Average Error: 4.0 → 0.8
Time: 2.1m
Precision: 64
Internal Precision: 1600
\[\frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c_n}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c_p} \cdot {\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c_n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;s \le -204638810005.92273 \lor \neg \left(s \le 487.4749094465218\right):\\ \;\;\;\;\frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c_p}}{(c_p \cdot \left((\frac{1}{2} \cdot t + \left(\log \frac{1}{2}\right))_*\right) + 1)_*} \cdot e^{c_n \cdot \left(\log_* (1 + \frac{-1}{1 + e^{-s}}) - \log_* (1 + \frac{-1}{e^{-t} + 1})\right)}\\ \mathbf{else}:\\ \;\;\;\;(\left(s - t\right) \cdot \left(\frac{1}{2} \cdot c_p\right) + 1)_* \cdot e^{c_n \cdot (s \cdot \left((\frac{-1}{8} \cdot s + \frac{-1}{2})_*\right) + \left(\frac{1}{2} \cdot t\right))_*}\\ \end{array}\]

Error

Bits error versus c_p

Bits error versus c_n

Bits error versus t

Bits error versus s

Target

Original4.0
Target2.1
Herbie0.8
\[{\left(\frac{1 + e^{-t}}{1 + e^{-s}}\right)}^{c_p} \cdot {\left(\frac{1 + e^{t}}{1 + e^{s}}\right)}^{c_n}\]

Derivation

  1. Split input into 2 regimes

  2. if s < -204638810005.92273 or 487.4749094465218 < s

    1. Initial program 3.6

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

    2. Initial simplification3.6

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

    4. Applied add-exp-log3.6

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

    5. Applied add-exp-log3.6

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

    6. Applied div-exp3.6

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

    7. Simplified3.3

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

    8. Taylor expanded around 0 0.9

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

    9. Simplified0.9

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

    if -204638810005.92273 < s < 487.4749094465218

    1. Initial program 4.4

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

    2. Initial simplification4.4

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

    4. Applied add-exp-log4.4

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

    5. Applied add-exp-log4.4

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

    6. Applied div-exp4.4

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

    7. Simplified3.5

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

    8. Taylor expanded around 0 3.2

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

    9. Simplified3.2

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

    10. Taylor expanded around 0 0.6

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

    11. Simplified0.6

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

  4. Final simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;s \le -204638810005.92273 \lor \neg \left(s \le 487.4749094465218\right):\\ \;\;\;\;\frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c_p}}{(c_p \cdot \left((\frac{1}{2} \cdot t + \left(\log \frac{1}{2}\right))_*\right) + 1)_*} \cdot e^{c_n \cdot \left(\log_* (1 + \frac{-1}{1 + e^{-s}}) - \log_* (1 + \frac{-1}{e^{-t} + 1})\right)}\\ \mathbf{else}:\\ \;\;\;\;(\left(s - t\right) \cdot \left(\frac{1}{2} \cdot c_p\right) + 1)_* \cdot e^{c_n \cdot (s \cdot \left((\frac{-1}{8} \cdot s + \frac{-1}{2})_*\right) + \left(\frac{1}{2} \cdot t\right))_*}\\ \end{array}\]

Runtime

Time bar (total: 2.1m)Debug logProfile

herbie shell --seed 2018258 +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))))