Initial program 3.9
\[\frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c_n}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c_p} \cdot {\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c_n}}\]
- Using strategy
rm Applied add-exp-log3.9
\[\leadsto \frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c_n}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c_p} \cdot {\color{blue}{\left(e^{\log \left(1 - \frac{1}{1 + e^{-t}}\right)}\right)}}^{c_n}}\]
Applied pow-exp3.9
\[\leadsto \frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c_n}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c_p} \cdot \color{blue}{e^{\log \left(1 - \frac{1}{1 + e^{-t}}\right) \cdot c_n}}}\]
Applied add-exp-log3.9
\[\leadsto \frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c_n}}{\color{blue}{e^{\log \left({\left(\frac{1}{1 + e^{-t}}\right)}^{c_p}\right)}} \cdot e^{\log \left(1 - \frac{1}{1 + e^{-t}}\right) \cdot c_n}}\]
Applied prod-exp3.9
\[\leadsto \frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c_n}}{\color{blue}{e^{\log \left({\left(\frac{1}{1 + e^{-t}}\right)}^{c_p}\right) + \log \left(1 - \frac{1}{1 + e^{-t}}\right) \cdot c_n}}}\]
Applied pow-to-exp3.9
\[\leadsto \frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c_p} \cdot \color{blue}{e^{\log \left(1 - \frac{1}{1 + e^{-s}}\right) \cdot c_n}}}{e^{\log \left({\left(\frac{1}{1 + e^{-t}}\right)}^{c_p}\right) + \log \left(1 - \frac{1}{1 + e^{-t}}\right) \cdot c_n}}\]
Applied pow-to-exp3.9
\[\leadsto \frac{\color{blue}{e^{\log \left(\frac{1}{1 + e^{-s}}\right) \cdot c_p}} \cdot e^{\log \left(1 - \frac{1}{1 + e^{-s}}\right) \cdot c_n}}{e^{\log \left({\left(\frac{1}{1 + e^{-t}}\right)}^{c_p}\right) + \log \left(1 - \frac{1}{1 + e^{-t}}\right) \cdot c_n}}\]
Applied prod-exp3.9
\[\leadsto \frac{\color{blue}{e^{\log \left(\frac{1}{1 + e^{-s}}\right) \cdot c_p + \log \left(1 - \frac{1}{1 + e^{-s}}\right) \cdot c_n}}}{e^{\log \left({\left(\frac{1}{1 + e^{-t}}\right)}^{c_p}\right) + \log \left(1 - \frac{1}{1 + e^{-t}}\right) \cdot c_n}}\]
Applied div-exp2.7
\[\leadsto \color{blue}{e^{\left(\log \left(\frac{1}{1 + e^{-s}}\right) \cdot c_p + \log \left(1 - \frac{1}{1 + e^{-s}}\right) \cdot c_n\right) - \left(\log \left({\left(\frac{1}{1 + e^{-t}}\right)}^{c_p}\right) + \log \left(1 - \frac{1}{1 + e^{-t}}\right) \cdot c_n\right)}}\]
Simplified1.5
\[\leadsto e^{\color{blue}{c_p \cdot \left(\log \left(e^{-t} + 1\right) - \log \left(e^{-s} + 1\right)\right) + c_n \cdot \left(\log \left(1 - \frac{1}{e^{-s} + 1}\right) - \log \left(1 - \frac{1}{e^{-t} + 1}\right)\right)}}\]
Taylor expanded around 0 0.5
\[\leadsto e^{c_p \cdot \left(\color{blue}{\left(\left(\log 2 + \frac{1}{8} \cdot {t}^{2}\right) - \frac{1}{2} \cdot t\right)} - \log \left(e^{-s} + 1\right)\right) + c_n \cdot \left(\log \left(1 - \frac{1}{e^{-s} + 1}\right) - \log \left(1 - \frac{1}{e^{-t} + 1}\right)\right)}\]
- Using strategy
rm Applied add-sqr-sqrt0.7
\[\leadsto e^{c_p \cdot \left(\left(\left(\log 2 + \frac{1}{8} \cdot {t}^{2}\right) - \frac{1}{2} \cdot t\right) - \log \color{blue}{\left(\sqrt{e^{-s} + 1} \cdot \sqrt{e^{-s} + 1}\right)}\right) + c_n \cdot \left(\log \left(1 - \frac{1}{e^{-s} + 1}\right) - \log \left(1 - \frac{1}{e^{-t} + 1}\right)\right)}\]
Applied log-prod0.7
\[\leadsto e^{c_p \cdot \left(\left(\left(\log 2 + \frac{1}{8} \cdot {t}^{2}\right) - \frac{1}{2} \cdot t\right) - \color{blue}{\left(\log \left(\sqrt{e^{-s} + 1}\right) + \log \left(\sqrt{e^{-s} + 1}\right)\right)}\right) + c_n \cdot \left(\log \left(1 - \frac{1}{e^{-s} + 1}\right) - \log \left(1 - \frac{1}{e^{-t} + 1}\right)\right)}\]
Final simplification0.7
\[\leadsto e^{c_p \cdot \left(\left(\left(\log 2 + \frac{1}{8} \cdot {t}^{2}\right) - t \cdot \frac{1}{2}\right) - \left(\log \left(\sqrt{e^{-s} + 1}\right) + \log \left(\sqrt{e^{-s} + 1}\right)\right)\right) + c_n \cdot \left(\log \left(1 - \frac{1}{e^{-s} + 1}\right) - \log \left(1 - \frac{1}{e^{-t} + 1}\right)\right)}\]
