Average Error: 29.0 → 0.0
Time: 37.4s
Precision: 64
Internal Precision: 576
\[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.0006290873782508793 \lor \neg \left(x \le 0.0009486447256490275\right):\\ \;\;\;\;\sqrt[3]{\left(\sqrt{\frac{2}{e^{-2 \cdot x} + 1}} \cdot \sqrt{\frac{2}{e^{-2 \cdot x} + 1}}\right) \cdot \left(\left(\sqrt{\frac{2}{e^{-2 \cdot x} + 1}} \cdot \sqrt{\frac{2}{e^{-2 \cdot x} + 1}}\right) \cdot \left(\sqrt{\frac{2}{e^{-2 \cdot x} + 1}} \cdot \sqrt{\frac{2}{e^{-2 \cdot x} + 1}}\right)\right)} - 1\\ \mathbf{else}:\\ \;\;\;\;x - \left(x \cdot \frac{1}{3}\right) \cdot \left(x \cdot x\right)\\ \end{array}\]

Error

Bits error versus x

Bits error versus y

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Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if x < -0.0006290873782508793 or 0.0009486447256490275 < x

    1. Initial program 0.0

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

    3. Applied add-sqr-sqrt0.8

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

    5. Applied add-cbrt-cube0.0

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

    if -0.0006290873782508793 < x < 0.0009486447256490275

    1. Initial program 59.1

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

    3. Applied add-sqr-sqrt59.1

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

    5. Applied add-cbrt-cube59.1

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

    6. Taylor expanded around 0 60.2

      \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \left({\left(\sqrt{2}\right)}^{2} \cdot x\right) + \frac{1}{2} \cdot {\left(\sqrt{2}\right)}^{2}\right) - \left(\frac{1}{6} \cdot \left({\left(\sqrt{2}\right)}^{2} \cdot {x}^{3}\right) + 1\right)}\]

    7. Simplified0.0

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.0006290873782508793 \lor \neg \left(x \le 0.0009486447256490275\right):\\ \;\;\;\;\sqrt[3]{\left(\sqrt{\frac{2}{e^{-2 \cdot x} + 1}} \cdot \sqrt{\frac{2}{e^{-2 \cdot x} + 1}}\right) \cdot \left(\left(\sqrt{\frac{2}{e^{-2 \cdot x} + 1}} \cdot \sqrt{\frac{2}{e^{-2 \cdot x} + 1}}\right) \cdot \left(\sqrt{\frac{2}{e^{-2 \cdot x} + 1}} \cdot \sqrt{\frac{2}{e^{-2 \cdot x} + 1}}\right)\right)} - 1\\ \mathbf{else}:\\ \;\;\;\;x - \left(x \cdot \frac{1}{3}\right) \cdot \left(x \cdot x\right)\\ \end{array}\]

Runtime

Time bar (total: 37.4s)Debug logProfile

herbie shell --seed 2018256 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  (- (/ 2 (+ 1 (exp (* -2 x)))) 1))