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Derivation

1. Initial program 3.8

   \[\frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c_n}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c_p} \cdot {\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c_n}}\]

2. Initial simplification3.8

   \[\leadsto \frac{{\left(1 - \frac{1}{e^{-s} + 1}\right)}^{c_n}}{{\left(1 - \frac{1}{e^{-t} + 1}\right)}^{c_n}} \cdot \frac{{\left(\frac{1}{e^{-s} + 1}\right)}^{c_p}}{{\left(\frac{1}{e^{-t} + 1}\right)}^{c_p}}\]
3. Using strategy rm

4. Applied pow-to-exp3.8

   \[\leadsto \frac{{\left(1 - \frac{1}{e^{-s} + 1}\right)}^{c_n}}{{\left(1 - \frac{1}{e^{-t} + 1}\right)}^{c_n}} \cdot \frac{{\left(\frac{1}{e^{-s} + 1}\right)}^{c_p}}{\color{blue}{e^{\log \left(\frac{1}{e^{-t} + 1}\right) \cdot c_p}}}\]

5. Applied add-exp-log3.8

   \[\leadsto \frac{{\left(1 - \frac{1}{e^{-s} + 1}\right)}^{c_n}}{{\left(1 - \frac{1}{e^{-t} + 1}\right)}^{c_n}} \cdot \frac{\color{blue}{e^{\log \left({\left(\frac{1}{e^{-s} + 1}\right)}^{c_p}\right)}}}{e^{\log \left(\frac{1}{e^{-t} + 1}\right) \cdot c_p}}\]

6. Applied div-exp3.6

   \[\leadsto \frac{{\left(1 - \frac{1}{e^{-s} + 1}\right)}^{c_n}}{{\left(1 - \frac{1}{e^{-t} + 1}\right)}^{c_n}} \cdot \color{blue}{e^{\log \left({\left(\frac{1}{e^{-s} + 1}\right)}^{c_p}\right) - \log \left(\frac{1}{e^{-t} + 1}\right) \cdot c_p}}\]

7. Applied add-exp-log3.6

   \[\leadsto \frac{{\left(1 - \frac{1}{e^{-s} + 1}\right)}^{c_n}}{\color{blue}{e^{\log \left({\left(1 - \frac{1}{e^{-t} + 1}\right)}^{c_n}\right)}}} \cdot e^{\log \left({\left(\frac{1}{e^{-s} + 1}\right)}^{c_p}\right) - \log \left(\frac{1}{e^{-t} + 1}\right) \cdot c_p}\]

8. Applied pow-to-exp3.6

   \[\leadsto \frac{\color{blue}{e^{\log \left(1 - \frac{1}{e^{-s} + 1}\right) \cdot c_n}}}{e^{\log \left({\left(1 - \frac{1}{e^{-t} + 1}\right)}^{c_n}\right)}} \cdot e^{\log \left({\left(\frac{1}{e^{-s} + 1}\right)}^{c_p}\right) - \log \left(\frac{1}{e^{-t} + 1}\right) \cdot c_p}\]

9. Applied div-exp3.6

   \[\leadsto \color{blue}{e^{\log \left(1 - \frac{1}{e^{-s} + 1}\right) \cdot c_n - \log \left({\left(1 - \frac{1}{e^{-t} + 1}\right)}^{c_n}\right)}} \cdot e^{\log \left({\left(\frac{1}{e^{-s} + 1}\right)}^{c_p}\right) - \log \left(\frac{1}{e^{-t} + 1}\right) \cdot c_p}\]

10. Applied prod-exp3.3

    \[\leadsto \color{blue}{e^{\left(\log \left(1 - \frac{1}{e^{-s} + 1}\right) \cdot c_n - \log \left({\left(1 - \frac{1}{e^{-t} + 1}\right)}^{c_n}\right)\right) + \left(\log \left({\left(\frac{1}{e^{-s} + 1}\right)}^{c_p}\right) - \log \left(\frac{1}{e^{-t} + 1}\right) \cdot c_p\right)}}\]

