Average Error: 29.4 → 0.0
Time: 12.7s
Precision: 64
Internal Precision: 1344
\[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.00845879631570754:\\ \;\;\;\;\sqrt[3]{\frac{2}{e^{-2 \cdot x} + 1} - 1} \cdot \left(\sqrt[3]{\frac{2}{e^{-2 \cdot x} + 1} - 1} \cdot \sqrt[3]{\sqrt[3]{\left(\frac{2}{e^{-2 \cdot x} + 1} - 1\right) \cdot \left(\left(\frac{2}{e^{-2 \cdot x} + 1} - 1\right) \cdot \left(\frac{2}{e^{-2 \cdot x} + 1} - 1\right)\right)}}\right)\\ \mathbf{elif}\;x \le 0.006796850790413319:\\ \;\;\;\;\left({x}^{5} \cdot \frac{2}{15} + x\right) - {x}^{3} \cdot \frac{1}{3}\\ \mathbf{else}:\\ \;\;\;\;(e^{\frac{\log 2 \cdot \log 2 - \log_* (1 + e^{-2 \cdot x}) \cdot \log_* (1 + e^{-2 \cdot x})}{\log_* (1 + e^{-2 \cdot x}) + \log 2}} - 1)^*\\ \end{array}\]

Error

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Bits error versus y

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Results

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Derivation

  1. Split input into 3 regimes

  2. if x < -0.00845879631570754

    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-cube-cbrt0.0

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

    6. Applied add-cbrt-cube0.0

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

    if -0.00845879631570754 < x < 0.006796850790413319

    1. Initial program 59.0

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

    2. Initial simplification59.0

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

    3. Taylor expanded around 0 0.0

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

    if 0.006796850790413319 < 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 flip--0.0

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.00845879631570754:\\ \;\;\;\;\sqrt[3]{\frac{2}{e^{-2 \cdot x} + 1} - 1} \cdot \left(\sqrt[3]{\frac{2}{e^{-2 \cdot x} + 1} - 1} \cdot \sqrt[3]{\sqrt[3]{\left(\frac{2}{e^{-2 \cdot x} + 1} - 1\right) \cdot \left(\left(\frac{2}{e^{-2 \cdot x} + 1} - 1\right) \cdot \left(\frac{2}{e^{-2 \cdot x} + 1} - 1\right)\right)}}\right)\\ \mathbf{elif}\;x \le 0.006796850790413319:\\ \;\;\;\;\left({x}^{5} \cdot \frac{2}{15} + x\right) - {x}^{3} \cdot \frac{1}{3}\\ \mathbf{else}:\\ \;\;\;\;(e^{\frac{\log 2 \cdot \log 2 - \log_* (1 + e^{-2 \cdot x}) \cdot \log_* (1 + e^{-2 \cdot x})}{\log_* (1 + e^{-2 \cdot x}) + \log 2}} - 1)^*\\ \end{array}\]

Runtime

Time bar (total: 12.7s)Debug logProfile

BaselineHerbieOracleSpan%
Regimes29.40.00.029.4100%
herbie shell --seed 2018339 +o rules:numerics
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  (- (/ 2 (+ 1 (exp (* -2 x)))) 1))