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Derivation

1. Initial program 4.0

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

3. Applied add-exp-log4.0

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

4. Applied pow-exp4.0

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

5. Applied add-exp-log4.0

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

6. Applied pow-exp4.0

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

7. Applied prod-exp4.0

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

8. Applied add-exp-log4.0

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

9. Applied add-exp-log4.0

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

10. Applied prod-exp4.0

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

11. Applied div-exp2.8

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

12. Simplified1.8

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

13. Taylor expanded around 0 0.7

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

14. Final simplification0.7

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

Runtime

Time bar (total: 2.5m)Debug logProfile

herbie shell --seed 2018296 
(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))))