Error

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

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

   6. Applied add-cube-cbrt1.3

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

   7. Applied add-sqr-sqrt1.3

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

   8. Applied times-frac1.3

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

   9. Applied associate-*r*1.3

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

   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

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

   8. Applied rec-exp3.9

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

   9. Applied pow-exp3.9

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

   10. Applied div-exp3.2

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

   11. Applied add-exp-log3.2

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

   12. Applied pow-to-exp3.2

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

   13. Applied div-exp3.2

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

   14. Applied prod-exp2.7

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

   15. Simplified1.8

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

   16. Taylor expanded around 0 0.4

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

   17. Simplified0.4

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

4. Final simplification0.6

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

Runtime

Time bar (total: 3.8m)Debug logProfile

herbie shell --seed 2018252 +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))))