Initial program 4.1
\[\frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c_n}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c_p} \cdot {\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c_n}}\]
Initial simplification4.1
\[\leadsto \frac{{\left(1 - \frac{1}{e^{-s} + 1}\right)}^{c_n}}{{\left(1 - \frac{1}{e^{-t} + 1}\right)}^{c_n}} \cdot \frac{{\left(\frac{1}{e^{-s} + 1}\right)}^{c_p}}{{\left(\frac{1}{e^{-t} + 1}\right)}^{c_p}}\]
- Using strategy
rm Applied add-exp-log4.1
\[\leadsto \frac{{\left(1 - \frac{1}{e^{-s} + 1}\right)}^{c_n}}{{\left(1 - \frac{1}{e^{-t} + 1}\right)}^{c_n}} \cdot \frac{{\left(\frac{1}{e^{-s} + 1}\right)}^{c_p}}{\color{blue}{e^{\log \left({\left(\frac{1}{e^{-t} + 1}\right)}^{c_p}\right)}}}\]
Applied add-exp-log4.1
\[\leadsto \frac{{\left(1 - \frac{1}{e^{-s} + 1}\right)}^{c_n}}{{\left(1 - \frac{1}{e^{-t} + 1}\right)}^{c_n}} \cdot \frac{{\left(\frac{1}{\color{blue}{e^{\log \left(e^{-s} + 1\right)}}}\right)}^{c_p}}{e^{\log \left({\left(\frac{1}{e^{-t} + 1}\right)}^{c_p}\right)}}\]
Applied rec-exp4.1
\[\leadsto \frac{{\left(1 - \frac{1}{e^{-s} + 1}\right)}^{c_n}}{{\left(1 - \frac{1}{e^{-t} + 1}\right)}^{c_n}} \cdot \frac{{\color{blue}{\left(e^{-\log \left(e^{-s} + 1\right)}\right)}}^{c_p}}{e^{\log \left({\left(\frac{1}{e^{-t} + 1}\right)}^{c_p}\right)}}\]
Applied pow-exp4.1
\[\leadsto \frac{{\left(1 - \frac{1}{e^{-s} + 1}\right)}^{c_n}}{{\left(1 - \frac{1}{e^{-t} + 1}\right)}^{c_n}} \cdot \frac{\color{blue}{e^{\left(-\log \left(e^{-s} + 1\right)\right) \cdot c_p}}}{e^{\log \left({\left(\frac{1}{e^{-t} + 1}\right)}^{c_p}\right)}}\]
Applied div-exp4.1
\[\leadsto \frac{{\left(1 - \frac{1}{e^{-s} + 1}\right)}^{c_n}}{{\left(1 - \frac{1}{e^{-t} + 1}\right)}^{c_n}} \cdot \color{blue}{e^{\left(-\log \left(e^{-s} + 1\right)\right) \cdot c_p - \log \left({\left(\frac{1}{e^{-t} + 1}\right)}^{c_p}\right)}}\]
Applied add-exp-log4.1
\[\leadsto \frac{{\left(1 - \frac{1}{e^{-s} + 1}\right)}^{c_n}}{\color{blue}{e^{\log \left({\left(1 - \frac{1}{e^{-t} + 1}\right)}^{c_n}\right)}}} \cdot e^{\left(-\log \left(e^{-s} + 1\right)\right) \cdot c_p - \log \left({\left(\frac{1}{e^{-t} + 1}\right)}^{c_p}\right)}\]
Applied pow-to-exp4.1
\[\leadsto \frac{\color{blue}{e^{\log \left(1 - \frac{1}{e^{-s} + 1}\right) \cdot c_n}}}{e^{\log \left({\left(1 - \frac{1}{e^{-t} + 1}\right)}^{c_n}\right)}} \cdot e^{\left(-\log \left(e^{-s} + 1\right)\right) \cdot c_p - \log \left({\left(\frac{1}{e^{-t} + 1}\right)}^{c_p}\right)}\]
Applied div-exp4.1
\[\leadsto \color{blue}{e^{\log \left(1 - \frac{1}{e^{-s} + 1}\right) \cdot c_n - \log \left({\left(1 - \frac{1}{e^{-t} + 1}\right)}^{c_n}\right)}} \cdot e^{\left(-\log \left(e^{-s} + 1\right)\right) \cdot c_p - \log \left({\left(\frac{1}{e^{-t} + 1}\right)}^{c_p}\right)}\]
Applied prod-exp4.1
\[\leadsto \color{blue}{e^{\left(\log \left(1 - \frac{1}{e^{-s} + 1}\right) \cdot c_n - \log \left({\left(1 - \frac{1}{e^{-t} + 1}\right)}^{c_n}\right)\right) + \left(\left(-\log \left(e^{-s} + 1\right)\right) \cdot c_p - \log \left({\left(\frac{1}{e^{-t} + 1}\right)}^{c_p}\right)\right)}}\]
Simplified1.6
\[\leadsto e^{\color{blue}{(\left(\log_* (1 + \frac{-1}{e^{-s} + 1}) - \log_* (1 + \frac{-1}{e^{-t} + 1})\right) \cdot c_n + \left(c_p \cdot \left(\log_* (1 + e^{-t}) - \log_* (1 + e^{-s})\right)\right))_*}}\]
Taylor expanded around 0 0.5
\[\leadsto e^{(\left(\log_* (1 + \frac{-1}{e^{-s} + 1}) - \log_* (1 + \frac{-1}{e^{-t} + 1})\right) \cdot c_n + \left(c_p \cdot \left(\color{blue}{\left(\left(\log 2 + \frac{1}{8} \cdot {t}^{2}\right) - \frac{1}{2} \cdot t\right)} - \log_* (1 + e^{-s})\right)\right))_*}\]
Simplified0.5
\[\leadsto e^{(\left(\log_* (1 + \frac{-1}{e^{-s} + 1}) - \log_* (1 + \frac{-1}{e^{-t} + 1})\right) \cdot c_n + \left(c_p \cdot \left(\color{blue}{(\left((\frac{1}{8} \cdot t + \frac{-1}{2})_*\right) \cdot t + \left(\log 2\right))_*} - \log_* (1 + e^{-s})\right)\right))_*}\]
Final simplification0.5
\[\leadsto e^{(\left(\log_* (1 + \frac{-1}{e^{-s} + 1}) - \log_* (1 + \frac{-1}{e^{-t} + 1})\right) \cdot c_n + \left(\left((\left((\frac{1}{8} \cdot t + \frac{-1}{2})_*\right) \cdot t + \left(\log 2\right))_* - \log_* (1 + e^{-s})\right) \cdot c_p\right))_*}\]
