Error

Target

Derivation

1. Initial program 3.7

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

3. Applied add-exp-log3.7

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

4. Applied add-exp-log3.7

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

5. Applied prod-exp3.7

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

6. Applied add-exp-log3.7

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

7. Applied pow-exp3.7

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

8. Applied add-exp-log3.7

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

9. Applied prod-exp3.7

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

10. Applied div-exp3.7

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

11. Applied simplify1.6

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

12. Taylor expanded around 0 0.6

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

13. Applied simplify2.1

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

15. Applied add-cube-cbrt2.1

    \[\leadsto \frac{{\left(\frac{1 - \frac{1}{1 + e^{-s}}}{1 - \frac{1}{e^{-t} + 1}}\right)}^{c_n}}{\color{blue}{\left(\sqrt[3]{\frac{{\left(1 + e^{-s}\right)}^{c_p}}{{\left(e^{c_p}\right)}^{\left(\left(\log 2 - \frac{1}{2} \cdot t\right) + \left(\frac{1}{8} \cdot t\right) \cdot t\right)}}} \cdot \sqrt[3]{\frac{{\left(1 + e^{-s}\right)}^{c_p}}{{\left(e^{c_p}\right)}^{\left(\left(\log 2 - \frac{1}{2} \cdot t\right) + \left(\frac{1}{8} \cdot t\right) \cdot t\right)}}}\right) \cdot \sqrt[3]{\frac{{\left(1 + e^{-s}\right)}^{c_p}}{{\left(e^{c_p}\right)}^{\left(\left(\log 2 - \frac{1}{2} \cdot t\right) + \left(\frac{1}{8} \cdot t\right) \cdot t\right)}}}}}\]

16. Applied add-cube-cbrt2.1

    \[\leadsto \frac{{\color{blue}{\left(\left(\sqrt[3]{\frac{1 - \frac{1}{1 + e^{-s}}}{1 - \frac{1}{e^{-t} + 1}}} \cdot \sqrt[3]{\frac{1 - \frac{1}{1 + e^{-s}}}{1 - \frac{1}{e^{-t} + 1}}}\right) \cdot \sqrt[3]{\frac{1 - \frac{1}{1 + e^{-s}}}{1 - \frac{1}{e^{-t} + 1}}}\right)}}^{c_n}}{\left(\sqrt[3]{\frac{{\left(1 + e^{-s}\right)}^{c_p}}{{\left(e^{c_p}\right)}^{\left(\left(\log 2 - \frac{1}{2} \cdot t\right) + \left(\frac{1}{8} \cdot t\right) \cdot t\right)}}} \cdot \sqrt[3]{\frac{{\left(1 + e^{-s}\right)}^{c_p}}{{\left(e^{c_p}\right)}^{\left(\left(\log 2 - \frac{1}{2} \cdot t\right) + \left(\frac{1}{8} \cdot t\right) \cdot t\right)}}}\right) \cdot \sqrt[3]{\frac{{\left(1 + e^{-s}\right)}^{c_p}}{{\left(e^{c_p}\right)}^{\left(\left(\log 2 - \frac{1}{2} \cdot t\right) + \left(\frac{1}{8} \cdot t\right) \cdot t\right)}}}}\]

17. Applied unpow-prod-down2.1

    \[\leadsto \frac{\color{blue}{{\left(\sqrt[3]{\frac{1 - \frac{1}{1 + e^{-s}}}{1 - \frac{1}{e^{-t} + 1}}} \cdot \sqrt[3]{\frac{1 - \frac{1}{1 + e^{-s}}}{1 - \frac{1}{e^{-t} + 1}}}\right)}^{c_n} \cdot {\left(\sqrt[3]{\frac{1 - \frac{1}{1 + e^{-s}}}{1 - \frac{1}{e^{-t} + 1}}}\right)}^{c_n}}}{\left(\sqrt[3]{\frac{{\left(1 + e^{-s}\right)}^{c_p}}{{\left(e^{c_p}\right)}^{\left(\left(\log 2 - \frac{1}{2} \cdot t\right) + \left(\frac{1}{8} \cdot t\right) \cdot t\right)}}} \cdot \sqrt[3]{\frac{{\left(1 + e^{-s}\right)}^{c_p}}{{\left(e^{c_p}\right)}^{\left(\left(\log 2 - \frac{1}{2} \cdot t\right) + \left(\frac{1}{8} \cdot t\right) \cdot t\right)}}}\right) \cdot \sqrt[3]{\frac{{\left(1 + e^{-s}\right)}^{c_p}}{{\left(e^{c_p}\right)}^{\left(\left(\log 2 - \frac{1}{2} \cdot t\right) + \left(\frac{1}{8} \cdot t\right) \cdot t\right)}}}}\]

