Average Error: 44.6 → 9.8
Time: 2.3m
Precision: 64
Internal precision: 128
\[\frac{{\left(\frac1{1 + e^{-s}}\right)}^{c_p} \cdot {\left(1 - \frac1{1 + e^{-s}}\right)}^{c_n}}{{\left(\frac1{1 + e^{-t}}\right)}^{c_p} \cdot {\left(1 - \frac1{1 + e^{-t}}\right)}^{c_n}}\]
⬇
\[\begin{array}{l} \mathbf{if}\;s \le -6.400609471484758:\\ \;\;\;\;\frac{{\left(\frac1{e^{-s} + 1}\right)}^{c_p}}{1 + \left(\frac{1}{2} \cdot t + \log \frac{1}{2}\right) \cdot c_p} \cdot \frac{{\left(1 - \frac1{e^{-s} + 1}\right)}^{c_n}}{{\left(1 - \frac1{1 + e^{-t}}\right)}^{c_n}}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(\log \left(1 - \frac1{e^{-s} + 1}\right) - \log \left(1 - \frac1{1 + e^{-t}}\right)\right) \cdot c_n - \frac{{\left(\log \left(\frac1{1 + e^{-t}}\right)\right)}^2 - {\left(\log \left(\frac1{e^{-s} + 1}\right)\right)}^2}{\log \left(\frac1{1 + e^{-t}}\right) + \log \left(\frac1{e^{-s} + 1}\right)} \cdot c_p}\\ \end{array}\]

Error

Bits error versus c_p

Bits error versus c_n

Bits error versus t

Bits error versus s

Target

Original44.6
Comparison27.7
Herbie9.8
\[ {\left(\frac{1 + e^{-t}}{1 + e^{-s}}\right)}^{c_p} \cdot {\left(\frac{1 + e^{t}}{1 + e^{s}}\right)}^{c_n} \]

Derivation

  1. Split input into 2 regimes.

  2. if s < -6.400609471484758

    1. Initial program 47.5

      \[\frac{{\left(\frac1{1 + e^{-s}}\right)}^{c_p} \cdot {\left(1 - \frac1{1 + e^{-s}}\right)}^{c_n}}{{\left(\frac1{1 + e^{-t}}\right)}^{c_p} \cdot {\left(1 - \frac1{1 + e^{-t}}\right)}^{c_n}}\]

    2. Applied taylor 1.2

      \[\leadsto \frac{{\left(\frac1{1 + e^{-s}}\right)}^{c_p} \cdot {\left(1 - \frac1{1 + e^{-s}}\right)}^{c_n}}{\left(\frac{1}{2} \cdot \left(t \cdot c_p\right) + \left(1 + c_p \cdot \log \frac{1}{2}\right)\right) \cdot {\left(1 - \frac1{1 + e^{-t}}\right)}^{c_n}}\]

    3. Taylor expanded around 0 1.2

      \[\leadsto \frac{{\left(\frac1{1 + e^{-s}}\right)}^{c_p} \cdot {\left(1 - \frac1{1 + e^{-s}}\right)}^{c_n}}{\color{blue}{\left(\frac{1}{2} \cdot \left(t \cdot c_p\right) + \left(1 + c_p \cdot \log \frac{1}{2}\right)\right)} \cdot {\left(1 - \frac1{1 + e^{-t}}\right)}^{c_n}}\]

    4. Applied simplify 1.2

      \[\leadsto \color{blue}{\frac{{\left(\frac1{e^{-s} + 1}\right)}^{c_p}}{1 + \left(\frac{1}{2} \cdot t + \log \frac{1}{2}\right) \cdot c_p} \cdot \frac{{\left(1 - \frac1{e^{-s} + 1}\right)}^{c_n}}{{\left(1 - \frac1{1 + e^{-t}}\right)}^{c_n}}}\]

    if -6.400609471484758 < s

    1. Initial program 43.4

      \[\frac{{\left(\frac1{1 + e^{-s}}\right)}^{c_p} \cdot {\left(1 - \frac1{1 + e^{-s}}\right)}^{c_n}}{{\left(\frac1{1 + e^{-t}}\right)}^{c_p} \cdot {\left(1 - \frac1{1 + e^{-t}}\right)}^{c_n}}\]
    2. Using strategy rm

    3. Applied add-exp-log 43.4

      \[\leadsto \frac{{\left(\frac1{1 + e^{-s}}\right)}^{c_p} \cdot {\left(1 - \frac1{1 + e^{-s}}\right)}^{c_n}}{{\left(\frac1{1 + e^{-t}}\right)}^{c_p} \cdot {\color{blue}{\left(e^{\log \left(1 - \frac1{1 + e^{-t}}\right)}\right)}}^{c_n}}\]

    4. Applied pow-exp 43.5

      \[\leadsto \frac{{\left(\frac1{1 + e^{-s}}\right)}^{c_p} \cdot {\left(1 - \frac1{1 + e^{-s}}\right)}^{c_n}}{{\left(\frac1{1 + e^{-t}}\right)}^{c_p} \cdot \color{blue}{e^{\log \left(1 - \frac1{1 + e^{-t}}\right) \cdot c_n}}}\]

    5. Applied add-exp-log 43.5

      \[\leadsto \frac{{\left(\frac1{1 + e^{-s}}\right)}^{c_p} \cdot {\left(1 - \frac1{1 + e^{-s}}\right)}^{c_n}}{{\color{blue}{\left(e^{\log \left(\frac1{1 + e^{-t}}\right)}\right)}}^{c_p} \cdot e^{\log \left(1 - \frac1{1 + e^{-t}}\right) \cdot c_n}}\]

