Average Error: 29.4 → 0.1
Time: 19.3s
Precision: 64
Internal Precision: 1344
\[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{2}{1 + e^{-2 \cdot x}} - 1 \le -1.0851674992178994 \cdot 10^{-07}:\\ \;\;\;\;\frac{\frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{2}{1 + e^{-2 \cdot x}} - 1}{\frac{2}{1 + e^{-2 \cdot x}} + 1}\\ \mathbf{elif}\;\frac{2}{1 + e^{-2 \cdot x}} - 1 \le 5.9324951686377854 \cdot 10^{-08}:\\ \;\;\;\;\left(x + {x}^{5} \cdot \frac{2}{15}\right) - {x}^{3} \cdot \frac{1}{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{2}{1 + e^{-2 \cdot x}} - 1}{\frac{2}{1 + e^{-2 \cdot x}} + 1}\\ \end{array}\]

Error

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Bits error versus y

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Results

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Derivation

  1. Split input into 2 regimes

  2. if (- (/ 2 (+ 1 (exp (* -2 x)))) 1) < -1.0851674992178994e-07 or 5.9324951686377854e-08 < (- (/ 2 (+ 1 (exp (* -2 x)))) 1)

    1. Initial program 0.2

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

    3. Applied flip--0.2

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

    if -1.0851674992178994e-07 < (- (/ 2 (+ 1 (exp (* -2 x)))) 1) < 5.9324951686377854e-08

    1. Initial program 59.8

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

    2. Taylor expanded around 0 0

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{1 + e^{-2 \cdot x}} - 1 \le -1.0851674992178994 \cdot 10^{-07}:\\ \;\;\;\;\frac{\frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{2}{1 + e^{-2 \cdot x}} - 1}{\frac{2}{1 + e^{-2 \cdot x}} + 1}\\ \mathbf{elif}\;\frac{2}{1 + e^{-2 \cdot x}} - 1 \le 5.9324951686377854 \cdot 10^{-08}:\\ \;\;\;\;\left(x + {x}^{5} \cdot \frac{2}{15}\right) - {x}^{3} \cdot \frac{1}{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{2}{1 + e^{-2 \cdot x}} - 1}{\frac{2}{1 + e^{-2 \cdot x}} + 1}\\ \end{array}\]

Runtime

Time bar (total: 19.3s)Debug logProfile

herbie shell --seed 2018216 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  (- (/ 2 (+ 1 (exp (* -2 x)))) 1))