Average Error: 29.4 → 0.0
Time: 2.7m
Precision: 64
Internal Precision: 1344
\[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.00716280177925632:\\ \;\;\;\;\log \left(e^{\frac{2}{e^{-2 \cdot x} + 1}}\right) - 1\\ \mathbf{elif}\;x \le 0.006855606559151648:\\ \;\;\;\;\left(\frac{2}{15} \cdot {x}^{5} + x\right) - {x}^{3} \cdot \frac{1}{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{2}{e^{-2 \cdot x} + 1}\right)}^{3} - {1}^{3}}{\frac{2}{e^{-2 \cdot x} + 1} \cdot \frac{2}{e^{-2 \cdot x} + 1} + \left(1 + \frac{2}{e^{-2 \cdot x} + 1}\right)}\\ \end{array}\]

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 < -0.00716280177925632

    1. Initial program 0.0

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

    3. Applied add-log-exp0.0

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

    if -0.00716280177925632 < x < 0.006855606559151648

    1. Initial program 59.2

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

    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 0.006855606559151648 < x

    1. Initial program 0.0

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

    3. 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

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.00716280177925632:\\ \;\;\;\;\log \left(e^{\frac{2}{e^{-2 \cdot x} + 1}}\right) - 1\\ \mathbf{elif}\;x \le 0.006855606559151648:\\ \;\;\;\;\left(\frac{2}{15} \cdot {x}^{5} + x\right) - {x}^{3} \cdot \frac{1}{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{2}{e^{-2 \cdot x} + 1}\right)}^{3} - {1}^{3}}{\frac{2}{e^{-2 \cdot x} + 1} \cdot \frac{2}{e^{-2 \cdot x} + 1} + \left(1 + \frac{2}{e^{-2 \cdot x} + 1}\right)}\\ \end{array}\]

Runtime

Time bar (total: 2.7m)Debug logProfile

herbie shell --seed 2018220 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  (- (/ 2 (+ 1 (exp (* -2 x)))) 1))