18. Applied times-frac2.1

    \[\leadsto \color{blue}{\frac{{\left(\sqrt[3]{\frac{1 - \frac{1}{1 + e^{-s}}}{1 - \frac{1}{e^{-t} + 1}}} \cdot \sqrt[3]{\frac{1 - \frac{1}{1 + e^{-s}}}{1 - \frac{1}{e^{-t} + 1}}}\right)}^{c_n}}{\sqrt[3]{\frac{{\left(1 + e^{-s}\right)}^{c_p}}{{\left(e^{c_p}\right)}^{\left(\left(\log 2 - \frac{1}{2} \cdot t\right) + \left(\frac{1}{8} \cdot t\right) \cdot t\right)}}} \cdot \sqrt[3]{\frac{{\left(1 + e^{-s}\right)}^{c_p}}{{\left(e^{c_p}\right)}^{\left(\left(\log 2 - \frac{1}{2} \cdot t\right) + \left(\frac{1}{8} \cdot t\right) \cdot t\right)}}}} \cdot \frac{{\left(\sqrt[3]{\frac{1 - \frac{1}{1 + e^{-s}}}{1 - \frac{1}{e^{-t} + 1}}}\right)}^{c_n}}{\sqrt[3]{\frac{{\left(1 + e^{-s}\right)}^{c_p}}{{\left(e^{c_p}\right)}^{\left(\left(\log 2 - \frac{1}{2} \cdot t\right) + \left(\frac{1}{8} \cdot t\right) \cdot t\right)}}}}}\]
19. Using strategy rm

20. Applied add-cube-cbrt2.1

    \[\leadsto \frac{{\left(\sqrt[3]{\frac{1 - \frac{1}{1 + e^{-s}}}{1 - \frac{1}{e^{-t} + 1}}} \cdot \sqrt[3]{\frac{1 - \frac{1}{1 + e^{-s}}}{1 - \frac{1}{e^{-t} + 1}}}\right)}^{c_n}}{\sqrt[3]{\frac{{\left(1 + e^{-s}\right)}^{c_p}}{{\left(e^{c_p}\right)}^{\left(\left(\log 2 - \frac{1}{2} \cdot t\right) + \left(\frac{1}{8} \cdot t\right) \cdot t\right)}}} \cdot \sqrt[3]{\frac{{\left(1 + e^{-s}\right)}^{c_p}}{{\left(e^{c_p}\right)}^{\left(\left(\log 2 - \frac{1}{2} \cdot t\right) + \left(\frac{1}{8} \cdot t\right) \cdot t\right)}}}} \cdot \frac{{\left(\sqrt[3]{\frac{1 - \frac{1}{1 + e^{-s}}}{1 - \frac{1}{e^{-t} + 1}}}\right)}^{c_n}}{\sqrt[3]{\frac{{\left(1 + e^{-s}\right)}^{c_p}}{{\color{blue}{\left(\left(\sqrt[3]{e^{c_p}} \cdot \sqrt[3]{e^{c_p}}\right) \cdot \sqrt[3]{e^{c_p}}\right)}}^{\left(\left(\log 2 - \frac{1}{2} \cdot t\right) + \left(\frac{1}{8} \cdot t\right) \cdot t\right)}}}}\]

