Average Error: 29.3 → 0.5
Time: 4.9s
Precision: 64
\[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \le -12083926.2470296454:\\ \;\;\;\;\frac{\left(\sqrt[3]{\frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{\frac{2}{\sqrt{1 + e^{-2 \cdot x}}}}{\sqrt{1 + e^{-2 \cdot x}}} - 1 \cdot 1} \cdot \sqrt[3]{\frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{\frac{2}{\sqrt{1 + e^{-2 \cdot x}}}}{\sqrt{1 + e^{-2 \cdot x}}} - 1 \cdot 1}\right) \cdot \sqrt[3]{\frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{\frac{2}{\sqrt{1 + e^{-2 \cdot x}}}}{\sqrt{1 + e^{-2 \cdot x}}} - 1 \cdot 1}}{\frac{2}{1 + e^{-2 \cdot x}} + 1}\\ \mathbf{elif}\;-2 \cdot x \le 4.592736067343327 \cdot 10^{-10}:\\ \;\;\;\;\frac{\left(1 \cdot {x}^{2} + 2 \cdot x\right) - 0.66666666666666696 \cdot {x}^{3}}{\frac{2}{1 + e^{-2 \cdot x}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{2}{1 + e^{-2 \cdot x}} - 1 \cdot 1}{\left(\sqrt[3]{2} \cdot \sqrt[3]{2}\right) \cdot \frac{\sqrt[3]{2}}{1 + e^{-2 \cdot x}} + 1}\\ \end{array}\]
\frac{2}{1 + e^{-2 \cdot x}} - 1
↓
\begin{array}{l}
\mathbf{if}\;-2 \cdot x \le -12083926.2470296454:\\
\;\;\;\;\frac{\left(\sqrt[3]{\frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{\frac{2}{\sqrt{1 + e^{-2 \cdot x}}}}{\sqrt{1 + e^{-2 \cdot x}}} - 1 \cdot 1} \cdot \sqrt[3]{\frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{\frac{2}{\sqrt{1 + e^{-2 \cdot x}}}}{\sqrt{1 + e^{-2 \cdot x}}} - 1 \cdot 1}\right) \cdot \sqrt[3]{\frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{\frac{2}{\sqrt{1 + e^{-2 \cdot x}}}}{\sqrt{1 + e^{-2 \cdot x}}} - 1 \cdot 1}}{\frac{2}{1 + e^{-2 \cdot x}} + 1}\\

\mathbf{elif}\;-2 \cdot x \le 4.592736067343327 \cdot 10^{-10}:\\
\;\;\;\;\frac{\left(1 \cdot {x}^{2} + 2 \cdot x\right) - 0.66666666666666696 \cdot {x}^{3}}{\frac{2}{1 + e^{-2 \cdot x}} + 1}\\

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

\end{array}
double code(double x, double y) {
	return ((2.0 / (1.0 + exp((-2.0 * x)))) - 1.0);
}
↓
double code(double x, double y) {
	double temp;
	if (((-2.0 * x) <= -12083926.247029645)) {
		temp = (((cbrt((((2.0 / (1.0 + exp((-2.0 * x)))) * ((2.0 / sqrt((1.0 + exp((-2.0 * x))))) / sqrt((1.0 + exp((-2.0 * x)))))) - (1.0 * 1.0))) * cbrt((((2.0 / (1.0 + exp((-2.0 * x)))) * ((2.0 / sqrt((1.0 + exp((-2.0 * x))))) / sqrt((1.0 + exp((-2.0 * x)))))) - (1.0 * 1.0)))) * cbrt((((2.0 / (1.0 + exp((-2.0 * x)))) * ((2.0 / sqrt((1.0 + exp((-2.0 * x))))) / sqrt((1.0 + exp((-2.0 * x)))))) - (1.0 * 1.0)))) / ((2.0 / (1.0 + exp((-2.0 * x)))) + 1.0));
	} else {
		double temp_1;
		if (((-2.0 * x) <= 4.5927360673433267e-10)) {
			temp_1 = ((((1.0 * pow(x, 2.0)) + (2.0 * x)) - (0.666666666666667 * pow(x, 3.0))) / ((2.0 / (1.0 + exp((-2.0 * x)))) + 1.0));
		} else {
			temp_1 = ((((2.0 / (1.0 + exp((-2.0 * x)))) * (2.0 / (1.0 + exp((-2.0 * x))))) - (1.0 * 1.0)) / (((cbrt(2.0) * cbrt(2.0)) * (cbrt(2.0) / (1.0 + exp((-2.0 * x))))) + 1.0));
		}
		temp = temp_1;
	}
	return temp;
}

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) < -12083926.247029645

    1. Initial program 0

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

    3. Applied flip--0

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

    5. Applied add-sqr-sqrt0

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

    6. Applied associate-/r*0

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

    8. Applied add-cube-cbrt0

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

    if -12083926.247029645 < (* -2.0 x) < 4.5927360673433267e-10

    1. Initial program 58.9

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

    3. Applied flip--58.9

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

    4. Taylor expanded around 0 0.7

      \[\leadsto \frac{\color{blue}{\left(1 \cdot {x}^{2} + 2 \cdot x\right) - 0.66666666666666696 \cdot {x}^{3}}}{\frac{2}{1 + e^{-2 \cdot x}} + 1}\]

    if 4.5927360673433267e-10 < (* -2.0 x)

    1. Initial program 0.4

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

    3. Applied flip--0.4

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

    5. Applied *-un-lft-identity0.4

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

    6. Applied add-cube-cbrt0.4

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

    7. Applied times-frac0.4

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

    8. Simplified0.4

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \le -12083926.2470296454:\\ \;\;\;\;\frac{\left(\sqrt[3]{\frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{\frac{2}{\sqrt{1 + e^{-2 \cdot x}}}}{\sqrt{1 + e^{-2 \cdot x}}} - 1 \cdot 1} \cdot \sqrt[3]{\frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{\frac{2}{\sqrt{1 + e^{-2 \cdot x}}}}{\sqrt{1 + e^{-2 \cdot x}}} - 1 \cdot 1}\right) \cdot \sqrt[3]{\frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{\frac{2}{\sqrt{1 + e^{-2 \cdot x}}}}{\sqrt{1 + e^{-2 \cdot x}}} - 1 \cdot 1}}{\frac{2}{1 + e^{-2 \cdot x}} + 1}\\ \mathbf{elif}\;-2 \cdot x \le 4.592736067343327 \cdot 10^{-10}:\\ \;\;\;\;\frac{\left(1 \cdot {x}^{2} + 2 \cdot x\right) - 0.66666666666666696 \cdot {x}^{3}}{\frac{2}{1 + e^{-2 \cdot x}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{2}{1 + e^{-2 \cdot x}} - 1 \cdot 1}{\left(\sqrt[3]{2} \cdot \sqrt[3]{2}\right) \cdot \frac{\sqrt[3]{2}}{1 + e^{-2 \cdot x}} + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020065 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2 (+ 1 (exp (* -2 x)))) 1))