Average Error: 29.1 → 0.2
Time: 4.3m
Precision: 64
Internal Precision: 1344
\[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{2}{1 + e^{-2 \cdot x}} - 1 \le -1.873316959395914 \cdot 10^{-08}:\\ \;\;\;\;(e^{\log 2 - \log_* (1 + e^{-2 \cdot x})} - 1)^*\\ \mathbf{elif}\;\frac{2}{1 + e^{-2 \cdot x}} - 1 \le 3.509697273692408 \cdot 10^{-09}:\\ \;\;\;\;\left(\frac{2}{15} \cdot {x}^{5} + x\right) - \frac{1}{3} \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;(e^{\frac{(\left(\log 2 \cdot \log 2\right) \cdot \left(\log 2\right) + \left(-{\left(\log_* (1 + e^{-2 \cdot x})\right)}^{3}\right))_* + (\left(-{\left(\log_* (1 + e^{-2 \cdot x})\right)}^{\left(\frac{1}{2} + 1\right)}\right) \cdot \left({\left(\log_* (1 + e^{-2 \cdot x})\right)}^{\left(\frac{1}{2} + 1\right)}\right) + \left({\left(\log_* (1 + e^{-2 \cdot x})\right)}^{3}\right))_*}{\log 2 \cdot \log 2 + \left(\log_* (1 + e^{-2 \cdot x}) \cdot \log_* (1 + e^{-2 \cdot x}) + \log 2 \cdot \log_* (1 + e^{-2 \cdot x})\right)}} - 1)^*\\ \end{array}\]

Error

Bits error versus x

Bits error versus y

Derivation

  1. Split input into 3 regimes

  2. if (- (/ 2 (+ 1 (exp (* -2 x)))) 1) < -1.873316959395914e-08

    1. Initial program 0.3

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
    2. Using strategy rm

    3. Applied add-exp-log0.3

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

    4. Applied expm1-def0.3

      \[\leadsto \color{blue}{(e^{\log \left(\frac{2}{1 + e^{-2 \cdot x}}\right)} - 1)^*}\]

    5. Simplified0.2

      \[\leadsto (e^{\color{blue}{\log 2 - \log_* (1 + e^{-2 \cdot x})}} - 1)^*\]

    if -1.873316959395914e-08 < (- (/ 2 (+ 1 (exp (* -2 x)))) 1) < 3.509697273692408e-09

    1. Initial program 59.9

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1\]

    2. Taylor expanded around 0 0

      \[\leadsto \color{blue}{\left(\frac{2}{15} \cdot {x}^{5} + x\right) - \frac{1}{3} \cdot {x}^{3}}\]

    if 3.509697273692408e-09 < (- (/ 2 (+ 1 (exp (* -2 x)))) 1)

    1. Initial program 0.4

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
    2. Using strategy rm

    3. Applied add-exp-log0.4

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

    4. Applied expm1-def0.4

      \[\leadsto \color{blue}{(e^{\log \left(\frac{2}{1 + e^{-2 \cdot x}}\right)} - 1)^*}\]

    5. Simplified0.3

      \[\leadsto (e^{\color{blue}{\log 2 - \log_* (1 + e^{-2 \cdot x})}} - 1)^*\]
    6. Using strategy rm

    7. Applied flip3--0.3

      \[\leadsto (e^{\color{blue}{\frac{{\left(\log 2\right)}^{3} - {\left(\log_* (1 + e^{-2 \cdot x})\right)}^{3}}{\log 2 \cdot \log 2 + \left(\log_* (1 + e^{-2 \cdot x}) \cdot \log_* (1 + e^{-2 \cdot x}) + \log 2 \cdot \log_* (1 + e^{-2 \cdot x})\right)}}} - 1)^*\]
    8. Using strategy rm

    9. Applied add-sqr-sqrt0.4

      \[\leadsto (e^{\frac{{\left(\log 2\right)}^{3} - {\color{blue}{\left(\sqrt{\log_* (1 + e^{-2 \cdot x})} \cdot \sqrt{\log_* (1 + e^{-2 \cdot x})}\right)}}^{3}}{\log 2 \cdot \log 2 + \left(\log_* (1 + e^{-2 \cdot x}) \cdot \log_* (1 + e^{-2 \cdot x}) + \log 2 \cdot \log_* (1 + e^{-2 \cdot x})\right)}} - 1)^*\]

    10. Applied unpow-prod-down0.4

      \[\leadsto (e^{\frac{{\left(\log 2\right)}^{3} - \color{blue}{{\left(\sqrt{\log_* (1 + e^{-2 \cdot x})}\right)}^{3} \cdot {\left(\sqrt{\log_* (1 + e^{-2 \cdot x})}\right)}^{3}}}{\log 2 \cdot \log 2 + \left(\log_* (1 + e^{-2 \cdot x}) \cdot \log_* (1 + e^{-2 \cdot x}) + \log 2 \cdot \log_* (1 + e^{-2 \cdot x})\right)}} - 1)^*\]

