Average Error: 29.4 → 0.4
Time: 21.8min
Precision: binary64
\[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \le -1162783.57812564005 \lor \neg \left(-2 \cdot x \le 2.2808535801405823 \cdot 10^{-5}\right):\\ \;\;\;\;\sqrt[3]{{\left(\sqrt[3]{{\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{3}} - 1\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot x - {x}^{3} \cdot \left(5.55112 \cdot 10^{-17} \cdot x + 0.33333333333333337\right)\\ \end{array}\]
\frac{2}{1 + e^{-2 \cdot x}} - 1
↓
\begin{array}{l}
\mathbf{if}\;-2 \cdot x \le -1162783.57812564005 \lor \neg \left(-2 \cdot x \le 2.2808535801405823 \cdot 10^{-5}\right):\\
\;\;\;\;\sqrt[3]{{\left(\sqrt[3]{{\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{3}} - 1\right)}^{3}}\\

\mathbf{else}:\\
\;\;\;\;1 \cdot x - {x}^{3} \cdot \left(5.55112 \cdot 10^{-17} \cdot x + 0.33333333333333337\right)\\

\end{array}
double code(double x, double y) {
	return ((double) ((2.0 / ((double) (1.0 + ((double) exp(((double) (-2.0 * x))))))) - 1.0));
}
↓
double code(double x, double y) {
	double VAR;
	if (((((double) (-2.0 * x)) <= -1162783.57812564) || !(((double) (-2.0 * x)) <= 2.2808535801405823e-05))) {
		VAR = ((double) cbrt(((double) pow(((double) (((double) cbrt(((double) pow((2.0 / ((double) (1.0 + ((double) exp(((double) (-2.0 * x))))))), 3.0)))) - 1.0)), 3.0))));
	} else {
		VAR = ((double) (((double) (1.0 * x)) - ((double) (((double) pow(x, 3.0)) * ((double) (((double) (5.551115123125783e-17 * x)) + 0.33333333333333337))))));
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if (* -2.0 x) < -1162783.57812564005 or 2.2808535801405823e-5 < (* -2.0 x)

    1. Initial program 0.0

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

    3. Applied add-cbrt-cube0.0

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

    4. Simplified0.0

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

    6. Applied add-cbrt-cube0.0

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

    7. Applied add-cbrt-cube0.0

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

    8. Applied cbrt-undiv0.0

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

    9. Simplified0.0

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

    if -1162783.57812564005 < (* -2.0 x) < 2.2808535801405823e-5

    1. Initial program 58.5

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

    2. Taylor expanded around 0 0.7

      \[\leadsto \color{blue}{1 \cdot x - \left(0.33333333333333337 \cdot {x}^{3} + 5.55112 \cdot 10^{-17} \cdot {x}^{4}\right)}\]

    3. Simplified0.7

      \[\leadsto \color{blue}{1 \cdot x - {x}^{3} \cdot \left(5.55112 \cdot 10^{-17} \cdot x + 0.33333333333333337\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \le -1162783.57812564005 \lor \neg \left(-2 \cdot x \le 2.2808535801405823 \cdot 10^{-5}\right):\\ \;\;\;\;\sqrt[3]{{\left(\sqrt[3]{{\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{3}} - 1\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot x - {x}^{3} \cdot \left(5.55112 \cdot 10^{-17} \cdot x + 0.33333333333333337\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020181 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))