Average Error: 28.6 → 0.0
Time: 37.0s
Precision: 64
Internal Precision: 128
\[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.006666873519218567 \lor \neg \left(x \le 0.006774998496990162\right):\\ \;\;\;\;(e^{\sqrt[3]{\left(\left(\log 2 - \log_* (1 + e^{-2 \cdot x})\right) \cdot \left(\log 2 - \log_* (1 + e^{-2 \cdot x})\right)\right) \cdot \left(\log 2 - \log \left(e^{\log_* (1 + e^{-2 \cdot x})}\right)\right)}} - 1)^*\\ \mathbf{else}:\\ \;\;\;\;\left({x}^{5} \cdot \frac{2}{15} + x\right) - \frac{1}{3} \cdot {x}^{3}\\ \end{array}\]

Error

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Bits error versus y

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Results

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Derivation

  1. Split input into 2 regimes

  2. if x < -0.006666873519218567 or 0.006774998496990162 < x

    1. Initial program 0.0

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

    2. Initial simplification0.0

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

    4. Applied add-exp-log0.0

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

    5. Applied expm1-def0.0

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

    6. Simplified0.0

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

    8. Applied add-cbrt-cube0.0

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

    10. Applied add-log-exp0.0

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

    if -0.006666873519218567 < x < 0.006774998496990162

    1. Initial program 58.7

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

    2. Initial simplification58.7

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

    3. Taylor expanded around 0 0.0

      \[\leadsto \color{blue}{\left(x + \frac{2}{15} \cdot {x}^{5}\right) - \frac{1}{3} \cdot {x}^{3}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.006666873519218567 \lor \neg \left(x \le 0.006774998496990162\right):\\ \;\;\;\;(e^{\sqrt[3]{\left(\left(\log 2 - \log_* (1 + e^{-2 \cdot x})\right) \cdot \left(\log 2 - \log_* (1 + e^{-2 \cdot x})\right)\right) \cdot \left(\log 2 - \log \left(e^{\log_* (1 + e^{-2 \cdot x})}\right)\right)}} - 1)^*\\ \mathbf{else}:\\ \;\;\;\;\left({x}^{5} \cdot \frac{2}{15} + x\right) - \frac{1}{3} \cdot {x}^{3}\\ \end{array}\]

Runtime

Time bar (total: 37.0s)Debug logProfile

herbie shell --seed 2018296 +o rules:numerics
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  (- (/ 2 (+ 1 (exp (* -2 x)))) 1))