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Target

Derivation

1. Split input into 2 regimes

2. if s < -733101611.240873

   1. Initial program 3.7

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

   2. Initial simplification3.7

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

   4. Applied add-exp-log3.7

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

   5. Applied pow-exp3.7

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

   6. Applied add-exp-log3.7

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

   7. Applied rec-exp3.7

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

   8. Applied pow-exp3.7

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

   9. Applied pow-to-exp3.7

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

   10. Applied div-exp2.6

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

   11. Applied pow-to-exp2.6

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

   12. Applied prod-exp2.6

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

   13. Applied div-exp1.4

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

   14. Simplified1.4

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

   15. Taylor expanded around 0 0.3

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

   16. Simplified0.3

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

   17. Taylor expanded around 0 1.3

       \[\leadsto e^{c_p \cdot \left(\left(t \cdot \left(\frac{-1}{2} + \frac{1}{8} \cdot t\right) + \log 2\right) - \log \left(e^{-s} + 1\right)\right) - c_n \cdot \left(\log \color{blue}{\left(\left(\frac{1}{48} \cdot {t}^{3} + \frac{1}{2}\right) - \frac{1}{4} \cdot t\right)} - \log \left(1 - \frac{1}{e^{-s} + 1}\right)\right)}\]

   if -733101611.240873 < s

   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

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

   4. Applied add-exp-log4.1

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

   5. Applied pow-exp4.1

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

   6. Applied add-exp-log4.1

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

   7. Applied rec-exp4.1

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

   8. Applied pow-exp4.1

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

   9. Applied pow-to-exp4.1

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

   10. Applied div-exp3.3

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

   11. Applied pow-to-exp3.3

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

   12. Applied prod-exp2.9

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

   13. Applied div-exp1.9

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

   14. Simplified1.9

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

   15. Taylor expanded around 0 0.7

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

   16. Simplified0.7

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

   17. Taylor expanded around 0 0.4

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

   18. Simplified0.4

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

4. Final simplification0.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;s \le -733101611.240873:\\ \;\;\;\;e^{c_p \cdot \left(\left(\log 2 + t \cdot \left(\frac{-1}{2} + \frac{1}{8} \cdot t\right)\right) - \log \left(e^{-s} + 1\right)\right) - \left(\log \left(\left(\frac{1}{2} + {t}^{3} \cdot \frac{1}{48}\right) - \frac{1}{4} \cdot t\right) - \log \left(1 - \frac{1}{e^{-s} + 1}\right)\right) \cdot c_n}\\ \mathbf{else}:\\ \;\;\;\;e^{c_p \cdot \left(\left(\log 2 + t \cdot \left(\frac{-1}{2} + \frac{1}{8} \cdot t\right)\right) - \left(\log 2 + s \cdot \left(\frac{-1}{2} + s \cdot \frac{1}{8}\right)\right)\right) - \left(\log \left(1 - \frac{1}{e^{-t} + 1}\right) - \log \left(1 - \frac{1}{e^{-s} + 1}\right)\right) \cdot c_n}\\ \end{array}\]

Runtime

Time bar (total: 3.7m)Debug logProfile

herbie shell --seed 2018234 
(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))))