Average Error: 3.8 → 0.7
Time: 4.3m
Precision: 64
Internal Precision: 2112
\[\frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c_n}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c_p} \cdot {\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c_n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;-s \le 752520684.5981619:\\ \;\;\;\;\frac{{\left(\sqrt[3]{\frac{1 - \frac{1}{1 + e^{-s}}}{1 - \frac{1}{e^{-t} + 1}}} \cdot \sqrt[3]{\frac{1 - \frac{1}{1 + e^{-s}}}{1 - \frac{1}{e^{-t} + 1}}}\right)}^{c_n}}{\sqrt[3]{e^{\left(c_p \cdot \frac{1}{8}\right) \cdot \left(s \cdot s\right) + \left(c_p \cdot \frac{1}{2}\right) \cdot \left(t - s\right)}} \cdot \sqrt[3]{e^{\left(c_p \cdot \frac{1}{8}\right) \cdot \left(s \cdot s\right) + \left(c_p \cdot \frac{1}{2}\right) \cdot \left(t - s\right)}}} \cdot \frac{{\left(\sqrt[3]{\frac{1 - \frac{1}{1 + e^{-s}}}{1 - \frac{1}{e^{-t} + 1}}}\right)}^{c_n}}{\sqrt[3]{e^{\left(c_p \cdot \frac{1}{8}\right) \cdot \left(s \cdot s\right) + \left(c_p \cdot \frac{1}{2}\right) \cdot \left(t - s\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c_n}}{{\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c_n}} \cdot \frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c_p}}{c_p \cdot \left(\log \frac{1}{2} + t \cdot \frac{1}{2}\right) + 1}\\ \end{array}\]

Error

Bits error versus c_p

Bits error versus c_n

Bits error versus t

Bits error versus s

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original3.8
Target2.0
Herbie0.7
\[{\left(\frac{1 + e^{-t}}{1 + e^{-s}}\right)}^{c_p} \cdot {\left(\frac{1 + e^{t}}{1 + e^{s}}\right)}^{c_n}\]

Derivation

  1. Split input into 2 regimes

  2. if (- s) < 752520684.5981619

    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. Using strategy rm

    3. Applied add-exp-log3.7

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

    4. Applied pow-exp3.8

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

    5. Applied add-exp-log3.8

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

    6. Applied rec-exp3.8

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

    7. Applied pow-exp3.8

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

    8. Applied prod-exp3.8

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

    9. Applied add-exp-log3.8

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

    10. Applied pow-exp3.8

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

    11. Applied add-exp-log3.8

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

    12. Applied rec-exp3.8

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

    13. Applied pow-exp3.7

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

    14. Applied prod-exp3.7

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

    15. Applied div-exp1.7

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

    16. Applied simplify1.7

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

    17. Taylor expanded around 0 0.3

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

    18. Applied simplify0.3

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

    20. Applied add-cube-cbrt0.3

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

    21. Applied add-cube-cbrt0.4

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

    22. Applied unpow-prod-down0.4

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

    23. Applied times-frac0.4

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

    if 752520684.5981619 < (- s)

    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. Taylor expanded around 0 1.6

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

    3. Applied simplify1.6

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

Runtime

Time bar (total: 4.3m)Debug logProfile

herbie shell --seed 2018198 
(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))))