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

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

4. Applied pow-to-exp3.8

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

5. Applied prod-exp3.8

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

6. Applied add-exp-log3.8

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

7. Applied div-exp3.3

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

8. Simplified1.6

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

9. Taylor expanded around 0 0.5

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

10. Simplified0.5

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

12. Applied log1p-udef0.5

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

13. Applied log1p-udef0.5

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

14. Applied diff-log0.5

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

16. Applied div-inv0.5

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

17. Applied log-prod0.5

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

18. Simplified0.5

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

19. Final simplification0.5

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

Runtime

Time bar (total: 1.5m)Debug logProfile

herbie shell --seed 2018295 +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))))