Average Error: 4.2 → 1.8
Time: 4.8m
Precision: 64
Internal Precision: 2368
\[\frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c_n}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c_p} \cdot {\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c_n}}\]
↓
\[e^{c_n \cdot \left(\log \left(1 - \frac{1}{1 + e^{-s}}\right) - \log \left(1 - \frac{1}{e^{-t} + 1}\right)\right) + \left(\log \left(e^{-t} + 1\right) \cdot c_p - \log \left(1 + e^{-s}\right) \cdot c_p\right)}\]

Error

Bits error versus c_p

Bits error versus c_n

Bits error versus t

Bits error versus s

Target

Original4.2
Target2.2
Herbie1.8
\[{\left(\frac{1 + e^{-t}}{1 + e^{-s}}\right)}^{c_p} \cdot {\left(\frac{1 + e^{t}}{1 + e^{s}}\right)}^{c_n}\]

Derivation

  1. 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. Using strategy rm

  3. Applied add-exp-log4.2

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

  4. Applied add-exp-log4.2

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

  5. Applied prod-exp4.2

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

  6. Applied add-exp-log4.2

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

  7. Applied pow-exp4.2

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

  8. Applied add-exp-log4.2

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

  9. Applied prod-exp4.2

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

  10. Applied div-exp4.2

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

  11. Applied simplify1.8

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

Runtime

Time bar (total: 4.8m)Debug logProfile

herbie shell --seed '#(1071725047 233389029 2036512464 3988615230 2972226563 1111574017)' 
(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))))