Average Error: 3.8 → 0.7
Time: 2.5m
Precision: 64
Internal Precision: 2112
\[\frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c_n}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c_p} \cdot {\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c_n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;s \le -748366900.8091519:\\ \;\;\;\;e^{(\left((s \cdot \left((\frac{-1}{8} \cdot s + \frac{-1}{2})_*\right) + \left(t \cdot \frac{1}{2}\right))_*\right) \cdot c_n + \left(c_p \cdot \left(\log_* (1 + e^{-t}) - \log_* (1 + e^{-s})\right)\right))_*}\\ \mathbf{else}:\\ \;\;\;\;e^{(\left(\log_* (1 + \frac{-1}{e^{-s} + 1}) - \log_* (1 + \frac{-1}{e^{-t} + 1})\right) \cdot c_n + \left(c_p \cdot (\left((t \cdot \frac{1}{8} + \frac{-1}{2})_*\right) \cdot t + \left(s \cdot \frac{1}{2}\right))_*\right))_*}\\ \end{array}\]

Error

Bits error versus c_p

Bits error versus c_n

Bits error versus t

Bits error versus s

Target

Original3.8
Target2.1
Herbie0.7
\[{\left(\frac{1 + e^{-t}}{1 + e^{-s}}\right)}^{c_p} \cdot {\left(\frac{1 + e^{t}}{1 + e^{s}}\right)}^{c_n}\]

Derivation

  1. Split input into 2 regimes

  2. if s < -748366900.8091519

    1. Initial program 3.5

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

    3. Applied pow-to-exp3.5

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

    4. Applied add-exp-log3.5

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

    5. Applied prod-exp3.5

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

    6. Applied add-exp-log3.5

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

    7. Applied add-exp-log3.5

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

    8. Applied rec-exp3.5

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

    9. Applied pow-exp3.5

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

    10. Applied prod-exp3.5

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

    11. Applied div-exp2.5

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

    12. Simplified1.6

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

    13. Taylor expanded around 0 1.5

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

    14. Simplified1.5

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

    if -748366900.8091519 < s

    1. Initial program 3.8

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

    3. Applied pow-to-exp3.8

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

    4. Applied add-exp-log3.9

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

    5. Applied prod-exp3.9

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

    6. Applied add-exp-log3.8

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

    7. Applied add-exp-log3.8

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

    8. Applied rec-exp3.8

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

    9. Applied pow-exp3.8

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

    10. Applied prod-exp3.8

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

    11. Applied div-exp3.5

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

    12. Simplified1.7

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

    13. Taylor expanded around 0 0.4

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

    14. Simplified0.4

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

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;s \le -748366900.8091519:\\ \;\;\;\;e^{(\left((s \cdot \left((\frac{-1}{8} \cdot s + \frac{-1}{2})_*\right) + \left(t \cdot \frac{1}{2}\right))_*\right) \cdot c_n + \left(c_p \cdot \left(\log_* (1 + e^{-t}) - \log_* (1 + e^{-s})\right)\right))_*}\\ \mathbf{else}:\\ \;\;\;\;e^{(\left(\log_* (1 + \frac{-1}{e^{-s} + 1}) - \log_* (1 + \frac{-1}{e^{-t} + 1})\right) \cdot c_n + \left(c_p \cdot (\left((t \cdot \frac{1}{8} + \frac{-1}{2})_*\right) \cdot t + \left(s \cdot \frac{1}{2}\right))_*\right))_*}\\ \end{array}\]

Runtime

Time bar (total: 2.5m)Debug logProfile

herbie shell --seed 2018254 +o rules:numerics
(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))))