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Target

Derivation

1. Split input into 3 regimes

2. if s < -204638810005.92273

   1. Initial program 4.0

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

   2. Initial simplification4.0

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

   4. Applied add-exp-log4.0

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

   5. Applied pow-exp4.0

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

   6. Applied pow-to-exp4.0

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

   7. Applied div-exp4.0

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

   8. Simplified4.0

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

   9. Taylor expanded around 0 2.5

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

   10. Simplified2.5

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

   if -204638810005.92273 < s < 487.4749094465218

   1. Initial program 4.4

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

   2. Initial simplification4.4

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

   4. Applied add-exp-log4.4

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

   5. Applied pow-exp4.4

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

   6. Applied pow-to-exp4.4

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

   7. Applied div-exp3.5

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

   8. Simplified3.5

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

   9. Taylor expanded around 0 3.2

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

   10. Simplified3.2

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

   11. Taylor expanded around 0 0.6

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

   12. Simplified0.6

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

   if 487.4749094465218 < s

   1. Initial program 3.3

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

   2. Initial simplification3.3

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

   4. Applied add-exp-log3.3

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

   5. Applied pow-exp3.3

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

   6. Applied pow-to-exp3.3

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

   7. Applied div-exp2.7

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

   8. Simplified2.7

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

   9. Taylor expanded around 0 0.3

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

   10. Simplified0.3

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

4. Final simplification1.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;s \le -204638810005.92273:\\ \;\;\;\;e^{c_n \cdot \left(\left(t - s\right) \cdot \frac{1}{2} + \left(s \cdot s\right) \cdot \frac{-1}{8}\right)} \cdot \frac{{\left(\frac{1}{e^{-s} + 1}\right)}^{c_p}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c_p}}\\ \mathbf{elif}\;s \le 487.4749094465218:\\ \;\;\;\;\left(\left(\frac{1}{2} \cdot c_p\right) \cdot \left(s - t\right) + 1\right) \cdot e^{c_n \cdot \left(\left(t - s\right) \cdot \frac{1}{2} + \left(s \cdot s\right) \cdot \frac{-1}{8}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{1}{e^{-s} + 1}\right)}^{c_p}}{1 + c_p \cdot \left(\log \frac{1}{2} + \frac{1}{2} \cdot t\right)} \cdot e^{c_n \cdot \left(\log \left(1 - \frac{1}{e^{-s} + 1}\right) - \log \left(1 - \frac{1}{1 + e^{-t}}\right)\right)}\\ \end{array}\]

Runtime

Time bar (total: 2.1m)Debug logProfile

herbie shell --seed 2018258 
(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))))