Average Error: 3.9 → 0.5
Time: 5.5m
Precision: 64
Internal Precision: 1856
\[\frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c_n}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c_p} \cdot {\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c_n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;e^{c_n \cdot \left(\log \left(1 - \frac{1}{1 + e^{-s}}\right) - \log \left(1 - \frac{1}{e^{-t} + 1}\right)\right) + \log \left({\left(\frac{1 + e^{-t}}{e^{-s} + 1}\right)}^{c_p}\right)} \le 10.110698407299443:\\ \;\;\;\;e^{c_n \cdot \left(\log \left(1 - \frac{1}{1 + e^{-s}}\right) - \log \left(1 - \frac{1}{e^{-t} + 1}\right)\right) + \log \left({\left(\frac{1 + e^{-t}}{e^{-s} + 1}\right)}^{c_p}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c_n}}{{\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c_n}} \cdot \frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c_p}}{c_p \cdot \left(\log \frac{1}{2} + t \cdot \frac{1}{2}\right) + 1}\\ \end{array}\]

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

Original3.9
Target1.9
Herbie0.5
\[{\left(\frac{1 + e^{-t}}{1 + e^{-s}}\right)}^{c_p} \cdot {\left(\frac{1 + e^{t}}{1 + e^{s}}\right)}^{c_n}\]

Derivation

  1. Split input into 2 regimes

  2. if (exp (+ (* c_n (- (log (- 1 (/ 1 (+ 1 (exp (- s)))))) (log (- 1 (/ 1 (+ (exp (- t)) 1)))))) (log (pow (/ (+ 1 (exp (- t))) (+ (exp (- s)) 1)) c_p)))) < 10.110698407299443

    1. Initial program 2.3

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

    3. Applied add-exp-log2.3

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

    4. Applied add-exp-log2.4

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

    5. Applied prod-exp2.4

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

    6. Applied add-exp-log2.4

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

    7. Applied pow-exp2.3

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

    8. Applied add-exp-log2.3

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

    9. Applied prod-exp2.3

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

    10. Applied div-exp2.3

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

    11. Applied simplify0.3

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

    13. Applied add-log-exp0.9

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

    14. Applied add-log-exp1.1

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

    15. Applied diff-log1.1

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

    16. Applied simplify0.3

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

    if 10.110698407299443 < (exp (+ (* c_n (- (log (- 1 (/ 1 (+ 1 (exp (- s)))))) (log (- 1 (/ 1 (+ (exp (- t)) 1)))))) (log (pow (/ (+ 1 (exp (- t))) (+ (exp (- s)) 1)) c_p))))

    1. Initial program 62.5

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

    2. Taylor expanded around 0 10.2

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

    3. Applied simplify10.2

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

Runtime

Time bar (total: 5.5m)Debug logProfile

herbie shell --seed 2018206 
(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))))