Average Error: 28.6 → 0.0
Time: 12.0s
Precision: 64
Internal Precision: 1344
\[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.0076802771654724:\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} - 1\\ \mathbf{elif}\;x \le 0.007723684741088805:\\ \;\;\;\;\left(\frac{2}{15} \cdot {x}^{5} + x\right) - {x}^{3} \cdot \frac{1}{3}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{\sqrt{\frac{2}{1 + e^{-2 \cdot x}} - 1}} \cdot \sqrt[3]{\sqrt{\frac{2}{1 + e^{-2 \cdot x}} - 1}}\right) \cdot \left(\sqrt[3]{\sqrt{\frac{2}{1 + e^{-2 \cdot x}} - 1}} \cdot \sqrt{\frac{2}{1 + e^{-2 \cdot x}} - 1}\right)\\ \end{array}\]

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 < -0.0076802771654724

    1. Initial program 0.0

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

    2. Taylor expanded around inf 0.0

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

    if -0.0076802771654724 < x < 0.007723684741088805

    1. Initial program 58.9

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

    2. 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 0.007723684741088805 < x

    1. Initial program 0.0

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

    2. Taylor expanded around inf 0.0

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

    4. Applied add-sqr-sqrt0.0

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

    6. Applied add-cube-cbrt0.0

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

    7. Applied associate-*l*0.0

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.0076802771654724:\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} - 1\\ \mathbf{elif}\;x \le 0.007723684741088805:\\ \;\;\;\;\left(\frac{2}{15} \cdot {x}^{5} + x\right) - {x}^{3} \cdot \frac{1}{3}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{\sqrt{\frac{2}{1 + e^{-2 \cdot x}} - 1}} \cdot \sqrt[3]{\sqrt{\frac{2}{1 + e^{-2 \cdot x}} - 1}}\right) \cdot \left(\sqrt[3]{\sqrt{\frac{2}{1 + e^{-2 \cdot x}} - 1}} \cdot \sqrt{\frac{2}{1 + e^{-2 \cdot x}} - 1}\right)\\ \end{array}\]

Runtime

Time bar (total: 12.0s)Debug logProfile

BaselineHerbieOracleSpan%
Regimes28.60.00.028.6100%
herbie shell --seed 2018286 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  (- (/ 2 (+ 1 (exp (* -2 x)))) 1))