Average Error: 4.0 → 0.9
Time: 1.7m
Precision: 64
Internal Precision: 2368
\[\frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c_n}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c_p} \cdot {\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c_n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;s \le -738404085.7526034:\\ \;\;\;\;\frac{{\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c_n}}{{\left(\left({t}^{3} \cdot \frac{1}{48} + \frac{1}{2}\right) - \frac{1}{4} \cdot t\right)}^{c_n}} \cdot \frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c_p}}{1 + \left(\log \frac{1}{2} + t \cdot \frac{1}{2}\right) \cdot c_p}\\ \mathbf{else}:\\ \;\;\;\;e^{c_n \cdot \left(\log \left(1 - \frac{1}{1 + e^{-s}}\right) - \log \left(1 - \frac{1}{1 + e^{-t}}\right)\right)} \cdot \frac{1 + c_p \cdot \left(\log \frac{1}{2} + s \cdot \frac{1}{2}\right)}{1 + \left(\log \frac{1}{2} + t \cdot \frac{1}{2}\right) \cdot c_p}\\ \end{array}\]

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

Original4.0
Target2.2
Herbie0.9
\[{\left(\frac{1 + e^{-t}}{1 + e^{-s}}\right)}^{c_p} \cdot {\left(\frac{1 + e^{t}}{1 + e^{s}}\right)}^{c_n}\]

Derivation

  1. Split input into 2 regimes

  2. if s < -738404085.7526034

    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. Taylor expanded around 0 1.4

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

    4. Simplified1.4

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

    5. Taylor expanded around 0 2.1

      \[\leadsto \frac{{\left(1 - \frac{1}{e^{-s} + 1}\right)}^{c_n}}{{\color{blue}{\left(\left(\frac{1}{48} \cdot {t}^{3} + \frac{1}{2}\right) - \frac{1}{4} \cdot t\right)}}^{c_n}} \cdot \frac{{\left(\frac{1}{e^{-s} + 1}\right)}^{c_p}}{1 + c_p \cdot \left(\log \frac{1}{2} + \frac{1}{2} \cdot t\right)}\]

    if -738404085.7526034 < 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. Taylor expanded around 0 3.3

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

    4. Simplified3.3

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

    6. Applied add-exp-log3.3

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

    7. Applied add-exp-log3.3

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

    8. Applied div-exp3.3

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

    9. Simplified2.3

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

    10. Taylor expanded around 0 0.5

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

    11. Simplified0.5

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

  4. Final simplification0.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;s \le -738404085.7526034:\\ \;\;\;\;\frac{{\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c_n}}{{\left(\left({t}^{3} \cdot \frac{1}{48} + \frac{1}{2}\right) - \frac{1}{4} \cdot t\right)}^{c_n}} \cdot \frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c_p}}{1 + \left(\log \frac{1}{2} + t \cdot \frac{1}{2}\right) \cdot c_p}\\ \mathbf{else}:\\ \;\;\;\;e^{c_n \cdot \left(\log \left(1 - \frac{1}{1 + e^{-s}}\right) - \log \left(1 - \frac{1}{1 + e^{-t}}\right)\right)} \cdot \frac{1 + c_p \cdot \left(\log \frac{1}{2} + s \cdot \frac{1}{2}\right)}{1 + \left(\log \frac{1}{2} + t \cdot \frac{1}{2}\right) \cdot c_p}\\ \end{array}\]

Runtime

Time bar (total: 1.7m)Debug logProfile

herbie shell --seed 2018273 
(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))))