21. Applied unpow-prod-down2.1

    \[\leadsto \frac{{\left(\sqrt[3]{\frac{1 - \frac{1}{1 + e^{-s}}}{1 - \frac{1}{e^{-t} + 1}}} \cdot \sqrt[3]{\frac{1 - \frac{1}{1 + e^{-s}}}{1 - \frac{1}{e^{-t} + 1}}}\right)}^{c_n}}{\sqrt[3]{\frac{{\left(1 + e^{-s}\right)}^{c_p}}{{\left(e^{c_p}\right)}^{\left(\left(\log 2 - \frac{1}{2} \cdot t\right) + \left(\frac{1}{8} \cdot t\right) \cdot t\right)}}} \cdot \sqrt[3]{\frac{{\left(1 + e^{-s}\right)}^{c_p}}{{\left(e^{c_p}\right)}^{\left(\left(\log 2 - \frac{1}{2} \cdot t\right) + \left(\frac{1}{8} \cdot t\right) \cdot t\right)}}}} \cdot \frac{{\left(\sqrt[3]{\frac{1 - \frac{1}{1 + e^{-s}}}{1 - \frac{1}{e^{-t} + 1}}}\right)}^{c_n}}{\sqrt[3]{\frac{{\left(1 + e^{-s}\right)}^{c_p}}{\color{blue}{{\left(\sqrt[3]{e^{c_p}} \cdot \sqrt[3]{e^{c_p}}\right)}^{\left(\left(\log 2 - \frac{1}{2} \cdot t\right) + \left(\frac{1}{8} \cdot t\right) \cdot t\right)} \cdot {\left(\sqrt[3]{e^{c_p}}\right)}^{\left(\left(\log 2 - \frac{1}{2} \cdot t\right) + \left(\frac{1}{8} \cdot t\right) \cdot t\right)}}}}}\]

22. Applied *-un-lft-identity2.1

    \[\leadsto \frac{{\left(\sqrt[3]{\frac{1 - \frac{1}{1 + e^{-s}}}{1 - \frac{1}{e^{-t} + 1}}} \cdot \sqrt[3]{\frac{1 - \frac{1}{1 + e^{-s}}}{1 - \frac{1}{e^{-t} + 1}}}\right)}^{c_n}}{\sqrt[3]{\frac{{\left(1 + e^{-s}\right)}^{c_p}}{{\left(e^{c_p}\right)}^{\left(\left(\log 2 - \frac{1}{2} \cdot t\right) + \left(\frac{1}{8} \cdot t\right) \cdot t\right)}}} \cdot \sqrt[3]{\frac{{\left(1 + e^{-s}\right)}^{c_p}}{{\left(e^{c_p}\right)}^{\left(\left(\log 2 - \frac{1}{2} \cdot t\right) + \left(\frac{1}{8} \cdot t\right) \cdot t\right)}}}} \cdot \frac{{\left(\sqrt[3]{\frac{1 - \frac{1}{1 + e^{-s}}}{1 - \frac{1}{e^{-t} + 1}}}\right)}^{c_n}}{\sqrt[3]{\frac{{\color{blue}{\left(1 \cdot \left(1 + e^{-s}\right)\right)}}^{c_p}}{{\left(\sqrt[3]{e^{c_p}} \cdot \sqrt[3]{e^{c_p}}\right)}^{\left(\left(\log 2 - \frac{1}{2} \cdot t\right) + \left(\frac{1}{8} \cdot t\right) \cdot t\right)} \cdot {\left(\sqrt[3]{e^{c_p}}\right)}^{\left(\left(\log 2 - \frac{1}{2} \cdot t\right) + \left(\frac{1}{8} \cdot t\right) \cdot t\right)}}}}\]

23. Applied unpow-prod-down2.1

    \[\leadsto \frac{{\left(\sqrt[3]{\frac{1 - \frac{1}{1 + e^{-s}}}{1 - \frac{1}{e^{-t} + 1}}} \cdot \sqrt[3]{\frac{1 - \frac{1}{1 + e^{-s}}}{1 - \frac{1}{e^{-t} + 1}}}\right)}^{c_n}}{\sqrt[3]{\frac{{\left(1 + e^{-s}\right)}^{c_p}}{{\left(e^{c_p}\right)}^{\left(\left(\log 2 - \frac{1}{2} \cdot t\right) + \left(\frac{1}{8} \cdot t\right) \cdot t\right)}}} \cdot \sqrt[3]{\frac{{\left(1 + e^{-s}\right)}^{c_p}}{{\left(e^{c_p}\right)}^{\left(\left(\log 2 - \frac{1}{2} \cdot t\right) + \left(\frac{1}{8} \cdot t\right) \cdot t\right)}}}} \cdot \frac{{\left(\sqrt[3]{\frac{1 - \frac{1}{1 + e^{-s}}}{1 - \frac{1}{e^{-t} + 1}}}\right)}^{c_n}}{\sqrt[3]{\frac{\color{blue}{{1}^{c_p} \cdot {\left(1 + e^{-s}\right)}^{c_p}}}{{\left(\sqrt[3]{e^{c_p}} \cdot \sqrt[3]{e^{c_p}}\right)}^{\left(\left(\log 2 - \frac{1}{2} \cdot t\right) + \left(\frac{1}{8} \cdot t\right) \cdot t\right)} \cdot {\left(\sqrt[3]{e^{c_p}}\right)}^{\left(\left(\log 2 - \frac{1}{2} \cdot t\right) + \left(\frac{1}{8} \cdot t\right) \cdot t\right)}}}}\]

