Average Error: 29.1 → 0.0
Time: 3.9s
Precision: binary64
\[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -8.65735454075148105 \cdot 10^{-4} \lor \neg \left(x \le 9.4691287847764767 \cdot 10^{-4}\right):\\ \;\;\;\;\frac{\frac{2}{\frac{{\left(\sqrt{1 + {\left(e^{-2}\right)}^{x}}\right)}^{4}}{2}} - 1 \cdot 1}{1 + \frac{2}{1 + {\left(e^{-2}\right)}^{x}}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1 - {x}^{3} \cdot \left(0.33333333333333337 + x \cdot 5.55112 \cdot 10^{-17}\right)\\ \end{array}\]
\frac{2}{1 + e^{-2 \cdot x}} - 1
↓
\begin{array}{l}
\mathbf{if}\;x \le -8.65735454075148105 \cdot 10^{-4} \lor \neg \left(x \le 9.4691287847764767 \cdot 10^{-4}\right):\\
\;\;\;\;\frac{\frac{2}{\frac{{\left(\sqrt{1 + {\left(e^{-2}\right)}^{x}}\right)}^{4}}{2}} - 1 \cdot 1}{1 + \frac{2}{1 + {\left(e^{-2}\right)}^{x}}}\\

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

\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 <= -0.0008657354540751481) || !(x <= 0.0009469128784776477))) {
		VAR = (((double) ((2.0 / (((double) pow(((double) sqrt(((double) (1.0 + ((double) pow(((double) exp(-2.0)), x)))))), 4.0)) / 2.0)) - ((double) (1.0 * 1.0)))) / ((double) (1.0 + (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) (0.33333333333333337 + ((double) (x * 5.551115123125783e-17))))))));
	}
	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 < -8.65735454075148105e-4 or 9.4691287847764767e-4 < x

    1. Initial program 0.0

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

    3. Applied add-sqr-sqrt0.1

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

    4. Applied associate-/r*0.1

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

    5. Simplified0.1

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

    7. Applied flip--0.1

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

    8. Simplified0.1

      \[\leadsto \frac{\color{blue}{\frac{2}{\frac{{\left(\sqrt{1 + {\left(e^{-2}\right)}^{x}}\right)}^{4}}{2}} - 1 \cdot 1}}{\frac{\frac{2}{\sqrt{1 + {\left(e^{-2}\right)}^{x}}}}{\sqrt{1 + e^{-2 \cdot x}}} + 1}\]

    9. Simplified0.1

      \[\leadsto \frac{\frac{2}{\frac{{\left(\sqrt{1 + {\left(e^{-2}\right)}^{x}}\right)}^{4}}{2}} - 1 \cdot 1}{\color{blue}{1 + \frac{2}{1 + {\left(e^{-2}\right)}^{x}}}}\]

    if -8.65735454075148105e-4 < x < 9.4691287847764767e-4

    1. Initial program 59.1

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

    2. Taylor expanded around 0 0.0

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

    3. Simplified0.0

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -8.65735454075148105 \cdot 10^{-4} \lor \neg \left(x \le 9.4691287847764767 \cdot 10^{-4}\right):\\ \;\;\;\;\frac{\frac{2}{\frac{{\left(\sqrt{1 + {\left(e^{-2}\right)}^{x}}\right)}^{4}}{2}} - 1 \cdot 1}{1 + \frac{2}{1 + {\left(e^{-2}\right)}^{x}}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1 - {x}^{3} \cdot \left(0.33333333333333337 + x \cdot 5.55112 \cdot 10^{-17}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020182 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))