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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

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

6. Applied rec-exp3.8

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

7. Applied pow-exp3.8

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

8. Applied div-exp2.9

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

9. Applied add-exp-log2.9

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

10. Applied prod-exp2.6

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

11. Simplified1.7

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

12. Taylor expanded around 0 0.6

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

13. Final simplification0.6

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

Runtime

Time bar (total: 2.8m)Debug logProfile

herbie shell --seed 2018274 
(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))))