Average Error: 29.4 → 0.0
Time: 7.6s
Precision: binary64
\[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -7.81243788670826279 \cdot 10^{-4}:\\ \;\;\;\;\left(\sqrt{\frac{2}{1 + {\left(e^{-2}\right)}^{x}}} + \sqrt{1}\right) \cdot \left(\sqrt{\frac{2}{1 + {\left(e^{-2}\right)}^{x}}} - \sqrt{1}\right)\\ \mathbf{elif}\;x \le 8.34697846928908103 \cdot 10^{-4}:\\ \;\;\;\;x \cdot 1 - {x}^{3} \cdot \left(x \cdot 5.55112 \cdot 10^{-17} + 0.33333333333333337\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{\left({\left(\frac{2}{1 + {\left(e^{-2}\right)}^{x}}\right)}^{3}\right)}^{3} - {\left({1}^{3}\right)}^{3}}{{\left(\frac{2}{1 + {\left(e^{-2}\right)}^{x}}\right)}^{6} + \left({1}^{6} + {\left(\frac{2}{1 + {\left(e^{-2}\right)}^{x}}\right)}^{3} \cdot {1}^{3}\right)}}{1 \cdot 1 + 2 \cdot \frac{1 + \frac{2}{1 + {\left(e^{-2}\right)}^{x}}}{1 + {\left(e^{-2}\right)}^{x}}}\\ \end{array}\]
\frac{2}{1 + e^{-2 \cdot x}} - 1
↓
\begin{array}{l}
\mathbf{if}\;x \le -7.81243788670826279 \cdot 10^{-4}:\\
\;\;\;\;\left(\sqrt{\frac{2}{1 + {\left(e^{-2}\right)}^{x}}} + \sqrt{1}\right) \cdot \left(\sqrt{\frac{2}{1 + {\left(e^{-2}\right)}^{x}}} - \sqrt{1}\right)\\

\mathbf{elif}\;x \le 8.34697846928908103 \cdot 10^{-4}:\\
\;\;\;\;x \cdot 1 - {x}^{3} \cdot \left(x \cdot 5.55112 \cdot 10^{-17} + 0.33333333333333337\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{{\left({\left(\frac{2}{1 + {\left(e^{-2}\right)}^{x}}\right)}^{3}\right)}^{3} - {\left({1}^{3}\right)}^{3}}{{\left(\frac{2}{1 + {\left(e^{-2}\right)}^{x}}\right)}^{6} + \left({1}^{6} + {\left(\frac{2}{1 + {\left(e^{-2}\right)}^{x}}\right)}^{3} \cdot {1}^{3}\right)}}{1 \cdot 1 + 2 \cdot \frac{1 + \frac{2}{1 + {\left(e^{-2}\right)}^{x}}}{1 + {\left(e^{-2}\right)}^{x}}}\\

\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 <= -0.0007812437886708263)) {
		VAR = ((double) (((double) (((double) sqrt(((double) (2.0 / ((double) (1.0 + ((double) pow(((double) exp(-2.0)), x)))))))) + ((double) sqrt(1.0)))) * ((double) (((double) sqrt(((double) (2.0 / ((double) (1.0 + ((double) pow(((double) exp(-2.0)), x)))))))) - ((double) sqrt(1.0))))));
	} else {
		double VAR_1;
		if ((x <= 0.0008346978469289081)) {
			VAR_1 = ((double) (((double) (x * 1.0)) - ((double) (((double) pow(x, 3.0)) * ((double) (((double) (x * 5.551115123125783e-17)) + 0.33333333333333337))))));
		} else {
			VAR_1 = ((double) (((double) (((double) (((double) pow(((double) pow(((double) (2.0 / ((double) (1.0 + ((double) pow(((double) exp(-2.0)), x)))))), 3.0)), 3.0)) - ((double) pow(((double) pow(1.0, 3.0)), 3.0)))) / ((double) (((double) pow(((double) (2.0 / ((double) (1.0 + ((double) pow(((double) exp(-2.0)), x)))))), 6.0)) + ((double) (((double) pow(1.0, 6.0)) + ((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) (((double) (1.0 * 1.0)) + ((double) (2.0 * ((double) (((double) (1.0 + ((double) (2.0 / ((double) (1.0 + ((double) pow(((double) exp(-2.0)), x)))))))) / ((double) (1.0 + ((double) pow(((double) exp(-2.0)), x))))))))))));
		}
		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 x < -7.81243788670826279e-4

    1. Initial program 0.0

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

    3. Applied add-sqr-sqrt0.0

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

    4. Applied add-sqr-sqrt0.0

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

    5. Applied difference-of-squares0.0

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

    6. Simplified0.0

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

    7. Simplified0.0

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

    if -7.81243788670826279e-4 < x < 8.34697846928908103e-4

    1. Initial program 59.3

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

    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

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

    if 8.34697846928908103e-4 < x

    1. Initial program 0.0

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

    3. Applied flip3--0.0

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

    4. Simplified0.0

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

    5. Simplified0.0

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

    7. Applied flip3--0.0

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

    8. Simplified0.0

      \[\leadsto \frac{\frac{{\left({\left(\frac{2}{1 + {\left(e^{-2}\right)}^{x}}\right)}^{3}\right)}^{3} - {\left({1}^{3}\right)}^{3}}{\color{blue}{{\left(\frac{2}{1 + {\left(e^{-2}\right)}^{x}}\right)}^{6} + \left({1}^{6} + {\left(\frac{2}{1 + {\left(e^{-2}\right)}^{x}}\right)}^{3} \cdot {1}^{3}\right)}}}{1 \cdot 1 + 2 \cdot \frac{1 + \frac{2}{1 + {\left(e^{-2}\right)}^{x}}}{1 + {\left(e^{-2}\right)}^{x}}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -7.81243788670826279 \cdot 10^{-4}:\\ \;\;\;\;\left(\sqrt{\frac{2}{1 + {\left(e^{-2}\right)}^{x}}} + \sqrt{1}\right) \cdot \left(\sqrt{\frac{2}{1 + {\left(e^{-2}\right)}^{x}}} - \sqrt{1}\right)\\ \mathbf{elif}\;x \le 8.34697846928908103 \cdot 10^{-4}:\\ \;\;\;\;x \cdot 1 - {x}^{3} \cdot \left(x \cdot 5.55112 \cdot 10^{-17} + 0.33333333333333337\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{\left({\left(\frac{2}{1 + {\left(e^{-2}\right)}^{x}}\right)}^{3}\right)}^{3} - {\left({1}^{3}\right)}^{3}}{{\left(\frac{2}{1 + {\left(e^{-2}\right)}^{x}}\right)}^{6} + \left({1}^{6} + {\left(\frac{2}{1 + {\left(e^{-2}\right)}^{x}}\right)}^{3} \cdot {1}^{3}\right)}}{1 \cdot 1 + 2 \cdot \frac{1 + \frac{2}{1 + {\left(e^{-2}\right)}^{x}}}{1 + {\left(e^{-2}\right)}^{x}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020184 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))