Average Error: 4.1 → 0.4
Time: 3.6m
Precision: 64
Internal Precision: 2624
\[\frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c_n}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c_p} \cdot {\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c_n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;s \le -748451056.7557516:\\ \;\;\;\;e^{c_n \cdot \left(\left(t - s\right) \cdot \frac{1}{2} + \left(s \cdot s\right) \cdot \frac{-1}{8}\right) + c_p \cdot \left(\left(\left(\log 2 + \frac{1}{8} \cdot {t}^{2}\right) - \frac{1}{2} \cdot t\right) - \log \left(1 + e^{-s}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{e}^{\left(\left(\log \left(1 - \frac{1}{1 + e^{-s}}\right) - \log \left(1 - \frac{1}{e^{-t} + 1}\right)\right) \cdot c_n + c_p \cdot \left(\left(t \cdot t\right) \cdot \frac{1}{8} + \frac{1}{2} \cdot \left(s - t\right)\right)\right)}\\ \end{array}\]

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

Original4.1
Target2.1
Herbie0.4
\[{\left(\frac{1 + e^{-t}}{1 + e^{-s}}\right)}^{c_p} \cdot {\left(\frac{1 + e^{t}}{1 + e^{s}}\right)}^{c_n}\]

Derivation

  1. Split input into 2 regimes

  2. if s < -748451056.7557516

    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

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

    4. Applied pow-to-exp3.7

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

    5. Applied pow-to-exp3.7

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

    6. Applied div-exp2.6

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

    7. Applied add-exp-log2.6

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

    8. Applied pow-to-exp2.6

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

    9. Applied div-exp2.6

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

    10. Applied prod-exp2.6

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

    11. Simplified1.5

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

    12. Taylor expanded around 0 0.3

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

    13. Taylor expanded around 0 0

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

    14. Simplified0

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

    if -748451056.7557516 < s

    1. Initial program 4.2

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

    2. Initial simplification4.2

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

    4. Applied pow-to-exp4.3

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

    5. Applied pow-to-exp4.2

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

    6. Applied div-exp3.5

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

    7. Applied add-exp-log3.5

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

    8. Applied pow-to-exp3.5

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

    9. Applied div-exp3.5

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

    10. Applied prod-exp3.1

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

    11. Simplified1.9

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

    12. Taylor expanded around 0 0.5

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

    13. Simplified0.5

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

    15. Applied *-un-lft-identity0.5

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

    16. Applied exp-prod0.5

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

    17. Simplified0.5

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;s \le -748451056.7557516:\\ \;\;\;\;e^{c_n \cdot \left(\left(t - s\right) \cdot \frac{1}{2} + \left(s \cdot s\right) \cdot \frac{-1}{8}\right) + c_p \cdot \left(\left(\left(\log 2 + \frac{1}{8} \cdot {t}^{2}\right) - \frac{1}{2} \cdot t\right) - \log \left(1 + e^{-s}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{e}^{\left(\left(\log \left(1 - \frac{1}{1 + e^{-s}}\right) - \log \left(1 - \frac{1}{e^{-t} + 1}\right)\right) \cdot c_n + c_p \cdot \left(\left(t \cdot t\right) \cdot \frac{1}{8} + \frac{1}{2} \cdot \left(s - t\right)\right)\right)}\\ \end{array}\]

Runtime

Time bar (total: 3.6m)Debug logProfile

herbie shell --seed 2018251 
(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))))