Average Error: 29.5 → 0.2
Time: 4.5s
Precision: 64
\[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \le -2.21614581479449901:\\ \;\;\;\;\log \left(e^{\frac{2}{\frac{{1}^{3} + {\left(e^{-2 \cdot x}\right)}^{3}}{e^{-2 \cdot x} \cdot \left(e^{-2 \cdot x} - 1\right) + 1 \cdot 1}} - 1}\right)\\ \mathbf{elif}\;-2 \cdot x \le 8.82007944170797845 \cdot 10^{-10}:\\ \;\;\;\;1 \cdot x - \left(5.55112 \cdot 10^{-17} \cdot {x}^{4} + 0.33333333333333337 \cdot {x}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\sqrt{1 + e^{-2 \cdot x}}}}{\sqrt{1 + e^{-2 \cdot x}}} - 1\\ \end{array}\]
\frac{2}{1 + e^{-2 \cdot x}} - 1
↓
\begin{array}{l}
\mathbf{if}\;-2 \cdot x \le -2.21614581479449901:\\
\;\;\;\;\log \left(e^{\frac{2}{\frac{{1}^{3} + {\left(e^{-2 \cdot x}\right)}^{3}}{e^{-2 \cdot x} \cdot \left(e^{-2 \cdot x} - 1\right) + 1 \cdot 1}} - 1}\right)\\

\mathbf{elif}\;-2 \cdot x \le 8.82007944170797845 \cdot 10^{-10}:\\
\;\;\;\;1 \cdot x - \left(5.55112 \cdot 10^{-17} \cdot {x}^{4} + 0.33333333333333337 \cdot {x}^{3}\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{2}{\sqrt{1 + e^{-2 \cdot x}}}}{\sqrt{1 + e^{-2 \cdot x}}} - 1\\

\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)) <= -2.216145814794499)) {
		VAR = ((double) log(((double) exp(((double) (((double) (2.0 / ((double) (((double) (((double) pow(1.0, 3.0)) + ((double) pow(((double) exp(((double) (-2.0 * x)))), 3.0)))) / ((double) (((double) (((double) exp(((double) (-2.0 * x)))) * ((double) (((double) exp(((double) (-2.0 * x)))) - 1.0)))) + ((double) (1.0 * 1.0)))))))) - 1.0))))));
	} else {
		double VAR_1;
		if ((((double) (-2.0 * x)) <= 8.820079441707978e-10)) {
			VAR_1 = ((double) (((double) (1.0 * x)) - ((double) (((double) (5.551115123125783e-17 * ((double) pow(x, 4.0)))) + ((double) (0.33333333333333337 * ((double) pow(x, 3.0))))))));
		} else {
			VAR_1 = ((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));
		}
		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 (* -2.0 x) < -2.216145814794499

    1. Initial program 0.0

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

    3. Applied flip3-+0.0

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

    4. Simplified0.0

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

    6. Applied add-log-exp0.0

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

    7. Applied add-log-exp0.0

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

    8. Applied diff-log0.0

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

    9. Simplified0.0

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

    if -2.216145814794499 < (* -2.0 x) < 8.820079441707978e-10

    1. Initial program 59.3

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

    2. Taylor expanded around 0 0.1

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

    if 8.820079441707978e-10 < (* -2.0 x)

    1. Initial program 0.4

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

    3. Applied add-sqr-sqrt0.4

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

    4. Applied associate-/r*0.4

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \le -2.21614581479449901:\\ \;\;\;\;\log \left(e^{\frac{2}{\frac{{1}^{3} + {\left(e^{-2 \cdot x}\right)}^{3}}{e^{-2 \cdot x} \cdot \left(e^{-2 \cdot x} - 1\right) + 1 \cdot 1}} - 1}\right)\\ \mathbf{elif}\;-2 \cdot x \le 8.82007944170797845 \cdot 10^{-10}:\\ \;\;\;\;1 \cdot x - \left(5.55112 \cdot 10^{-17} \cdot {x}^{4} + 0.33333333333333337 \cdot {x}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\sqrt{1 + e^{-2 \cdot x}}}}{\sqrt{1 + e^{-2 \cdot x}}} - 1\\ \end{array}\]

Reproduce

herbie shell --seed 2020131 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))