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Target

Derivation

1. Split input into 2 regimes

2. if (/ 1 (+ 1 (exp (- s)))) < 1.8336260487680146e-300 or 0.5000333288116081 < (/ 1 (+ 1 (exp (- s))))

   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. Taylor expanded around 0 2.1

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

   3. Applied simplify2.1

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

   if 1.8336260487680146e-300 < (/ 1 (+ 1 (exp (- s)))) < 0.5000333288116081

   1. Initial program 3.4

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

   3. Applied add-exp-log3.4

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

   4. Applied pow-exp3.4

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

   5. Applied add-exp-log3.4

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

   6. Applied rec-exp3.4

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

   7. Applied pow-exp3.4

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

   8. Applied prod-exp3.4

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

   9. Applied add-exp-log3.4

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

   10. Applied pow-exp3.4

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

   11. Applied add-exp-log3.4

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

   12. Applied rec-exp3.4

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

   13. Applied pow-exp3.4

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

   14. Applied prod-exp3.4

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

   15. Applied div-exp1.4

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

   16. Applied simplify1.4

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

   17. Taylor expanded around 0 1.2

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

   18. Applied simplify1.2

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

   20. Applied sub-neg1.2

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

   21. Applied exp-sum1.2

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

   22. Applied associate-*l*1.2

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

   23. Applied simplify1.2

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

4. Applied simplify1.7

   \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;\frac{1}{e^{-s} + 1} \le 1.8336260487680146 \cdot 10^{-300} \lor \neg \left(\frac{1}{e^{-s} + 1} \le 0.5000333288116081\right):\\ \;\;\;\;\frac{{\left(\frac{1}{e^{-s} + 1}\right)}^{c_p}}{{\left(1 - \frac{1}{e^{-t} + 1}\right)}^{c_n}} \cdot \frac{{\left(1 - \frac{1}{e^{-s} + 1}\right)}^{c_n}}{c_p \cdot \left(\log \frac{1}{2} + t \cdot \frac{1}{2}\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(\frac{1}{2} \cdot c_n\right) \cdot \left(t - s\right)} \cdot \frac{{\left(\frac{e^{-t} + 1}{e^{-s} + 1}\right)}^{c_p}}{e^{\left(s \cdot s\right) \cdot \left(\frac{1}{8} \cdot c_n\right)}}\\ \end{array}}\]

Runtime

Time bar (total: 5.1m)Debug logProfile

herbie shell --seed 2018296 
(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))))