Error

Target

Derivation

1. Initial program 4.1

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

3. Applied add-exp-log4.1

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

4. Applied pow-to-exp4.1

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

5. Applied prod-exp4.1

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

6. Applied add-exp-log4.1

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

7. Applied add-exp-log4.1

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

8. Applied prod-exp4.1

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

9. Applied div-exp3.4

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

10. Simplified1.8

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

11. Taylor expanded around 0 0.6

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

12. Simplified0.6

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

13. Final simplification0.6

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

Runtime

Time bar (total: 3.1m)Debug logProfile

herbie shell --seed 2018263 +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))))