Average Error: 3.9 → 1.1
Time: 8.4m
Precision: 64
Internal Precision: 1664
\[\frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c_n}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c_p} \cdot {\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c_n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -5.002354531330843 \cdot 10^{-66}:\\ \;\;\;\;\frac{{\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c_n}}{{\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c_n}} \cdot \frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c_p}}{\left(\log \frac{1}{2} + t \cdot \frac{1}{2}\right) \cdot c_p + 1}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(\log \left(1 + e^{-s}\right) - \log \left(e^{-t} + 1\right)\right) \cdot \left(-c_p\right) + \left(\left(\log \left(1 - \frac{1}{1 + e^{-s}}\right) - \log \left(\sqrt[3]{1 - \frac{1}{e^{-t} + 1}} \cdot \sqrt[3]{1 \cdot 1 - \frac{1}{e^{-t} + 1} \cdot \frac{1}{e^{-t} + 1}}\right)\right) + \left(\log \left(\sqrt[3]{1 + \frac{1}{e^{-t} + 1}}\right) - \log \left(\sqrt[3]{1 - \frac{1}{e^{-t} + 1}}\right)\right)\right) \cdot c_n}\\ \end{array}\]

Error

Bits error versus c_p

Bits error versus c_n

Bits error versus t

Bits error versus s

Target

Original3.9
Target2.0
Herbie1.1
\[{\left(\frac{1 + e^{-t}}{1 + e^{-s}}\right)}^{c_p} \cdot {\left(\frac{1 + e^{t}}{1 + e^{s}}\right)}^{c_n}\]

Derivation

  1. Split input into 2 regimes

  2. if t < -5.002354531330843e-66

    1. Initial program 11.6

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

    2. Taylor expanded around 0 3.1

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

    3. Applied simplify3.1

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

    if -5.002354531330843e-66 < t

    1. Initial program 2.7

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

    3. Applied add-exp-log2.7

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

    4. Applied pow-exp2.8

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

    5. Applied add-exp-log2.8

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

    6. Applied rec-exp2.8

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

    7. Applied pow-exp2.8

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

    8. Applied prod-exp2.8

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

    9. Applied add-exp-log2.8

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

    10. Applied pow-exp2.8

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

    11. Applied add-exp-log2.8

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

    12. Applied rec-exp2.8

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

    13. Applied pow-exp2.7

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

    14. Applied prod-exp2.7

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

    15. Applied div-exp0.6

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

    16. Applied simplify0.6

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

    18. Applied add-cube-cbrt0.8

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

    19. Applied log-prod0.8

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

    20. Applied associate--r+0.8

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

    22. Applied flip--0.8

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

    23. Applied cbrt-div0.8

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

    24. Applied associate-*r/0.8

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

    25. Applied log-div0.8

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

    26. Applied associate--r-0.8

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

    27. Applied associate--l+0.8

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

Runtime

Time bar (total: 8.4m)Debug logProfile

herbie shell --seed '#(1063185673 2139736501 2393378123 1907444849 1070993796 1007244912)' 
(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))))