    6. Applied pow-exp 43.5

      \[\leadsto \frac{{\left(\frac1{1 + e^{-s}}\right)}^{c_p} \cdot {\left(1 - \frac1{1 + e^{-s}}\right)}^{c_n}}{\color{blue}{e^{\log \left(\frac1{1 + e^{-t}}\right) \cdot c_p}} \cdot e^{\log \left(1 - \frac1{1 + e^{-t}}\right) \cdot c_n}}\]

    7. Applied prod-exp 43.5

      \[\leadsto \frac{{\left(\frac1{1 + e^{-s}}\right)}^{c_p} \cdot {\left(1 - \frac1{1 + e^{-s}}\right)}^{c_n}}{\color{blue}{e^{\log \left(\frac1{1 + e^{-t}}\right) \cdot c_p + \log \left(1 - \frac1{1 + e^{-t}}\right) \cdot c_n}}}\]

    8. Applied add-exp-log 43.5

      \[\leadsto \frac{{\left(\frac1{1 + e^{-s}}\right)}^{c_p} \cdot {\color{blue}{\left(e^{\log \left(1 - \frac1{1 + e^{-s}}\right)}\right)}}^{c_n}}{e^{\log \left(\frac1{1 + e^{-t}}\right) \cdot c_p + \log \left(1 - \frac1{1 + e^{-t}}\right) \cdot c_n}}\]

    9. Applied pow-exp 43.5

      \[\leadsto \frac{{\left(\frac1{1 + e^{-s}}\right)}^{c_p} \cdot \color{blue}{e^{\log \left(1 - \frac1{1 + e^{-s}}\right) \cdot c_n}}}{e^{\log \left(\frac1{1 + e^{-t}}\right) \cdot c_p + \log \left(1 - \frac1{1 + e^{-t}}\right) \cdot c_n}}\]

    10. Applied add-exp-log 43.5

      \[\leadsto \frac{{\color{blue}{\left(e^{\log \left(\frac1{1 + e^{-s}}\right)}\right)}}^{c_p} \cdot e^{\log \left(1 - \frac1{1 + e^{-s}}\right) \cdot c_n}}{e^{\log \left(\frac1{1 + e^{-t}}\right) \cdot c_p + \log \left(1 - \frac1{1 + e^{-t}}\right) \cdot c_n}}\]

    11. Applied pow-exp 43.4

      \[\leadsto \frac{\color{blue}{e^{\log \left(\frac1{1 + e^{-s}}\right) \cdot c_p}} \cdot e^{\log \left(1 - \frac1{1 + e^{-s}}\right) \cdot c_n}}{e^{\log \left(\frac1{1 + e^{-t}}\right) \cdot c_p + \log \left(1 - \frac1{1 + e^{-t}}\right) \cdot c_n}}\]

    12. Applied prod-exp 43.4

      \[\leadsto \frac{\color{blue}{e^{\log \left(\frac1{1 + e^{-s}}\right) \cdot c_p + \log \left(1 - \frac1{1 + e^{-s}}\right) \cdot c_n}}}{e^{\log \left(\frac1{1 + e^{-t}}\right) \cdot c_p + \log \left(1 - \frac1{1 + e^{-t}}\right) \cdot c_n}}\]

    13. Applied div-exp 13.4

      \[\leadsto \color{blue}{e^{\left(\log \left(\frac1{1 + e^{-s}}\right) \cdot c_p + \log \left(1 - \frac1{1 + e^{-s}}\right) \cdot c_n\right) - \left(\log \left(\frac1{1 + e^{-t}}\right) \cdot c_p + \log \left(1 - \frac1{1 + e^{-t}}\right) \cdot c_n\right)}}\]

    14. Applied simplify 13.4

      \[\leadsto e^{\color{blue}{\left(\log \left(1 - \frac1{e^{-s} + 1}\right) - \log \left(1 - \frac1{1 + e^{-t}}\right)\right) \cdot c_n - \left(\log \left(\frac1{1 + e^{-t}}\right) - \log \left(\frac1{e^{-s} + 1}\right)\right) \cdot c_p}}\]
    15. Using strategy rm

    16. Applied flip-- 13.4

      \[\leadsto e^{\left(\log \left(1 - \frac1{e^{-s} + 1}\right) - \log \left(1 - \frac1{1 + e^{-t}}\right)\right) \cdot c_n - \color{blue}{\frac{{\left(\log \left(\frac1{1 + e^{-t}}\right)\right)}^2 - {\left(\log \left(\frac1{e^{-s} + 1}\right)\right)}^2}{\log \left(\frac1{1 + e^{-t}}\right) + \log \left(\frac1{e^{-s} + 1}\right)}} \cdot c_p}\]
  3. Recombined 2 regimes into one program.
  4. Removed slow pow expressions

Runtime

Time bar (total: 2.3m) Debug log

Please include this information when filing a bug report:

herbie --seed '#(344842374 1768352507 1215022084 3232447583 2417745204 3721860693)'
(FPCore (c_p c_n t s)
  :name "Harley's example"
  :pre (and (< 0 c_p) (< 0 c_n))

  :target
  (* (pow (/ (+ 1 (exp (- t))) (+ 1 (exp (- s)))) c_p) (pow (/ (+ 1 (exp t)) (+ 1 (exp s))) c_n))

  (/ (* (pow (/ (+ 1 (exp (- s)))) c_p) (pow (- 1 (/ (+ 1 (exp (- s))))) c_n)) (* (pow (/ (+ 1 (exp (- t)))) c_p) (pow (- 1 (/ (+ 1 (exp (- t))))) c_n))))