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

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

5. Applied prod-exp3.9

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

6. Applied pow-to-exp3.9

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

7. Applied add-exp-log3.9

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

8. Applied pow-exp3.9

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

9. Applied prod-exp3.9

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

10. Applied div-exp2.7

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

11. Simplified1.6

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

12. Taylor expanded around 0 0.6

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

13. Simplified0.6

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

15. Applied fma-udef0.6

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

16. Applied associate--l+0.6

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

17. Final simplification0.6

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

Runtime

Time bar (total: 2.9m)Debug logProfile

herbie shell --seed 2018227 +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))))