Average Error: 4.0 → 0.6
Time: 2.8m
Precision: 64
Internal Precision: 2368
\[\frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c_n}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c_p} \cdot {\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c_n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;s \le -754458506.6971726:\\ \;\;\;\;e^{(c_n \cdot \left(\log_* (1 + \frac{-1}{1 + e^{-s}}) - \log_* (1 + \frac{-1}{e^{-t} + 1})\right) + \left(c_p \cdot \left(\log_* (1 + e^{-t}) - \log_* (1 + e^{-s})\right)\right))_*}\\ \mathbf{else}:\\ \;\;\;\;e^{(c_n \cdot \left(\log_* (1 + \frac{-1}{1 + e^{-s}}) - \log_* (1 + \frac{-1}{e^{-t} + 1})\right) + \left((\left((t \cdot \frac{1}{8} + \frac{-1}{2})_*\right) \cdot t + \left(s \cdot \frac{1}{2}\right))_* \cdot c_p\right))_*}\\ \end{array}\]

Error

Bits error versus c_p

Bits error versus c_n

Bits error versus t

Bits error versus s

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 add-exp-log3.7

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

    5. Applied add-exp-log3.7

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

    6. Applied div-exp3.7

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

    7. Applied add-exp-log3.7

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

    8. Applied prod-exp3.7

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

    9. Simplified1.4

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

    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 add-exp-log4.1

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

    5. Applied add-exp-log4.0

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

    6. Applied div-exp4.0

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

    7. Applied add-exp-log4.0

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

    8. Applied prod-exp4.0

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

    9. Simplified2.0

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

    10. Taylor expanded around 0 0.4

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

    11. Simplified0.4

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

  4. Final simplification0.6

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

Runtime

Time bar (total: 2.8m)Debug logProfile

herbie shell --seed 2018248 +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))))