Average Error: 29.6 → 0.5
Time: 4.5s
Precision: 64
\[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \le -0.01756904497623861 \lor \neg \left(-2 \cdot x \le 2.50044575718993887 \cdot 10^{-18}\right):\\ \;\;\;\;\frac{\left(-1 \cdot 1\right) + \frac{\frac{2}{1 + e^{-2 \cdot x}} \cdot 2}{1 + e^{-2 \cdot x}}}{\frac{\frac{2}{\sqrt{1 + e^{-2 \cdot x}}}}{\sqrt{1 + e^{-2 \cdot x}}} + 1}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot x - \left(5.55112 \cdot 10^{-17} \cdot {x}^{4} + 0.33333333333333337 \cdot {x}^{3}\right)\\ \end{array}\]
\frac{2}{1 + e^{-2 \cdot x}} - 1
↓
\begin{array}{l}
\mathbf{if}\;-2 \cdot x \le -0.01756904497623861 \lor \neg \left(-2 \cdot x \le 2.50044575718993887 \cdot 10^{-18}\right):\\
\;\;\;\;\frac{\left(-1 \cdot 1\right) + \frac{\frac{2}{1 + e^{-2 \cdot x}} \cdot 2}{1 + e^{-2 \cdot x}}}{\frac{\frac{2}{\sqrt{1 + e^{-2 \cdot x}}}}{\sqrt{1 + e^{-2 \cdot x}}} + 1}\\

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

\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)) <= -0.01756904497623861) || !(((double) (-2.0 * x)) <= 2.500445757189939e-18))) {
		VAR = ((double) (((double) (((double) -(((double) (1.0 * 1.0)))) + ((double) (((double) (((double) (2.0 / ((double) (1.0 + ((double) exp(((double) (-2.0 * x)))))))) * 2.0)) / ((double) (1.0 + ((double) exp(((double) (-2.0 * x)))))))))) / ((double) (((double) (((double) (2.0 / ((double) sqrt(((double) (1.0 + ((double) exp(((double) (-2.0 * x)))))))))) / ((double) sqrt(((double) (1.0 + ((double) exp(((double) (-2.0 * x)))))))))) + 1.0))));
	} else {
		VAR = ((double) (((double) (1.0 * x)) - ((double) (((double) (5.551115123125783e-17 * ((double) pow(x, 4.0)))) + ((double) (0.33333333333333337 * ((double) pow(x, 3.0))))))));
	}
	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 (* -2.0 x) < -0.01756904497623861 or 2.500445757189939e-18 < (* -2.0 x)

    1. Initial program 0.9

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

    3. Applied add-sqr-sqrt0.9

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

    4. Applied associate-/r*0.9

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

    6. Applied flip--0.9

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

    7. Simplified0.9

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

    if -0.01756904497623861 < (* -2.0 x) < 2.500445757189939e-18

    1. Initial program 59.8

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

    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. Recombined 2 regimes into one program.

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \le -0.01756904497623861 \lor \neg \left(-2 \cdot x \le 2.50044575718993887 \cdot 10^{-18}\right):\\ \;\;\;\;\frac{\left(-1 \cdot 1\right) + \frac{\frac{2}{1 + e^{-2 \cdot x}} \cdot 2}{1 + e^{-2 \cdot x}}}{\frac{\frac{2}{\sqrt{1 + e^{-2 \cdot x}}}}{\sqrt{1 + e^{-2 \cdot x}}} + 1}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot x - \left(5.55112 \cdot 10^{-17} \cdot {x}^{4} + 0.33333333333333337 \cdot {x}^{3}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020123 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2 (+ 1 (exp (* -2 x)))) 1))