Average Error: 3.8 → 2.5
Time: 2.5m
Precision: 64
Internal Precision: 1600
\[\frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c_n}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c_p} \cdot {\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c_n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;c_p \le 3.258119948767867 \cdot 10^{-44}:\\ \;\;\;\;\frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c_n}}{{\left(\sqrt[3]{\left(1 - \frac{1}{e^{-t} + 1}\right) \cdot \left(\left(1 - \frac{1}{e^{-t} + 1}\right) \cdot \left(1 - \frac{1}{e^{-t} + 1}\right)\right)}\right)}^{c_n} \cdot {\left(\frac{1}{e^{-t} + 1}\right)}^{c_p}}\\ \mathbf{else}:\\ \;\;\;\;e^{c_p \cdot \left(\left(t \cdot t\right) \cdot \frac{1}{8} + \left(s - t\right) \cdot \frac{1}{2}\right) - c_n \cdot \left(\log \left(1 - \frac{1}{e^{-t} + 1}\right) - \log \left(1 - \frac{1}{1 + e^{-s}}\right)\right)}\\ \end{array}\]

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

Original3.8
Target1.8
Herbie2.5
\[{\left(\frac{1 + e^{-t}}{1 + e^{-s}}\right)}^{c_p} \cdot {\left(\frac{1 + e^{t}}{1 + e^{s}}\right)}^{c_n}\]

Derivation

  1. Split input into 2 regimes

  2. if c_p < 3.258119948767867e-44

    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-cbrt-cube2.7

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

    if 3.258119948767867e-44 < c_p

    1. Initial program 9.5

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

    3. Applied add-exp-log9.6

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

    4. Applied add-exp-log9.6

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

    5. Applied pow-exp9.6

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

    6. Applied add-exp-log9.6

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

    7. Applied pow-exp9.6

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

    8. Applied prod-exp9.6

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

    9. Applied div-exp9.6

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

    10. Simplified1.6

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

    11. Taylor expanded around 0 1.7

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

    12. Simplified1.7

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

  4. Final simplification2.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;c_p \le 3.258119948767867 \cdot 10^{-44}:\\ \;\;\;\;\frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c_n}}{{\left(\sqrt[3]{\left(1 - \frac{1}{e^{-t} + 1}\right) \cdot \left(\left(1 - \frac{1}{e^{-t} + 1}\right) \cdot \left(1 - \frac{1}{e^{-t} + 1}\right)\right)}\right)}^{c_n} \cdot {\left(\frac{1}{e^{-t} + 1}\right)}^{c_p}}\\ \mathbf{else}:\\ \;\;\;\;e^{c_p \cdot \left(\left(t \cdot t\right) \cdot \frac{1}{8} + \left(s - t\right) \cdot \frac{1}{2}\right) - c_n \cdot \left(\log \left(1 - \frac{1}{e^{-t} + 1}\right) - \log \left(1 - \frac{1}{1 + e^{-s}}\right)\right)}\\ \end{array}\]

Runtime

Time bar (total: 2.5m)Debug logProfile

BaselineHerbieOracleSpan%
Regimes3.92.50.53.340.3%
herbie shell --seed 2018286 
(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))))