Average Error: 29.3 → 0.0
Time: 3.4s
Precision: binary64
\[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -0.001125542754715476:\\ \;\;\;\;\sqrt{2} \cdot \frac{\sqrt{2}}{1 + {\left(e^{-2}\right)}^{x}} - 1\\ \mathbf{elif}\;x \leq 0.0007063694993854554:\\ \;\;\;\;x \cdot 1 - {x}^{3} \cdot \left(x \cdot 5.551115123125783 \cdot 10^{-17} + 0.33333333333333337\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(e^{2 \cdot \frac{2}{{\left(1 + {\left(e^{-2}\right)}^{x}\right)}^{2}} - 1 \cdot 1}\right)}{1 + \frac{2}{1 + {\left(e^{-2}\right)}^{x}}}\\ \end{array}\]
\frac{2}{1 + e^{-2 \cdot x}} - 1
↓
\begin{array}{l}
\mathbf{if}\;x \leq -0.001125542754715476:\\
\;\;\;\;\sqrt{2} \cdot \frac{\sqrt{2}}{1 + {\left(e^{-2}\right)}^{x}} - 1\\

\mathbf{elif}\;x \leq 0.0007063694993854554:\\
\;\;\;\;x \cdot 1 - {x}^{3} \cdot \left(x \cdot 5.551115123125783 \cdot 10^{-17} + 0.33333333333333337\right)\\

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

\end{array}
double code(double x, double y) {
	return ((double) ((2.0 / ((double) (1.0 + ((double) exp(((double) (-2.0 * x))))))) - 1.0));
}
↓
double code(double x, double y) {
	double VAR;
	if ((x <= -0.001125542754715476)) {
		VAR = ((double) (((double) (((double) sqrt(2.0)) * (((double) sqrt(2.0)) / ((double) (1.0 + ((double) pow(((double) exp(-2.0)), x))))))) - 1.0));
	} else {
		double VAR_1;
		if ((x <= 0.0007063694993854554)) {
			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) log(((double) exp(((double) (((double) (2.0 * (2.0 / ((double) pow(((double) (1.0 + ((double) pow(((double) exp(-2.0)), x)))), 2.0))))) - ((double) (1.0 * 1.0)))))))) / ((double) (1.0 + (2.0 / ((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 < -0.0011255427547154761

    1. Initial program 0.0

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

    3. Applied *-un-lft-identity0.0

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

    4. Applied add-sqr-sqrt0.0

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

    5. Applied times-frac0.0

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

    6. Simplified0.0

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

    7. Simplified0.0

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

    if -0.0011255427547154761 < x < 7.0636949938545539e-4

    1. Initial program 59.2

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

    2. Taylor expanded around 0 0.0

      \[\leadsto \color{blue}{1 \cdot x - \left(5.551115123125783 \cdot 10^{-17} \cdot {x}^{4} + 0.33333333333333337 \cdot {x}^{3}\right)}\]

    3. Simplified0.0

      \[\leadsto \color{blue}{1 \cdot x - {x}^{3} \cdot \left(x \cdot 5.551115123125783 \cdot 10^{-17} + 0.33333333333333337\right)}\]

    if 7.0636949938545539e-4 < x

    1. Initial program 0.0

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

    3. Applied flip--0.0

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

    4. Simplified0.0

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

    5. Simplified0.0

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

    7. Applied add-log-exp0.0

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

    8. Applied add-log-exp0.0

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

    9. Applied diff-log0.0

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

    10. Simplified0.0

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.001125542754715476:\\ \;\;\;\;\sqrt{2} \cdot \frac{\sqrt{2}}{1 + {\left(e^{-2}\right)}^{x}} - 1\\ \mathbf{elif}\;x \leq 0.0007063694993854554:\\ \;\;\;\;x \cdot 1 - {x}^{3} \cdot \left(x \cdot 5.551115123125783 \cdot 10^{-17} + 0.33333333333333337\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(e^{2 \cdot \frac{2}{{\left(1 + {\left(e^{-2}\right)}^{x}\right)}^{2}} - 1 \cdot 1}\right)}{1 + \frac{2}{1 + {\left(e^{-2}\right)}^{x}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020196 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))