    11. Applied add-cube-cbrt4.6

      \[\leadsto (e^{\frac{{\color{blue}{\left(\left(\sqrt[3]{\log 2} \cdot \sqrt[3]{\log 2}\right) \cdot \sqrt[3]{\log 2}\right)}}^{3} - {\left(\sqrt{\log_* (1 + e^{-2 \cdot x})}\right)}^{3} \cdot {\left(\sqrt{\log_* (1 + e^{-2 \cdot x})}\right)}^{3}}{\log 2 \cdot \log 2 + \left(\log_* (1 + e^{-2 \cdot x}) \cdot \log_* (1 + e^{-2 \cdot x}) + \log 2 \cdot \log_* (1 + e^{-2 \cdot x})\right)}} - 1)^*\]

    12. Applied unpow-prod-down4.7

      \[\leadsto (e^{\frac{\color{blue}{{\left(\sqrt[3]{\log 2} \cdot \sqrt[3]{\log 2}\right)}^{3} \cdot {\left(\sqrt[3]{\log 2}\right)}^{3}} - {\left(\sqrt{\log_* (1 + e^{-2 \cdot x})}\right)}^{3} \cdot {\left(\sqrt{\log_* (1 + e^{-2 \cdot x})}\right)}^{3}}{\log 2 \cdot \log 2 + \left(\log_* (1 + e^{-2 \cdot x}) \cdot \log_* (1 + e^{-2 \cdot x}) + \log 2 \cdot \log_* (1 + e^{-2 \cdot x})\right)}} - 1)^*\]

    13. Applied prod-diff4.7

      \[\leadsto (e^{\frac{\color{blue}{(\left({\left(\sqrt[3]{\log 2} \cdot \sqrt[3]{\log 2}\right)}^{3}\right) \cdot \left({\left(\sqrt[3]{\log 2}\right)}^{3}\right) + \left(-{\left(\sqrt{\log_* (1 + e^{-2 \cdot x})}\right)}^{3} \cdot {\left(\sqrt{\log_* (1 + e^{-2 \cdot x})}\right)}^{3}\right))_* + (\left(-{\left(\sqrt{\log_* (1 + e^{-2 \cdot x})}\right)}^{3}\right) \cdot \left({\left(\sqrt{\log_* (1 + e^{-2 \cdot x})}\right)}^{3}\right) + \left({\left(\sqrt{\log_* (1 + e^{-2 \cdot x})}\right)}^{3} \cdot {\left(\sqrt{\log_* (1 + e^{-2 \cdot x})}\right)}^{3}\right))_*}}{\log 2 \cdot \log 2 + \left(\log_* (1 + e^{-2 \cdot x}) \cdot \log_* (1 + e^{-2 \cdot x}) + \log 2 \cdot \log_* (1 + e^{-2 \cdot x})\right)}} - 1)^*\]

    14. Simplified0.3

      \[\leadsto (e^{\frac{\color{blue}{(\left(\log 2 \cdot \log 2\right) \cdot \left(\log 2\right) + \left(-{\left(\log_* (1 + e^{x \cdot -2})\right)}^{3}\right))_*} + (\left(-{\left(\sqrt{\log_* (1 + e^{-2 \cdot x})}\right)}^{3}\right) \cdot \left({\left(\sqrt{\log_* (1 + e^{-2 \cdot x})}\right)}^{3}\right) + \left({\left(\sqrt{\log_* (1 + e^{-2 \cdot x})}\right)}^{3} \cdot {\left(\sqrt{\log_* (1 + e^{-2 \cdot x})}\right)}^{3}\right))_*}{\log 2 \cdot \log 2 + \left(\log_* (1 + e^{-2 \cdot x}) \cdot \log_* (1 + e^{-2 \cdot x}) + \log 2 \cdot \log_* (1 + e^{-2 \cdot x})\right)}} - 1)^*\]

    15. Simplified0.4

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{1 + e^{-2 \cdot x}} - 1 \le -1.873316959395914 \cdot 10^{-08}:\\ \;\;\;\;(e^{\log 2 - \log_* (1 + e^{-2 \cdot x})} - 1)^*\\ \mathbf{elif}\;\frac{2}{1 + e^{-2 \cdot x}} - 1 \le 3.509697273692408 \cdot 10^{-09}:\\ \;\;\;\;\left(\frac{2}{15} \cdot {x}^{5} + x\right) - \frac{1}{3} \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;(e^{\frac{(\left(\log 2 \cdot \log 2\right) \cdot \left(\log 2\right) + \left(-{\left(\log_* (1 + e^{-2 \cdot x})\right)}^{3}\right))_* + (\left(-{\left(\log_* (1 + e^{-2 \cdot x})\right)}^{\left(\frac{1}{2} + 1\right)}\right) \cdot \left({\left(\log_* (1 + e^{-2 \cdot x})\right)}^{\left(\frac{1}{2} + 1\right)}\right) + \left({\left(\log_* (1 + e^{-2 \cdot x})\right)}^{3}\right))_*}{\log 2 \cdot \log 2 + \left(\log_* (1 + e^{-2 \cdot x}) \cdot \log_* (1 + e^{-2 \cdot x}) + \log 2 \cdot \log_* (1 + e^{-2 \cdot x})\right)}} - 1)^*\\ \end{array}\]

Runtime

Time bar (total: 4.3m)Debug logProfile

herbie shell --seed 2018214 +o rules:numerics
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  (- (/ 2 (+ 1 (exp (* -2 x)))) 1))