Error

Target

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. Using strategy rm

3. Applied add-exp-log3.8

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

4. Applied pow-exp3.8

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

5. Applied add-exp-log3.8

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

6. Applied rec-exp3.8

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

7. Applied pow-exp3.8

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

8. Applied prod-exp3.8

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

9. Applied add-exp-log3.8

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

10. Applied pow-exp3.8

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

11. Applied add-exp-log3.8

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

12. Applied prod-exp3.8

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

13. Applied div-exp2.2

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

14. Applied simplify1.6

    \[\leadsto e^{\color{blue}{(\left(\log_* (1 + \frac{-1}{1 + e^{-s}}) - \log_* (1 + \frac{-1}{e^{-t} + 1})\right) \cdot c_n + \left(c_p \cdot \log_* (1 + e^{-t}) - \log_* (1 + e^{-s}) \cdot c_p\right))_*}}\]
15. Using strategy rm

16. Applied add-log-exp1.6

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

17. Applied add-log-exp1.6

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

18. Applied diff-log1.6

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

19. Applied simplify1.6

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

21. Applied add-cube-cbrt1.6

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

22. Applied log-prod1.6

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

24. Applied log-prod1.6

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

Runtime

Time bar (total: 3.0m)Debug logProfile

herbie shell --seed 2018193 +o rules:numerics
(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))))