Initial program 4.1
\[\frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c_n}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c_p} \cdot {\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c_n}}\]
- Using strategy
rm Applied pow-to-exp4.1
\[\leadsto \frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c_n}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c_p} \cdot \color{blue}{e^{\log \left(1 - \frac{1}{1 + e^{-t}}\right) \cdot c_n}}}\]
Applied add-exp-log4.1
\[\leadsto \frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c_n}}{{\left(\frac{1}{\color{blue}{e^{\log \left(1 + e^{-t}\right)}}}\right)}^{c_p} \cdot e^{\log \left(1 - \frac{1}{1 + e^{-t}}\right) \cdot c_n}}\]
Applied rec-exp4.1
\[\leadsto \frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c_n}}{{\color{blue}{\left(e^{-\log \left(1 + e^{-t}\right)}\right)}}^{c_p} \cdot e^{\log \left(1 - \frac{1}{1 + e^{-t}}\right) \cdot c_n}}\]
Applied pow-exp4.1
\[\leadsto \frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c_n}}{\color{blue}{e^{\left(-\log \left(1 + e^{-t}\right)\right) \cdot c_p}} \cdot e^{\log \left(1 - \frac{1}{1 + e^{-t}}\right) \cdot c_n}}\]
Applied prod-exp4.1
\[\leadsto \frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c_n}}{\color{blue}{e^{\left(-\log \left(1 + e^{-t}\right)\right) \cdot c_p + \log \left(1 - \frac{1}{1 + e^{-t}}\right) \cdot c_n}}}\]
Applied add-exp-log4.1
\[\leadsto \frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c_p} \cdot \color{blue}{e^{\log \left({\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c_n}\right)}}}{e^{\left(-\log \left(1 + e^{-t}\right)\right) \cdot c_p + \log \left(1 - \frac{1}{1 + e^{-t}}\right) \cdot c_n}}\]
Applied add-exp-log4.1
\[\leadsto \frac{\color{blue}{e^{\log \left({\left(\frac{1}{1 + e^{-s}}\right)}^{c_p}\right)}} \cdot e^{\log \left({\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c_n}\right)}}{e^{\left(-\log \left(1 + e^{-t}\right)\right) \cdot c_p + \log \left(1 - \frac{1}{1 + e^{-t}}\right) \cdot c_n}}\]
Applied prod-exp4.1
\[\leadsto \frac{\color{blue}{e^{\log \left({\left(\frac{1}{1 + e^{-s}}\right)}^{c_p}\right) + \log \left({\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c_n}\right)}}}{e^{\left(-\log \left(1 + e^{-t}\right)\right) \cdot c_p + \log \left(1 - \frac{1}{1 + e^{-t}}\right) \cdot c_n}}\]
Applied div-exp2.9
\[\leadsto \color{blue}{e^{\left(\log \left({\left(\frac{1}{1 + e^{-s}}\right)}^{c_p}\right) + \log \left({\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c_n}\right)\right) - \left(\left(-\log \left(1 + e^{-t}\right)\right) \cdot c_p + \log \left(1 - \frac{1}{1 + e^{-t}}\right) \cdot c_n\right)}}\]
Simplified1.8
\[\leadsto e^{\color{blue}{\left(\log \left(1 - \frac{1}{e^{-s} + 1}\right) - \log \left(1 - \frac{1}{e^{-t} + 1}\right)\right) \cdot c_n - c_p \cdot \left(\log \left(e^{-s} + 1\right) - \log \left(e^{-t} + 1\right)\right)}}\]
Taylor expanded around 0 0.6
\[\leadsto e^{\left(\log \left(1 - \frac{1}{e^{-s} + 1}\right) - \log \left(1 - \frac{1}{e^{-t} + 1}\right)\right) \cdot c_n - c_p \cdot \left(\log \left(e^{-s} + 1\right) - \color{blue}{\left(\left(\log 2 + \frac{1}{8} \cdot {t}^{2}\right) - \frac{1}{2} \cdot t\right)}\right)}\]
Final simplification0.6
\[\leadsto e^{\left(\log \left(1 - \frac{1}{e^{-s} + 1}\right) - \log \left(1 - \frac{1}{1 + e^{-t}}\right)\right) \cdot c_n - \left(\log \left(e^{-s} + 1\right) - \left(\left(\log 2 + {t}^{2} \cdot \frac{1}{8}\right) - \frac{1}{2} \cdot t\right)\right) \cdot c_p}\]
