Average Error: 29.5 → 0.0
Time: 4.9s
Precision: binary64
\[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -9.66599909477714012 \cdot 10^{-4} \lor \neg \left(x \le 8.6546567923037718 \cdot 10^{-4}\right):\\ \;\;\;\;\frac{\sqrt[3]{{\left(\frac{2}{1 + {\left(e^{-2}\right)}^{x}}\right)}^{3} - {1}^{3}}}{\sqrt[3]{{\left(\frac{2}{1 + {\left(e^{-2}\right)}^{x}}\right)}^{2} + 1 \cdot \left(1 + \frac{2}{1 + {\left(e^{-2}\right)}^{x}}\right)}} \cdot \left(\frac{\sqrt[3]{{\left(\frac{2}{1 + {\left(e^{-2}\right)}^{x}}\right)}^{3} - {1}^{3}}}{\sqrt[3]{{\left(\frac{2}{1 + {\left(e^{-2}\right)}^{x}}\right)}^{2} + 1 \cdot \left(1 + \frac{2}{1 + {\left(e^{-2}\right)}^{x}}\right)}} \cdot \frac{\sqrt[3]{{\left(\frac{2}{1 + {\left(e^{-2}\right)}^{x}}\right)}^{3} - {1}^{3}}}{\sqrt[3]{{\left(\frac{2}{1 + {\left(e^{-2}\right)}^{x}}\right)}^{2} + 1 \cdot \left(1 + \frac{2}{1 + {\left(e^{-2}\right)}^{x}}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1 - {x}^{3} \cdot \left(x \cdot 5.55112 \cdot 10^{-17} + 0.33333333333333337\right)\\ \end{array}\]
\frac{2}{1 + e^{-2 \cdot x}} - 1
↓
\begin{array}{l}
\mathbf{if}\;x \le -9.66599909477714012 \cdot 10^{-4} \lor \neg \left(x \le 8.6546567923037718 \cdot 10^{-4}\right):\\
\;\;\;\;\frac{\sqrt[3]{{\left(\frac{2}{1 + {\left(e^{-2}\right)}^{x}}\right)}^{3} - {1}^{3}}}{\sqrt[3]{{\left(\frac{2}{1 + {\left(e^{-2}\right)}^{x}}\right)}^{2} + 1 \cdot \left(1 + \frac{2}{1 + {\left(e^{-2}\right)}^{x}}\right)}} \cdot \left(\frac{\sqrt[3]{{\left(\frac{2}{1 + {\left(e^{-2}\right)}^{x}}\right)}^{3} - {1}^{3}}}{\sqrt[3]{{\left(\frac{2}{1 + {\left(e^{-2}\right)}^{x}}\right)}^{2} + 1 \cdot \left(1 + \frac{2}{1 + {\left(e^{-2}\right)}^{x}}\right)}} \cdot \frac{\sqrt[3]{{\left(\frac{2}{1 + {\left(e^{-2}\right)}^{x}}\right)}^{3} - {1}^{3}}}{\sqrt[3]{{\left(\frac{2}{1 + {\left(e^{-2}\right)}^{x}}\right)}^{2} + 1 \cdot \left(1 + \frac{2}{1 + {\left(e^{-2}\right)}^{x}}\right)}}\right)\\

\mathbf{else}:\\
\;\;\;\;x \cdot 1 - {x}^{3} \cdot \left(x \cdot 5.55112 \cdot 10^{-17} + 0.33333333333333337\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 (((x <= -0.000966599909477714) || !(x <= 0.0008654656792303772))) {
		VAR = ((double) (((double) (((double) cbrt(((double) (((double) pow(((double) (2.0 / ((double) (1.0 + ((double) pow(((double) exp(-2.0)), x)))))), 3.0)) - ((double) pow(1.0, 3.0)))))) / ((double) cbrt(((double) (((double) pow(((double) (2.0 / ((double) (1.0 + ((double) pow(((double) exp(-2.0)), x)))))), 2.0)) + ((double) (1.0 * ((double) (1.0 + ((double) (2.0 / ((double) (1.0 + ((double) pow(((double) exp(-2.0)), x)))))))))))))))) * ((double) (((double) (((double) cbrt(((double) (((double) pow(((double) (2.0 / ((double) (1.0 + ((double) pow(((double) exp(-2.0)), x)))))), 3.0)) - ((double) pow(1.0, 3.0)))))) / ((double) cbrt(((double) (((double) pow(((double) (2.0 / ((double) (1.0 + ((double) pow(((double) exp(-2.0)), x)))))), 2.0)) + ((double) (1.0 * ((double) (1.0 + ((double) (2.0 / ((double) (1.0 + ((double) pow(((double) exp(-2.0)), x)))))))))))))))) * ((double) (((double) cbrt(((double) (((double) pow(((double) (2.0 / ((double) (1.0 + ((double) pow(((double) exp(-2.0)), x)))))), 3.0)) - ((double) pow(1.0, 3.0)))))) / ((double) cbrt(((double) (((double) pow(((double) (2.0 / ((double) (1.0 + ((double) pow(((double) exp(-2.0)), x)))))), 2.0)) + ((double) (1.0 * ((double) (1.0 + ((double) (2.0 / ((double) (1.0 + ((double) pow(((double) exp(-2.0)), x))))))))))))))))))));
	} else {
		VAR = ((double) (((double) (x * 1.0)) - ((double) (((double) pow(x, 3.0)) * ((double) (((double) (x * 5.551115123125783e-17)) + 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 x < -9.66599909477714012e-4 or 8.6546567923037718e-4 < x

    1. Initial program 0.1

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

    3. Applied flip3--0.1

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

    4. Simplified0.1

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

    5. Simplified0.1

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

    7. Applied add-cube-cbrt1.3

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

    8. Applied add-cube-cbrt0.1

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

    9. Applied times-frac0.1

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

    10. Simplified0.1

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

    11. Simplified0.1

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

    if -9.66599909477714012e-4 < x < 8.6546567923037718e-4

    1. Initial program 59.1

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

    2. Taylor expanded around 0 0.0

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

    3. Simplified0.0

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -9.66599909477714012 \cdot 10^{-4} \lor \neg \left(x \le 8.6546567923037718 \cdot 10^{-4}\right):\\ \;\;\;\;\frac{\sqrt[3]{{\left(\frac{2}{1 + {\left(e^{-2}\right)}^{x}}\right)}^{3} - {1}^{3}}}{\sqrt[3]{{\left(\frac{2}{1 + {\left(e^{-2}\right)}^{x}}\right)}^{2} + 1 \cdot \left(1 + \frac{2}{1 + {\left(e^{-2}\right)}^{x}}\right)}} \cdot \left(\frac{\sqrt[3]{{\left(\frac{2}{1 + {\left(e^{-2}\right)}^{x}}\right)}^{3} - {1}^{3}}}{\sqrt[3]{{\left(\frac{2}{1 + {\left(e^{-2}\right)}^{x}}\right)}^{2} + 1 \cdot \left(1 + \frac{2}{1 + {\left(e^{-2}\right)}^{x}}\right)}} \cdot \frac{\sqrt[3]{{\left(\frac{2}{1 + {\left(e^{-2}\right)}^{x}}\right)}^{3} - {1}^{3}}}{\sqrt[3]{{\left(\frac{2}{1 + {\left(e^{-2}\right)}^{x}}\right)}^{2} + 1 \cdot \left(1 + \frac{2}{1 + {\left(e^{-2}\right)}^{x}}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1 - {x}^{3} \cdot \left(x \cdot 5.55112 \cdot 10^{-17} + 0.33333333333333337\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020179 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))