\left(\frac{{1}^{3} + {\left(e^{x}\right)}^{3}}{1 \cdot 1 + \left(e^{x} \cdot e^{x} - 1 \cdot e^{x}\right)}\right)} - x \cdot y\]
- Using strategy
rm Applied *-un-lft-identity0.5
\[\leadsto \log \left(\frac{{1}^{3} + {\left(e^{x}\right)}^{3}}{1 \cdot 1 + \left(\color{blue}{\left(1 \cdot e^{x}\right)} \cdot e^{x} - 1 \cdot e^{x}\right)}\right) - x \cdot y\]
Applied associate-*l*0.5
\[\leadsto \log \left(\frac{{1}^{3} + {\left(e^{x}\right)}^{3}}{1 \cdot 1 + \left(\color{blue}{1 \cdot \left(e^{x} \cdot e^{x}\right)} - 1 \cdot e^{x}\right)}\right) - x \cdot y\]
Applied distribute-lft-out--0.5
\[\leadsto \log \left(\frac{{1}^{3} + {\left(e^{x}\right)}^{3}}{1 \cdot 1 + \color{blue}{1 \cdot \left(e^{x} \cdot e^{x} - e^{x}\right)}}\right) - x \cdot y\]
Applied distribute-lft-out0.5
\[\leadsto \log \left(\frac{{1}^{3} + {\left(e^{x}\right)}^{3}}{\color{blue}{1 \cdot \left(1 + \left(e^{x} \cdot e^{x} - e^{x}\right)\right)}}\right) - x \cdot y\]
Applied add-sqr-sqrt1.3
\[\leadsto \log \left(\frac{\color{blue}{\sqrt{{1}^{3} + {\left(e^{x}\right)}^{3}} \cdot \sqrt{{1}^{3} + {\left(e^{x}\right)}^{3}}}}{1 \cdot \left(1 + \left(e^{x} \cdot e^{x} - e^{x}\right)\right)}\right) - x \cdot y\]
Applied times-frac1.3
\[\leadsto \log \color{blue}{\left(\frac{\sqrt{{1}^{3} + {\left(e^{x}\right)}^{3}}}{1} \cdot \frac{\sqrt{{1}^{3} + {\left(e^{x}\right)}^{3}}}{1 + \left(e^{x} \cdot e^{x} - e^{x}\right)}\right)} - x \cdot y\]
Applied log-prod1.0
\[\leadsto \color{blue}{\left(\log \left(\frac{\sqrt{{1}^{3} + {\left(e^{x}\right)}^{3}}}{1}\right) + \log \left(\frac{\sqrt{{1}^{3} + {\left(e^{x}\right)}^{3}}}{1 + \left(e^{x} \cdot e^{x} - e^{x}\right)}\right)\right)} - x \cdot y\]
- Using strategy
rm Applied *-un-lft-identity1.0
\[\leadsto \left(\log \left(\frac{\sqrt{{1}^{3} + {\left(e^{x}\right)}^{3}}}{1}\right) + \log \left(\frac{\sqrt{{1}^{3} + {\left(e^{x}\right)}^{3}}}{1 + \color{blue}{1 \cdot \left(e^{x} \cdot e^{x} - e^{x}\right)}}\right)\right) - x \cdot y\]
Applied *-un-lft-identity1.0
\[\leadsto \left(\log \left(\frac{\sqrt{{1}^{3} + {\left(e^{x}\right)}^{3}}}{1}\right) + \log \left(\frac{\sqrt{{1}^{3} + {\left(e^{x}\right)}^{3}}}{\color{blue}{1 \cdot 1} + 1 \cdot \left(e^{x} \cdot e^{x} - e^{x}\right)}\right)\right) - x \cdot y\]
Applied distribute-lft-out1.0
\[\leadsto \left(\log \left(\frac{\sqrt{{1}^{3} + {\left(e^{x}\right)}^{3}}}{1}\right) + \log \left(\frac{\sqrt{{1}^{3} + {\left(e^{x}\right)}^{3}}}{\color{blue}{1 \cdot \left(1 + \left(e^{x} \cdot e^{x} - e^{x}\right)\right)}}\right)\right) - x \cdot y\]
Applied add-sqr-sqrt0.5
\[\leadsto \left(\log \left(\frac{\sqrt{{1}^{3} + {\left(e^{x}\right)}^{3}}}{1}\right) + \log \left(\frac{\color{blue}{\sqrt{\sqrt{{1}^{3} + {\left(e^{x}\right)}^{3}}} \cdot \sqrt{\sqrt{{1}^{3} + {\left(e^{x}\right)}^{3}}}}}{1 \cdot \left(1 + \left(e^{x} \cdot e^{x} - e^{x}\right)\right)}\right)\right) - x \cdot y\]
Applied times-frac0.5
\[\leadsto \left(\log \left(\frac{\sqrt{{1}^{3} + {\left(e^{x}\right)}^{3}}}{1}\right) + \log \color{blue}{\left(\frac{\sqrt{\sqrt{{1}^{3} + {\left(e^{x}\right)}^{3}}}}{1} \cdot \frac{\sqrt{\sqrt{{1}^{3} + {\left(e^{x}\right)}^{3}}}}{1 + \left(e^{x} \cdot e^{x} - e^{x}\right)}\right)}\right) - x \cdot y\]
Applied log-prod0.5
\[\leadsto \left(\log \left(\frac{\sqrt{{1}^{3} + {\left(e^{x}\right)}^{3}}}{1}\right) + \color{blue}{\left(\log \left(\frac{\sqrt{\sqrt{{1}^{3} + {\left(e^{x}\right)}^{3}}}}{1}\right) + \log \left(\frac{\sqrt{\sqrt{{1}^{3} + {\left(e^{x}\right)}^{3}}}}{1 + \left(e^{x} \cdot e^{x} - e^{x}\right)}\right)\right)}\right) - x \cdot y\]
Final simplification0.5
\[\leadsto \left(\log \left(\sqrt{{\left(e^{x}\right)}^{3} + 1}\right) + \left(\log \left(\sqrt{\sqrt{{\left(e^{x}\right)}^{3} + 1}}\right) + \log \left(\frac{\sqrt{\sqrt{{\left(e^{x}\right)}^{3} + 1}}}{\left(e^{x} \cdot e^{x} - e^{x}\right) + 1}\right)\right)\right) - y \cdot x\]