11. Simplified1.6

    \[\leadsto e^{\color{blue}{\left(\log \left(e^{-t} + 1\right) - \log \left(e^{-s} + 1\right)\right) \cdot c_p + \left(\log \left(1 - \frac{1}{e^{-s} + 1}\right) - \log \left(1 - \frac{1}{e^{-t} + 1}\right)\right) \cdot c_n}}\]

12. Taylor expanded around 0 0.5

    \[\leadsto e^{\left(\color{blue}{\left(\left(\log 2 + \frac{1}{8} \cdot {t}^{2}\right) - \frac{1}{2} \cdot t\right)} - \log \left(e^{-s} + 1\right)\right) \cdot c_p + \left(\log \left(1 - \frac{1}{e^{-s} + 1}\right) - \log \left(1 - \frac{1}{e^{-t} + 1}\right)\right) \cdot c_n}\]
13. Using strategy rm

14. Applied flip--0.5

    \[\leadsto e^{\left(\left(\left(\log 2 + \frac{1}{8} \cdot {t}^{2}\right) - \frac{1}{2} \cdot t\right) - \log \left(e^{-s} + 1\right)\right) \cdot c_p + \left(\log \left(1 - \frac{1}{e^{-s} + 1}\right) - \log \color{blue}{\left(\frac{1 \cdot 1 - \frac{1}{e^{-t} + 1} \cdot \frac{1}{e^{-t} + 1}}{1 + \frac{1}{e^{-t} + 1}}\right)}\right) \cdot c_n}\]

15. Applied log-div0.5

    \[\leadsto e^{\left(\left(\left(\log 2 + \frac{1}{8} \cdot {t}^{2}\right) - \frac{1}{2} \cdot t\right) - \log \left(e^{-s} + 1\right)\right) \cdot c_p + \left(\log \left(1 - \frac{1}{e^{-s} + 1}\right) - \color{blue}{\left(\log \left(1 \cdot 1 - \frac{1}{e^{-t} + 1} \cdot \frac{1}{e^{-t} + 1}\right) - \log \left(1 + \frac{1}{e^{-t} + 1}\right)\right)}\right) \cdot c_n}\]

16. Simplified0.5

    \[\leadsto e^{\left(\left(\left(\log 2 + \frac{1}{8} \cdot {t}^{2}\right) - \frac{1}{2} \cdot t\right) - \log \left(e^{-s} + 1\right)\right) \cdot c_p + \left(\log \left(1 - \frac{1}{e^{-s} + 1}\right) - \left(\color{blue}{\log \left(1 - \frac{1}{\left(1 + e^{-t}\right) \cdot \left(1 + e^{-t}\right)}\right)} - \log \left(1 + \frac{1}{e^{-t} + 1}\right)\right)\right) \cdot c_n}\]

17. Final simplification0.5

    \[\leadsto e^{\left(\left(\left(\log 2 + {t}^{2} \cdot \frac{1}{8}\right) - t \cdot \frac{1}{2}\right) - \log \left(e^{-s} + 1\right)\right) \cdot c_p + c_n \cdot \left(\log \left(1 - \frac{1}{e^{-s} + 1}\right) - \left(\log \left(1 - \frac{1}{\left(1 + e^{-t}\right) \cdot \left(1 + e^{-t}\right)}\right) - \log \left(\frac{1}{1 + e^{-t}} + 1\right)\right)\right)}\]

Runtime

Time bar (total: 1.8m)Debug logProfile

herbie shell --seed 2018295 
(FPCore (c_p c_n t s)
  :name "Harley's example"
  :pre (and (< 0 c_p) (< 0 c_n))

  :herbie-target
  (* (pow (/ (+ 1 (exp (- t))) (+ 1 (exp (- s)))) c_p) (pow (/ (+ 1 (exp t)) (+ 1 (exp s))) c_n))

  (/ (* (pow (/ 1 (+ 1 (exp (- s)))) c_p) (pow (- 1 (/ 1 (+ 1 (exp (- s))))) c_n)) (* (pow (/ 1 (+ 1 (exp (- t)))) c_p) (pow (- 1 (/ 1 (+ 1 (exp (- t))))) c_n))))