Error

Derivation

1. Split input into 3 regimes

2. if x < -0.007284216709661482

   1. Initial program 0.0

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

   2. Taylor expanded around -inf 0.0

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

   if -0.007284216709661482 < x < 0.005885669270588273

   1. Initial program 59.0

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

   2. Taylor expanded around -inf 59.0

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

   3. 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 0.005885669270588273 < x

   1. Initial program 0.0

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

   3. Applied flip3-+0.0

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

   4. Applied associate-/r/0.0

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

   5. Applied fma-neg0.0

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

   6. Simplified0.0

      \[\leadsto (\color{blue}{\left(\frac{2}{1 + e^{x \cdot -6}}\right)} \cdot \left(1 \cdot 1 + \left(e^{-2 \cdot x} \cdot e^{-2 \cdot x} - 1 \cdot e^{-2 \cdot x}\right)\right) + \left(-1\right))_*\]
   7. Using strategy rm

   8. Applied add-cbrt-cube0.0

      \[\leadsto \color{blue}{\sqrt[3]{\left((\left(\frac{2}{1 + e^{x \cdot -6}}\right) \cdot \left(1 \cdot 1 + \left(e^{-2 \cdot x} \cdot e^{-2 \cdot x} - 1 \cdot e^{-2 \cdot x}\right)\right) + \left(-1\right))_* \cdot (\left(\frac{2}{1 + e^{x \cdot -6}}\right) \cdot \left(1 \cdot 1 + \left(e^{-2 \cdot x} \cdot e^{-2 \cdot x} - 1 \cdot e^{-2 \cdot x}\right)\right) + \left(-1\right))_*\right) \cdot (\left(\frac{2}{1 + e^{x \cdot -6}}\right) \cdot \left(1 \cdot 1 + \left(e^{-2 \cdot x} \cdot e^{-2 \cdot x} - 1 \cdot e^{-2 \cdot x}\right)\right) + \left(-1\right))_*}}\]
3. Recombined 3 regimes into one program.

4. Final simplification0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.007284216709661482:\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} - 1\\ \mathbf{elif}\;x \le 0.005885669270588273:\\ \;\;\;\;\left(\frac{2}{15} \cdot {x}^{5} + x\right) - {x}^{3} \cdot \frac{1}{3}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\left((\left(\frac{2}{e^{-6 \cdot x} + 1}\right) \cdot \left(1 + \left(e^{-2 \cdot x} \cdot e^{-2 \cdot x} - e^{-2 \cdot x}\right)\right) + -1)_* \cdot (\left(\frac{2}{e^{-6 \cdot x} + 1}\right) \cdot \left(1 + \left(e^{-2 \cdot x} \cdot e^{-2 \cdot x} - e^{-2 \cdot x}\right)\right) + -1)_*\right) \cdot (\left(\frac{2}{e^{-6 \cdot x} + 1}\right) \cdot \left(1 + \left(e^{-2 \cdot x} \cdot e^{-2 \cdot x} - e^{-2 \cdot x}\right)\right) + -1)_*}\\ \end{array}\]

Reproduce

herbie shell --seed 2018365 +o rules:numerics
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  (- (/ 2 (+ 1 (exp (* -2 x)))) 1))

Details

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