Initial program 3.7
\[\frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c_n}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c_p} \cdot {\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c_n}}\]
Initial simplification3.7
\[\leadsto \frac{{\left(1 - \frac{1}{e^{-s} + 1}\right)}^{c_n}}{{\left(1 - \frac{1}{e^{-t} + 1}\right)}^{c_n}} \cdot \frac{{\left(\frac{1}{e^{-s} + 1}\right)}^{c_p}}{{\left(\frac{1}{e^{-t} + 1}\right)}^{c_p}}\]
- Using strategy
rm Applied pow-to-exp3.7
\[\leadsto \frac{{\left(1 - \frac{1}{e^{-s} + 1}\right)}^{c_n}}{{\left(1 - \frac{1}{e^{-t} + 1}\right)}^{c_n}} \cdot \frac{{\left(\frac{1}{e^{-s} + 1}\right)}^{c_p}}{\color{blue}{e^{\log \left(\frac{1}{e^{-t} + 1}\right) \cdot c_p}}}\]
Applied add-exp-log3.7
\[\leadsto \frac{{\left(1 - \frac{1}{e^{-s} + 1}\right)}^{c_n}}{{\left(1 - \frac{1}{e^{-t} + 1}\right)}^{c_n}} \cdot \frac{{\left(\frac{1}{\color{blue}{e^{\log \left(e^{-s} + 1\right)}}}\right)}^{c_p}}{e^{\log \left(\frac{1}{e^{-t} + 1}\right) \cdot c_p}}\]
Applied rec-exp3.7
\[\leadsto \frac{{\left(1 - \frac{1}{e^{-s} + 1}\right)}^{c_n}}{{\left(1 - \frac{1}{e^{-t} + 1}\right)}^{c_n}} \cdot \frac{{\color{blue}{\left(e^{-\log \left(e^{-s} + 1\right)}\right)}}^{c_p}}{e^{\log \left(\frac{1}{e^{-t} + 1}\right) \cdot c_p}}\]
Applied pow-exp3.7
\[\leadsto \frac{{\left(1 - \frac{1}{e^{-s} + 1}\right)}^{c_n}}{{\left(1 - \frac{1}{e^{-t} + 1}\right)}^{c_n}} \cdot \frac{\color{blue}{e^{\left(-\log \left(e^{-s} + 1\right)\right) \cdot c_p}}}{e^{\log \left(\frac{1}{e^{-t} + 1}\right) \cdot c_p}}\]
Applied div-exp3.0
\[\leadsto \frac{{\left(1 - \frac{1}{e^{-s} + 1}\right)}^{c_n}}{{\left(1 - \frac{1}{e^{-t} + 1}\right)}^{c_n}} \cdot \color{blue}{e^{\left(-\log \left(e^{-s} + 1\right)\right) \cdot c_p - \log \left(\frac{1}{e^{-t} + 1}\right) \cdot c_p}}\]
Applied pow-to-exp3.0
\[\leadsto \frac{{\left(1 - \frac{1}{e^{-s} + 1}\right)}^{c_n}}{\color{blue}{e^{\log \left(1 - \frac{1}{e^{-t} + 1}\right) \cdot c_n}}} \cdot e^{\left(-\log \left(e^{-s} + 1\right)\right) \cdot c_p - \log \left(\frac{1}{e^{-t} + 1}\right) \cdot c_p}\]
Applied add-exp-log3.0
\[\leadsto \frac{{\color{blue}{\left(e^{\log \left(1 - \frac{1}{e^{-s} + 1}\right)}\right)}}^{c_n}}{e^{\log \left(1 - \frac{1}{e^{-t} + 1}\right) \cdot c_n}} \cdot e^{\left(-\log \left(e^{-s} + 1\right)\right) \cdot c_p - \log \left(\frac{1}{e^{-t} + 1}\right) \cdot c_p}\]
Applied pow-exp3.0
\[\leadsto \frac{\color{blue}{e^{\log \left(1 - \frac{1}{e^{-s} + 1}\right) \cdot c_n}}}{e^{\log \left(1 - \frac{1}{e^{-t} + 1}\right) \cdot c_n}} \cdot e^{\left(-\log \left(e^{-s} + 1\right)\right) \cdot c_p - \log \left(\frac{1}{e^{-t} + 1}\right) \cdot c_p}\]
Applied div-exp2.2
\[\leadsto \color{blue}{e^{\log \left(1 - \frac{1}{e^{-s} + 1}\right) \cdot c_n - \log \left(1 - \frac{1}{e^{-t} + 1}\right) \cdot c_n}} \cdot e^{\left(-\log \left(e^{-s} + 1\right)\right) \cdot c_p - \log \left(\frac{1}{e^{-t} + 1}\right) \cdot c_p}\]
Applied prod-exp1.5
\[\leadsto \color{blue}{e^{\left(\log \left(1 - \frac{1}{e^{-s} + 1}\right) \cdot c_n - \log \left(1 - \frac{1}{e^{-t} + 1}\right) \cdot c_n\right) + \left(\left(-\log \left(e^{-s} + 1\right)\right) \cdot c_p - \log \left(\frac{1}{e^{-t} + 1}\right) \cdot c_p\right)}}\]
Simplified1.5
\[\leadsto e^{\color{blue}{c_n \cdot \left(\log \left(1 - \frac{1}{e^{-s} + 1}\right) - \log \left(1 - \frac{1}{e^{-t} + 1}\right)\right) - c_p \cdot \left(\log \left(e^{-s} + 1\right) - \log \left(e^{-t} + 1\right)\right)}}\]
Taylor expanded around 0 0.5
\[\leadsto e^{c_n \cdot \left(\log \left(1 - \frac{1}{e^{-s} + 1}\right) - \log \left(1 - \frac{1}{e^{-t} + 1}\right)\right) - c_p \cdot \left(\log \left(e^{-s} + 1\right) - \color{blue}{\left(\left(\log 2 + \frac{1}{8} \cdot {t}^{2}\right) - \frac{1}{2} \cdot t\right)}\right)}\]
Final simplification0.5
\[\leadsto e^{c_n \cdot \left(\log \left(1 - \frac{1}{1 + e^{-s}}\right) - \log \left(1 - \frac{1}{e^{-t} + 1}\right)\right) - \left(\log \left(1 + e^{-s}\right) - \left(\left(\log 2 + {t}^{2} \cdot \frac{1}{8}\right) - t \cdot \frac{1}{2}\right)\right) \cdot c_p}\]
