Average Error: 29.4 → 0.3
Time: 4.4s
Precision: 64
\[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \le -0.03841085747037655:\\ \;\;\;\;\sqrt[3]{{\left(\frac{2}{1 + e^{-2 \cdot x}} - 1\right)}^{3}}\\ \mathbf{elif}\;-2 \cdot x \le 3.94349320920764624 \cdot 10^{-16}:\\ \;\;\;\;1 \cdot x - \left(5.55112 \cdot 10^{-17} \cdot {x}^{4} + 0.33333333333333337 \cdot {x}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(e^{\frac{2 \cdot 2}{{\left(\sqrt{1 + e^{-2 \cdot x}}\right)}^{3}} \cdot {\left(\frac{\sqrt[3]{2}}{\sqrt{1 + e^{-2 \cdot x}}}\right)}^{3} - {1}^{3}}\right)}{\frac{2}{e^{-2 \cdot x} + 1} \cdot \left(1 + \frac{2}{e^{-2 \cdot x} + 1}\right) + 1 \cdot 1}\\ \end{array}\]
\frac{2}{1 + e^{-2 \cdot x}} - 1
↓
\begin{array}{l}
\mathbf{if}\;-2 \cdot x \le -0.03841085747037655:\\
\;\;\;\;\sqrt[3]{{\left(\frac{2}{1 + e^{-2 \cdot x}} - 1\right)}^{3}}\\

\mathbf{elif}\;-2 \cdot x \le 3.94349320920764624 \cdot 10^{-16}:\\
\;\;\;\;1 \cdot x - \left(5.55112 \cdot 10^{-17} \cdot {x}^{4} + 0.33333333333333337 \cdot {x}^{3}\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{\log \left(e^{\frac{2 \cdot 2}{{\left(\sqrt{1 + e^{-2 \cdot x}}\right)}^{3}} \cdot {\left(\frac{\sqrt[3]{2}}{\sqrt{1 + e^{-2 \cdot x}}}\right)}^{3} - {1}^{3}}\right)}{\frac{2}{e^{-2 \cdot x} + 1} \cdot \left(1 + \frac{2}{e^{-2 \cdot x} + 1}\right) + 1 \cdot 1}\\

\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)) <= -0.03841085747037655)) {
		VAR = ((double) cbrt(((double) pow(((double) (((double) (2.0 / ((double) (1.0 + ((double) exp(((double) (-2.0 * x)))))))) - 1.0)), 3.0))));
	} else {
		double VAR_1;
		if ((((double) (-2.0 * x)) <= 3.943493209207646e-16)) {
			VAR_1 = ((double) (((double) (1.0 * x)) - ((double) (((double) (5.551115123125783e-17 * ((double) pow(x, 4.0)))) + ((double) (0.33333333333333337 * ((double) pow(x, 3.0))))))));
		} else {
			VAR_1 = ((double) (((double) log(((double) exp(((double) (((double) (((double) (((double) (2.0 * 2.0)) / ((double) pow(((double) sqrt(((double) (1.0 + ((double) exp(((double) (-2.0 * x)))))))), 3.0)))) * ((double) pow(((double) (((double) cbrt(2.0)) / ((double) sqrt(((double) (1.0 + ((double) exp(((double) (-2.0 * x)))))))))), 3.0)))) - ((double) pow(1.0, 3.0)))))))) / ((double) (((double) (((double) (2.0 / ((double) (((double) exp(((double) (-2.0 * x)))) + 1.0)))) * ((double) (1.0 + ((double) (2.0 / ((double) (((double) exp(((double) (-2.0 * x)))) + 1.0)))))))) + ((double) (1.0 * 1.0))))));
		}
		VAR = VAR_1;
	}
	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 3 regimes

  2. if (* -2.0 x) < -0.03841085747037655

    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

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

    if -0.03841085747037655 < (* -2.0 x) < 3.943493209207646e-16

    1. Initial program 59.7

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

    2. Taylor expanded around 0 0.1

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

    if 3.943493209207646e-16 < (* -2.0 x)

    1. Initial program 1.0

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

    3. Applied flip3--1.0

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

    4. Simplified1.0

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

    6. Applied add-sqr-sqrt1.0

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

    7. Applied add-cube-cbrt1.1

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

    8. Applied times-frac1.2

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

    9. Applied unpow-prod-down1.2

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

    10. Simplified1.1

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

    12. Applied add-log-exp1.1

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

    13. Applied add-log-exp1.1

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

    14. Applied diff-log1.1

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

    15. Simplified1.1

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \le -0.03841085747037655:\\ \;\;\;\;\sqrt[3]{{\left(\frac{2}{1 + e^{-2 \cdot x}} - 1\right)}^{3}}\\ \mathbf{elif}\;-2 \cdot x \le 3.94349320920764624 \cdot 10^{-16}:\\ \;\;\;\;1 \cdot x - \left(5.55112 \cdot 10^{-17} \cdot {x}^{4} + 0.33333333333333337 \cdot {x}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(e^{\frac{2 \cdot 2}{{\left(\sqrt{1 + e^{-2 \cdot x}}\right)}^{3}} \cdot {\left(\frac{\sqrt[3]{2}}{\sqrt{1 + e^{-2 \cdot x}}}\right)}^{3} - {1}^{3}}\right)}{\frac{2}{e^{-2 \cdot x} + 1} \cdot \left(1 + \frac{2}{e^{-2 \cdot x} + 1}\right) + 1 \cdot 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020122 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2 (+ 1 (exp (* -2 x)))) 1))