24. Applied times-frac2.1

    \[\leadsto \frac{{\left(\sqrt[3]{\frac{1 - \frac{1}{1 + e^{-s}}}{1 - \frac{1}{e^{-t} + 1}}} \cdot \sqrt[3]{\frac{1 - \frac{1}{1 + e^{-s}}}{1 - \frac{1}{e^{-t} + 1}}}\right)}^{c_n}}{\sqrt[3]{\frac{{\left(1 + e^{-s}\right)}^{c_p}}{{\left(e^{c_p}\right)}^{\left(\left(\log 2 - \frac{1}{2} \cdot t\right) + \left(\frac{1}{8} \cdot t\right) \cdot t\right)}}} \cdot \sqrt[3]{\frac{{\left(1 + e^{-s}\right)}^{c_p}}{{\left(e^{c_p}\right)}^{\left(\left(\log 2 - \frac{1}{2} \cdot t\right) + \left(\frac{1}{8} \cdot t\right) \cdot t\right)}}}} \cdot \frac{{\left(\sqrt[3]{\frac{1 - \frac{1}{1 + e^{-s}}}{1 - \frac{1}{e^{-t} + 1}}}\right)}^{c_n}}{\sqrt[3]{\color{blue}{\frac{{1}^{c_p}}{{\left(\sqrt[3]{e^{c_p}} \cdot \sqrt[3]{e^{c_p}}\right)}^{\left(\left(\log 2 - \frac{1}{2} \cdot t\right) + \left(\frac{1}{8} \cdot t\right) \cdot t\right)}} \cdot \frac{{\left(1 + e^{-s}\right)}^{c_p}}{{\left(\sqrt[3]{e^{c_p}}\right)}^{\left(\left(\log 2 - \frac{1}{2} \cdot t\right) + \left(\frac{1}{8} \cdot t\right) \cdot t\right)}}}}}\]

25. Applied simplify2.1

    \[\leadsto \frac{{\left(\sqrt[3]{\frac{1 - \frac{1}{1 + e^{-s}}}{1 - \frac{1}{e^{-t} + 1}}} \cdot \sqrt[3]{\frac{1 - \frac{1}{1 + e^{-s}}}{1 - \frac{1}{e^{-t} + 1}}}\right)}^{c_n}}{\sqrt[3]{\frac{{\left(1 + e^{-s}\right)}^{c_p}}{{\left(e^{c_p}\right)}^{\left(\left(\log 2 - \frac{1}{2} \cdot t\right) + \left(\frac{1}{8} \cdot t\right) \cdot t\right)}}} \cdot \sqrt[3]{\frac{{\left(1 + e^{-s}\right)}^{c_p}}{{\left(e^{c_p}\right)}^{\left(\left(\log 2 - \frac{1}{2} \cdot t\right) + \left(\frac{1}{8} \cdot t\right) \cdot t\right)}}}} \cdot \frac{{\left(\sqrt[3]{\frac{1 - \frac{1}{1 + e^{-s}}}{1 - \frac{1}{e^{-t} + 1}}}\right)}^{c_n}}{\sqrt[3]{\color{blue}{\frac{1}{{\left(\sqrt[3]{e^{c_p}} \cdot \sqrt[3]{e^{c_p}}\right)}^{\left(\left(\log 2 - \frac{1}{2} \cdot t\right) + \left(t \cdot t\right) \cdot \frac{1}{8}\right)}}} \cdot \frac{{\left(1 + e^{-s}\right)}^{c_p}}{{\left(\sqrt[3]{e^{c_p}}\right)}^{\left(\left(\log 2 - \frac{1}{2} \cdot t\right) + \left(\frac{1}{8} \cdot t\right) \cdot t\right)}}}}\]

Runtime

Time bar (total: 7.6m)Debug logProfile

herbie shell --seed '#(1071501266 3581234924 1086666455 2685055582 1243441566 1802958749)' 
(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))))