Average Error: 29.2 → 0.0
Time: 27.9s
Precision: 64
Internal Precision: 1344
\[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.007894789801005202:\\ \;\;\;\;\frac{\frac{2}{\sqrt{e^{-2 \cdot x} + 1}}}{\sqrt{e^{-2 \cdot x} + 1}} - 1\\ \mathbf{elif}\;x \le 0.007152501184390637:\\ \;\;\;\;\left(\frac{2}{15} \cdot {x}^{5} + x\right) - {x}^{3} \cdot \frac{1}{3}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\left(\sqrt[3]{\log \left(e^{\frac{2}{e^{-2 \cdot x} + 1} - 1}\right)} \cdot \sqrt[3]{\log \left(e^{\frac{2}{e^{-2 \cdot x} + 1} - 1}\right)}\right) \cdot \sqrt[3]{\log \left(e^{\frac{2}{e^{-2 \cdot x} + 1} - 1}\right)}} \cdot \left(\sqrt[3]{\log \left(e^{\frac{2}{e^{-2 \cdot x} + 1} - 1}\right)} \cdot \sqrt[3]{\log \left(e^{\frac{2}{e^{-2 \cdot x} + 1} - 1}\right)}\right)\\ \end{array}\]

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 < -0.007894789801005202

    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-sqr-sqrt0.0

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

    5. Applied associate-/r*0.0

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

    if -0.007894789801005202 < x < 0.007152501184390637

    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 0.007152501184390637 < 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-log-exp0.0

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

    6. Applied add-cube-cbrt0.0

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

    8. Applied add-cube-cbrt0.0

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.007894789801005202:\\ \;\;\;\;\frac{\frac{2}{\sqrt{e^{-2 \cdot x} + 1}}}{\sqrt{e^{-2 \cdot x} + 1}} - 1\\ \mathbf{elif}\;x \le 0.007152501184390637:\\ \;\;\;\;\left(\frac{2}{15} \cdot {x}^{5} + x\right) - {x}^{3} \cdot \frac{1}{3}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\left(\sqrt[3]{\log \left(e^{\frac{2}{e^{-2 \cdot x} + 1} - 1}\right)} \cdot \sqrt[3]{\log \left(e^{\frac{2}{e^{-2 \cdot x} + 1} - 1}\right)}\right) \cdot \sqrt[3]{\log \left(e^{\frac{2}{e^{-2 \cdot x} + 1} - 1}\right)}} \cdot \left(\sqrt[3]{\log \left(e^{\frac{2}{e^{-2 \cdot x} + 1} - 1}\right)} \cdot \sqrt[3]{\log \left(e^{\frac{2}{e^{-2 \cdot x} + 1} - 1}\right)}\right)\\ \end{array}\]

Runtime

Time bar (total: 27.9s)Debug logProfile

herbie shell --seed 2018242 +o rules:numerics
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  (- (/ 2 (+ 1 (exp (* -2 x)))) 1))