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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -0.0074891919992706064 \lor \neg \left(x \le 0.007577753449784084\right):\\ \;\;\;\;\frac{{\left(\frac{2}{e^{-2 \cdot x} + 1}\right)}^{3} - {1}^{3}}{\frac{2}{e^{-2 \cdot x} + 1} \cdot \frac{2}{e^{-2 \cdot x} + 1} + \left(1 + \frac{2}{e^{-2 \cdot x} + 1}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left({x}^{5} \cdot \frac{2}{15} + x\right) - \frac{1}{3} \cdot {x}^{3}\\ \end{array}\]

Derivation

Split input into 2 regimes
if x < -0.0074891919992706064 or 0.007577753449784084 < x
1. Initial program 0.0
  \[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
2. Using strategy rm
3. Applied flip3--0.0
  \[\leadsto \color{blue}{\frac{{\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{3} - {1}^{3}}{\frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{2}{1 + e^{-2 \cdot x}} + \left(1 \cdot 1 + \frac{2}{1 + e^{-2 \cdot x}} \cdot 1\right)}}\]
if -0.0074891919992706064 < x < 0.007577753449784084
1. Initial program 58.9
  \[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
2. Taylor expanded around 0 0.0
  \[\leadsto \color{blue}{\left(x + \frac{2}{15} \cdot {x}^{5}\right) - \frac{1}{3} \cdot {x}^{3}}\]
Recombined 2 regimes into one program.
Final simplification0.0
\[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.0074891919992706064 \lor \neg \left(x \le 0.007577753449784084\right):\\ \;\;\;\;\frac{{\left(\frac{2}{e^{-2 \cdot x} + 1}\right)}^{3} - {1}^{3}}{\frac{2}{e^{-2 \cdot x} + 1} \cdot \frac{2}{e^{-2 \cdot x} + 1} + \left(1 + \frac{2}{e^{-2 \cdot x} + 1}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left({x}^{5} \cdot \frac{2}{15} + x\right) - \frac{1}{3} \cdot {x}^{3}\\ \end{array}\]

Runtime

herbie shell --seed 2018225 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  (- (/ 2 (+ 1 (exp (* -2 x)))) 1))

Error

Try it out

Derivation

`if x < -0.0074891919992706064 or 0.007577753449784084 < x`

`if -0.0074891919992706064 < x < 0.007577753449784084`

Runtime

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