Average Error: 4.0 → 0.6
Time: 2.9m
Precision: 64
Internal Precision: 2368
\[\frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c_n}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c_p} \cdot {\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c_n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;s \le -754458506.6971726:\\ \;\;\;\;e^{c_n \cdot \left(\log \left(1 - \frac{1}{1 + e^{-s}}\right) - \log \left(1 - \frac{1}{1 + e^{-t}}\right)\right) - \left(\log \left(1 + e^{-s}\right) - \log \left(1 + e^{-t}\right)\right) \cdot c_p}\\ \mathbf{else}:\\ \;\;\;\;e^{c_n \cdot \left(\log \left(1 - \frac{1}{1 + e^{-s}}\right) - \log \left(1 - \frac{1}{1 + e^{-t}}\right)\right) - c_p \cdot \left(\frac{1}{2} \cdot \left(t - s\right) + s \cdot \left(\frac{1}{8} \cdot s\right)\right)}\\ \end{array}\]

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

Original4.0
Target2.1
Herbie0.6
\[{\left(\frac{1 + e^{-t}}{1 + e^{-s}}\right)}^{c_p} \cdot {\left(\frac{1 + e^{t}}{1 + e^{s}}\right)}^{c_n}\]

Derivation

  1. Split input into 2 regimes

  2. if s < -754458506.6971726

    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-exp2.5

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

    7. Applied add-exp-log2.5

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

    8. Applied prod-exp2.5

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

    9. Simplified1.4

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

    if -754458506.6971726 < 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. Initial simplification4.0

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

    4. Applied pow-to-exp4.1

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

    5. Applied pow-to-exp4.0

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

    6. Applied div-exp3.4

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

    7. Applied add-exp-log3.4

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

    8. Applied prod-exp3.1

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

    9. Simplified2.0

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

    10. Taylor expanded around 0 0.4

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

    11. Simplified0.4

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

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;s \le -754458506.6971726:\\ \;\;\;\;e^{c_n \cdot \left(\log \left(1 - \frac{1}{1 + e^{-s}}\right) - \log \left(1 - \frac{1}{1 + e^{-t}}\right)\right) - \left(\log \left(1 + e^{-s}\right) - \log \left(1 + e^{-t}\right)\right) \cdot c_p}\\ \mathbf{else}:\\ \;\;\;\;e^{c_n \cdot \left(\log \left(1 - \frac{1}{1 + e^{-s}}\right) - \log \left(1 - \frac{1}{1 + e^{-t}}\right)\right) - c_p \cdot \left(\frac{1}{2} \cdot \left(t - s\right) + s \cdot \left(\frac{1}{8} \cdot s\right)\right)}\\ \end{array}\]

Runtime

Time bar (total: 2.9m)Debug logProfile

herbie shell --seed 2018248 
(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))))