Error

Target

Derivation

1. Split input into 2 regimes

2. if s < -9.580224664025417e-26

   1. Initial program 4.6

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

   2. Initial simplification4.6

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

   4. Applied add-exp-log4.6

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

   5. Applied add-exp-log4.6

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

   6. Applied pow-exp4.6

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

   7. Applied div-exp4.6

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

   8. Applied pow-to-exp4.6

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

   9. Applied add-exp-log4.6

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

   10. Applied div-exp4.5

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

   11. Applied prod-exp3.5

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

   12. Simplified2.1

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

   13. Taylor expanded around 0 1.8

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

   14. Simplified1.8

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

   if -9.580224664025417e-26 < s

   1. Initial program 3.6

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

   2. Initial simplification3.6

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

   4. Applied add-exp-log3.6

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

   5. Applied add-exp-log3.6

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

   6. Applied pow-exp3.6

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

   7. Applied div-exp3.6

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

   8. Applied pow-to-exp3.6

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

   9. Applied add-exp-log3.6

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

   10. Applied div-exp3.2

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

   11. Applied prod-exp3.2

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

   12. Simplified1.3

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

   13. Taylor expanded around 0 0.3

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

   14. Simplified0.3

       \[\leadsto e^{(\left(\log_* (1 + \frac{-1}{e^{-s} + 1}) - \log_* (1 + \frac{-1}{e^{-t} + 1})\right) \cdot c_n + \left(\color{blue}{(\left((t \cdot \frac{1}{8} + \frac{-1}{2})_*\right) \cdot t + \left(\frac{1}{2} \cdot s\right))_*} \cdot c_p\right))_*}\]
3. Recombined 2 regimes into one program.

4. Final simplification0.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;s \le -9.580224664025417 \cdot 10^{-26}:\\ \;\;\;\;e^{(\left((s \cdot \left((\frac{-1}{8} \cdot s + \frac{-1}{2})_*\right) + \left(t \cdot \frac{1}{2}\right))_*\right) \cdot c_n + \left(c_p \cdot \left(\log_* (1 + e^{-t}) - \log_* (1 + e^{-s})\right)\right))_*}\\ \mathbf{else}:\\ \;\;\;\;e^{(\left(\log_* (1 + \frac{-1}{1 + e^{-s}}) - \log_* (1 + \frac{-1}{1 + e^{-t}})\right) \cdot c_n + \left((\left((t \cdot \frac{1}{8} + \frac{-1}{2})_*\right) \cdot t + \left(s \cdot \frac{1}{2}\right))_* \cdot c_p\right))_*}\\ \end{array}\]

Runtime

Time bar (total: 2.1m)Debug logProfile

herbie shell --seed 2018286 +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))))