Average Error: 29.5 → 0.0
Time: 2.5m
Precision: 64
Internal Precision: 1344
\[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.00744990134718954 \lor \neg \left(x \le 0.007411501740066481\right):\\ \;\;\;\;\frac{{\left(\log \left(e^{\frac{2}{e^{-2 \cdot x} + 1}}\right)\right)}^{3} - {1}^{3}}{\log \left(e^{\frac{2}{e^{-2 \cdot x} + 1}}\right) \cdot \log \left(e^{\frac{2}{e^{-2 \cdot x} + 1}}\right) + \left(1 + \log \left(e^{\frac{2}{e^{-2 \cdot x} + 1}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{2}{15} \cdot {x}^{5} - {x}^{3} \cdot \frac{1}{3}\right) + x\\ \end{array}\]

Error

Bits error versus x

Bits error versus y

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if x < -0.00744990134718954 or 0.007411501740066481 < x

    1. Initial program 0.0

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

    3. Applied add-log-exp0.0

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

    5. Applied flip3--0.0

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

    if -0.00744990134718954 < x < 0.007411501740066481

    1. Initial program 59.1

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

    2. Taylor expanded around 0 0.0

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

    4. Applied associate--l+0.0

      \[\leadsto \color{blue}{x + \left(\frac{2}{15} \cdot {x}^{5} - \frac{1}{3} \cdot {x}^{3}\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.00744990134718954 \lor \neg \left(x \le 0.007411501740066481\right):\\ \;\;\;\;\frac{{\left(\log \left(e^{\frac{2}{e^{-2 \cdot x} + 1}}\right)\right)}^{3} - {1}^{3}}{\log \left(e^{\frac{2}{e^{-2 \cdot x} + 1}}\right) \cdot \log \left(e^{\frac{2}{e^{-2 \cdot x} + 1}}\right) + \left(1 + \log \left(e^{\frac{2}{e^{-2 \cdot x} + 1}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{2}{15} \cdot {x}^{5} - {x}^{3} \cdot \frac{1}{3}\right) + x\\ \end{array}\]

Runtime

Time bar (total: 2.5m)Debug logProfile

herbie shell --seed 2018219 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  (- (/ 2 (+ 1 (exp (* -2 x)))) 1))