Average Error: 29.1 → 0.6
Time: 5.5s
Precision: binary64
\[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \le -115103951900.492599 \lor \neg \left(-2 \cdot x \le 1.45645500891533392 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{\sqrt[3]{{\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{6}}}{\frac{2}{1 + e^{-2 \cdot x}} + 1} - \frac{1 \cdot 1}{\frac{2}{1 + e^{-2 \cdot x}} + 1}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot x - \left(5.55112 \cdot 10^{-17} \cdot {x}^{4} + 0.33333333333333337 \cdot {x}^{3}\right)\\ \end{array}\]
\frac{2}{1 + e^{-2 \cdot x}} - 1
↓
\begin{array}{l}
\mathbf{if}\;-2 \cdot x \le -115103951900.492599 \lor \neg \left(-2 \cdot x \le 1.45645500891533392 \cdot 10^{-5}\right):\\
\;\;\;\;\frac{\sqrt[3]{{\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{6}}}{\frac{2}{1 + e^{-2 \cdot x}} + 1} - \frac{1 \cdot 1}{\frac{2}{1 + e^{-2 \cdot x}} + 1}\\

\mathbf{else}:\\
\;\;\;\;1 \cdot x - \left(5.55112 \cdot 10^{-17} \cdot {x}^{4} + 0.33333333333333337 \cdot {x}^{3}\right)\\

\end{array}
double code(double x, double y) {
	return ((double) (((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)) <= -115103951900.4926) || !(((double) (-2.0 * x)) <= 1.456455008915334e-05))) {
		VAR = ((double) (((double) (((double) cbrt(((double) pow(((double) (2.0 / ((double) (1.0 + ((double) exp(((double) (-2.0 * x)))))))), 6.0)))) / ((double) (((double) (2.0 / ((double) (1.0 + ((double) exp(((double) (-2.0 * x)))))))) + 1.0)))) - ((double) (((double) (1.0 * 1.0)) / ((double) (((double) (2.0 / ((double) (1.0 + ((double) exp(((double) (-2.0 * x)))))))) + 1.0))))));
	} else {
		VAR = ((double) (((double) (1.0 * x)) - ((double) (((double) (5.551115123125783e-17 * ((double) pow(x, 4.0)))) + ((double) (0.33333333333333337 * ((double) pow(x, 3.0))))))));
	}
	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) < -115103951900.492599 or 1.45645500891533392e-5 < (* -2.0 x)

    1. Initial program 0.1

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

    3. Applied flip--0.1

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

    5. Applied add-cbrt-cube0.1

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

    6. Applied add-cbrt-cube0.1

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

    7. Applied cbrt-undiv0.1

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

    8. Applied add-cbrt-cube0.1

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

    9. Applied add-cbrt-cube0.1

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

    10. Applied cbrt-undiv0.1

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

    11. Applied cbrt-unprod0.1

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

    12. Simplified0.1

      \[\leadsto \frac{\sqrt[3]{\color{blue}{{\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{6}}} - 1 \cdot 1}{\frac{2}{1 + e^{-2 \cdot x}} + 1}\]
    13. Using strategy rm

    14. Applied div-sub0.1

      \[\leadsto \color{blue}{\frac{\sqrt[3]{{\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{6}}}{\frac{2}{1 + e^{-2 \cdot x}} + 1} - \frac{1 \cdot 1}{\frac{2}{1 + e^{-2 \cdot x}} + 1}}\]

    if -115103951900.492599 < (* -2.0 x) < 1.45645500891533392e-5

    1. Initial program 57.9

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

    2. Taylor expanded around 0 1.1

      \[\leadsto \color{blue}{1 \cdot x - \left(5.55112 \cdot 10^{-17} \cdot {x}^{4} + 0.33333333333333337 \cdot {x}^{3}\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \le -115103951900.492599 \lor \neg \left(-2 \cdot x \le 1.45645500891533392 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{\sqrt[3]{{\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{6}}}{\frac{2}{1 + e^{-2 \cdot x}} + 1} - \frac{1 \cdot 1}{\frac{2}{1 + e^{-2 \cdot x}} + 1}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot x - \left(5.55112 \cdot 10^{-17} \cdot {x}^{4} + 0.33333333333333337 \cdot {x}^{3}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020168 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))