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Target

Derivation

1. Split input into 2 regimes

2. if s < -518424806.8107358

   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. Initial simplification4.1

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

   3. Taylor expanded around 0 1.3

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

   4. Simplified1.3

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

   6. Applied add-cube-cbrt1.3

      \[\leadsto \frac{{\left(1 - \frac{1}{e^{-s} + 1}\right)}^{c_n}}{{\left(1 - \frac{1}{e^{-t} + 1}\right)}^{c_n}} \cdot \frac{{\left(\frac{1}{e^{-s} + 1}\right)}^{c_p}}{\color{blue}{\left(\sqrt[3]{1 + c_p \cdot \left(\log \frac{1}{2} + \frac{1}{2} \cdot t\right)} \cdot \sqrt[3]{1 + c_p \cdot \left(\log \frac{1}{2} + \frac{1}{2} \cdot t\right)}\right) \cdot \sqrt[3]{1 + c_p \cdot \left(\log \frac{1}{2} + \frac{1}{2} \cdot t\right)}}}\]

   7. Applied add-sqr-sqrt1.3

      \[\leadsto \frac{{\left(1 - \frac{1}{e^{-s} + 1}\right)}^{c_n}}{{\left(1 - \frac{1}{e^{-t} + 1}\right)}^{c_n}} \cdot \frac{\color{blue}{\sqrt{{\left(\frac{1}{e^{-s} + 1}\right)}^{c_p}} \cdot \sqrt{{\left(\frac{1}{e^{-s} + 1}\right)}^{c_p}}}}{\left(\sqrt[3]{1 + c_p \cdot \left(\log \frac{1}{2} + \frac{1}{2} \cdot t\right)} \cdot \sqrt[3]{1 + c_p \cdot \left(\log \frac{1}{2} + \frac{1}{2} \cdot t\right)}\right) \cdot \sqrt[3]{1 + c_p \cdot \left(\log \frac{1}{2} + \frac{1}{2} \cdot t\right)}}\]

   8. Applied times-frac1.3

      \[\leadsto \frac{{\left(1 - \frac{1}{e^{-s} + 1}\right)}^{c_n}}{{\left(1 - \frac{1}{e^{-t} + 1}\right)}^{c_n}} \cdot \color{blue}{\left(\frac{\sqrt{{\left(\frac{1}{e^{-s} + 1}\right)}^{c_p}}}{\sqrt[3]{1 + c_p \cdot \left(\log \frac{1}{2} + \frac{1}{2} \cdot t\right)} \cdot \sqrt[3]{1 + c_p \cdot \left(\log \frac{1}{2} + \frac{1}{2} \cdot t\right)}} \cdot \frac{\sqrt{{\left(\frac{1}{e^{-s} + 1}\right)}^{c_p}}}{\sqrt[3]{1 + c_p \cdot \left(\log \frac{1}{2} + \frac{1}{2} \cdot t\right)}}\right)}\]

   9. Applied associate-*r*1.3

      \[\leadsto \color{blue}{\left(\frac{{\left(1 - \frac{1}{e^{-s} + 1}\right)}^{c_n}}{{\left(1 - \frac{1}{e^{-t} + 1}\right)}^{c_n}} \cdot \frac{\sqrt{{\left(\frac{1}{e^{-s} + 1}\right)}^{c_p}}}{\sqrt[3]{1 + c_p \cdot \left(\log \frac{1}{2} + \frac{1}{2} \cdot t\right)} \cdot \sqrt[3]{1 + c_p \cdot \left(\log \frac{1}{2} + \frac{1}{2} \cdot t\right)}}\right) \cdot \frac{\sqrt{{\left(\frac{1}{e^{-s} + 1}\right)}^{c_p}}}{\sqrt[3]{1 + c_p \cdot \left(\log \frac{1}{2} + \frac{1}{2} \cdot t\right)}}}\]

   if -518424806.8107358 < s

   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. Initial simplification3.9

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

   4. Applied add-exp-log3.9

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

   5. Applied rec-exp3.9

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

   6. Applied pow-exp3.9

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

   7. Applied add-exp-log3.9

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

   8. Applied div-exp3.8

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

   9. Applied add-exp-log3.8

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

   10. Applied pow-exp3.9

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

   11. Applied pow-to-exp3.8

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

   12. Applied div-exp3.0

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

   13. Applied prod-exp2.5

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

   14. Simplified1.8

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

   15. Taylor expanded around 0 0.4

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

   16. Simplified0.4

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

4. Final simplification0.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;s \le -518424806.8107358:\\ \;\;\;\;\left(\frac{\sqrt{{\left(\frac{1}{1 + e^{-s}}\right)}^{c_p}}}{\sqrt[3]{1 + c_p \cdot \left(t \cdot \frac{1}{2} + \log \frac{1}{2}\right)} \cdot \sqrt[3]{1 + c_p \cdot \left(t \cdot \frac{1}{2} + \log \frac{1}{2}\right)}} \cdot \frac{{\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c_n}}{{\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c_n}}\right) \cdot \frac{\sqrt{{\left(\frac{1}{1 + e^{-s}}\right)}^{c_p}}}{\sqrt[3]{1 + c_p \cdot \left(t \cdot \frac{1}{2} + \log \frac{1}{2}\right)}}\\ \mathbf{else}:\\ \;\;\;\;e^{c_p \cdot \left(\left(t \cdot t\right) \cdot \frac{1}{8} + \left(s - t\right) \cdot \frac{1}{2}\right) - \left(\log \left(1 - \frac{1}{1 + e^{-t}}\right) - \log \left(1 - \frac{1}{1 + e^{-s}}\right)\right) \cdot c_n}\\ \end{array}\]

Runtime

Time bar (total: 4.0m)Debug logProfile

herbie shell --seed 2018252 
(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))))