Average Error: 29.8 → 0.0
Time: 4.2m
Precision: 64
Internal Precision: 1344
\[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.007037697980788856:\\ \;\;\;\;\left(\sqrt{\frac{2}{e^{-2 \cdot x} + 1}} + 1\right) \cdot \left(\sqrt{\sqrt[3]{\frac{2}{e^{-2 \cdot x} + 1} \cdot \left(\frac{2}{e^{-2 \cdot x} + 1} \cdot \frac{2}{e^{-2 \cdot x} + 1}\right)}} - 1\right)\\ \mathbf{elif}\;x \le 0.007019152014457255:\\ \;\;\;\;\left({x}^{5} \cdot \frac{2}{15} + x\right) - {x}^{3} \cdot \frac{1}{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{2}{e^{-2 \cdot x} + 1}\right)}^{3} - {1}^{3}}{\left(\frac{2}{e^{-2 \cdot x} + 1} + 1\right) + \frac{2}{e^{-2 \cdot x} + 1} \cdot \frac{2}{e^{-2 \cdot x} + 1}}\\ \end{array}\]

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 < -0.007037697980788856

    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-cbrt-cube0.0

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

    6. Applied add-sqr-sqrt0.0

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

    7. Applied difference-of-sqr-10.0

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

    8. Simplified0.0

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

    if -0.007037697980788856 < x < 0.007019152014457255

    1. Initial program 59.2

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

    2. Initial simplification59.2

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

    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 0.007019152014457255 < 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 flip3--0.0

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.007037697980788856:\\ \;\;\;\;\left(\sqrt{\frac{2}{e^{-2 \cdot x} + 1}} + 1\right) \cdot \left(\sqrt{\sqrt[3]{\frac{2}{e^{-2 \cdot x} + 1} \cdot \left(\frac{2}{e^{-2 \cdot x} + 1} \cdot \frac{2}{e^{-2 \cdot x} + 1}\right)}} - 1\right)\\ \mathbf{elif}\;x \le 0.007019152014457255:\\ \;\;\;\;\left({x}^{5} \cdot \frac{2}{15} + x\right) - {x}^{3} \cdot \frac{1}{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{2}{e^{-2 \cdot x} + 1}\right)}^{3} - {1}^{3}}{\left(\frac{2}{e^{-2 \cdot x} + 1} + 1\right) + \frac{2}{e^{-2 \cdot x} + 1} \cdot \frac{2}{e^{-2 \cdot x} + 1}}\\ \end{array}\]

Runtime

Time bar (total: 4.2m)Debug logProfile

herbie shell --seed 2018230 +o rules:numerics
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  (- (/ 2 (+ 1 (exp (* -2 x)))) 1))