Error

Target

Derivation

1. Initial program 3.9

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

3. Applied pow-to-exp3.9

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

4. Applied add-exp-log3.9

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

5. Applied pow-exp3.9

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

6. Applied prod-exp3.9

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

7. Applied pow-to-exp3.9

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

8. Applied add-exp-log3.9

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

9. Applied pow-exp3.9

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

10. Applied prod-exp3.9

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

11. Applied div-exp1.6

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

12. Simplified1.6

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

13. Taylor expanded around 0 1.4

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

14. Simplified1.4

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

15. Taylor expanded around 0 0.7

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

16. Simplified0.7

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

17. Final simplification0.7

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

Runtime

Time bar (total: 2.7m)Debug logProfile

herbie shell --seed 2018250 +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))))