Initial program 3.8
\[\frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c_n}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c_p} \cdot {\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c_n}}\]
Initial simplification3.8
\[\leadsto \frac{{\left(1 - \frac{1}{e^{-s} + 1}\right)}^{c_n}}{{\left(1 - \frac{1}{e^{-t} + 1}\right)}^{c_n}} \cdot \frac{{\left(\frac{1}{e^{-s} + 1}\right)}^{c_p}}{{\left(\frac{1}{e^{-t} + 1}\right)}^{c_p}}\]
- Using strategy
rm Applied add-exp-log3.8
\[\leadsto \frac{{\left(1 - \frac{1}{e^{-s} + 1}\right)}^{c_n}}{{\left(1 - \frac{1}{e^{-t} + 1}\right)}^{c_n}} \cdot \frac{{\left(\frac{1}{e^{-s} + 1}\right)}^{c_p}}{\color{blue}{e^{\log \left({\left(\frac{1}{e^{-t} + 1}\right)}^{c_p}\right)}}}\]
Applied pow-to-exp3.8
\[\leadsto \frac{{\left(1 - \frac{1}{e^{-s} + 1}\right)}^{c_n}}{{\left(1 - \frac{1}{e^{-t} + 1}\right)}^{c_n}} \cdot \frac{\color{blue}{e^{\log \left(\frac{1}{e^{-s} + 1}\right) \cdot c_p}}}{e^{\log \left({\left(\frac{1}{e^{-t} + 1}\right)}^{c_p}\right)}}\]
Applied div-exp3.8
\[\leadsto \frac{{\left(1 - \frac{1}{e^{-s} + 1}\right)}^{c_n}}{{\left(1 - \frac{1}{e^{-t} + 1}\right)}^{c_n}} \cdot \color{blue}{e^{\log \left(\frac{1}{e^{-s} + 1}\right) \cdot c_p - \log \left({\left(\frac{1}{e^{-t} + 1}\right)}^{c_p}\right)}}\]
Applied add-exp-log3.8
\[\leadsto \frac{{\left(1 - \frac{1}{e^{-s} + 1}\right)}^{c_n}}{{\color{blue}{\left(e^{\log \left(1 - \frac{1}{e^{-t} + 1}\right)}\right)}}^{c_n}} \cdot e^{\log \left(\frac{1}{e^{-s} + 1}\right) \cdot c_p - \log \left({\left(\frac{1}{e^{-t} + 1}\right)}^{c_p}\right)}\]
Applied pow-exp3.8
\[\leadsto \frac{{\left(1 - \frac{1}{e^{-s} + 1}\right)}^{c_n}}{\color{blue}{e^{\log \left(1 - \frac{1}{e^{-t} + 1}\right) \cdot c_n}}} \cdot e^{\log \left(\frac{1}{e^{-s} + 1}\right) \cdot c_p - \log \left({\left(\frac{1}{e^{-t} + 1}\right)}^{c_p}\right)}\]
Applied add-exp-log3.8
\[\leadsto \frac{\color{blue}{e^{\log \left({\left(1 - \frac{1}{e^{-s} + 1}\right)}^{c_n}\right)}}}{e^{\log \left(1 - \frac{1}{e^{-t} + 1}\right) \cdot c_n}} \cdot e^{\log \left(\frac{1}{e^{-s} + 1}\right) \cdot c_p - \log \left({\left(\frac{1}{e^{-t} + 1}\right)}^{c_p}\right)}\]
Applied div-exp3.6
\[\leadsto \color{blue}{e^{\log \left({\left(1 - \frac{1}{e^{-s} + 1}\right)}^{c_n}\right) - \log \left(1 - \frac{1}{e^{-t} + 1}\right) \cdot c_n}} \cdot e^{\log \left(\frac{1}{e^{-s} + 1}\right) \cdot c_p - \log \left({\left(\frac{1}{e^{-t} + 1}\right)}^{c_p}\right)}\]
Applied prod-exp3.3
\[\leadsto \color{blue}{e^{\left(\log \left({\left(1 - \frac{1}{e^{-s} + 1}\right)}^{c_n}\right) - \log \left(1 - \frac{1}{e^{-t} + 1}\right) \cdot c_n\right) + \left(\log \left(\frac{1}{e^{-s} + 1}\right) \cdot c_p - \log \left({\left(\frac{1}{e^{-t} + 1}\right)}^{c_p}\right)\right)}}\]
Simplified1.7
\[\leadsto e^{\color{blue}{c_n \cdot \left(\log \left(1 - \frac{1}{e^{-s} + 1}\right) - \log \left(1 - \frac{1}{e^{-t} + 1}\right)\right) + \left(\log \left(e^{-t} + 1\right) - \log \left(e^{-s} + 1\right)\right) \cdot c_p}}\]
Taylor expanded around 0 0.6
\[\leadsto e^{c_n \cdot \left(\log \left(1 - \frac{1}{e^{-s} + 1}\right) - \log \left(1 - \frac{1}{e^{-t} + 1}\right)\right) + \left(\color{blue}{\left(\left(\log 2 + \frac{1}{8} \cdot {t}^{2}\right) - \frac{1}{2} \cdot t\right)} - \log \left(e^{-s} + 1\right)\right) \cdot c_p}\]
Final simplification0.6
\[\leadsto e^{c_p \cdot \left(\left(\left(\frac{1}{8} \cdot {t}^{2} + \log 2\right) - t \cdot \frac{1}{2}\right) - \log \left(1 + e^{-s}\right)\right) + c_n \cdot \left(\log \left(1 - \frac{1}{1 + e^{-s}}\right) - \log \left(1 - \frac{1}{e^{-t} + 1}\right)\right)}\]
