Average Error: 29.4 → 0.3
Time: 4.0s
Precision: 64
\[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \le -0.03841085747037655:\\ \;\;\;\;e^{\log \left(\mathsf{fma}\left(\frac{2}{{1}^{3} + {\left(e^{-2 \cdot x}\right)}^{3}}, 1 \cdot 1 + \left(e^{-2 \cdot x} \cdot e^{-2 \cdot x} - 1 \cdot e^{-2 \cdot x}\right), -1\right)\right)}\\ \mathbf{elif}\;-2 \cdot x \le 3.94349320920764624 \cdot 10^{-16}:\\ \;\;\;\;\mathsf{fma}\left(1, x, -\mathsf{fma}\left(5.55112 \cdot 10^{-17}, {x}^{4}, 0.33333333333333337 \cdot {x}^{3}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(\frac{2}{1 + e^{-2 \cdot x}} - 1\right)}^{3}}\\ \end{array}\]
\frac{2}{1 + e^{-2 \cdot x}} - 1
↓
\begin{array}{l}
\mathbf{if}\;-2 \cdot x \le -0.03841085747037655:\\
\;\;\;\;e^{\log \left(\mathsf{fma}\left(\frac{2}{{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\right)\right)}\\

\mathbf{elif}\;-2 \cdot x \le 3.94349320920764624 \cdot 10^{-16}:\\
\;\;\;\;\mathsf{fma}\left(1, x, -\mathsf{fma}\left(5.55112 \cdot 10^{-17}, {x}^{4}, 0.33333333333333337 \cdot {x}^{3}\right)\right)\\

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

\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)) <= -0.03841085747037655)) {
		VAR = ((double) exp(((double) log(((double) fma(((double) (2.0 / ((double) (((double) pow(1.0, 3.0)) + ((double) pow(((double) exp(((double) (-2.0 * x)))), 3.0)))))), ((double) (((double) (1.0 * 1.0)) + ((double) (((double) (((double) exp(((double) (-2.0 * x)))) * ((double) exp(((double) (-2.0 * x)))))) - ((double) (1.0 * ((double) exp(((double) (-2.0 * x)))))))))), ((double) -(1.0))))))));
	} else {
		double VAR_1;
		if ((((double) (-2.0 * x)) <= 3.943493209207646e-16)) {
			VAR_1 = ((double) fma(1.0, x, ((double) -(((double) fma(5.551115123125783e-17, ((double) pow(x, 4.0)), ((double) (0.33333333333333337 * ((double) pow(x, 3.0))))))))));
		} else {
			VAR_1 = ((double) cbrt(((double) pow(((double) (((double) (2.0 / ((double) (1.0 + ((double) exp(((double) (-2.0 * x)))))))) - 1.0)), 3.0))));
		}
		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) < -0.03841085747037655

    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. Applied associate-/r/0.0

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

    5. Applied fma-neg0.0

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

    7. Applied add-exp-log0.0

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

    if -0.03841085747037655 < (* -2.0 x) < 3.943493209207646e-16

    1. Initial program 59.7

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

    2. Taylor expanded around 0 0.1

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

    3. Simplified0.1

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

    if 3.943493209207646e-16 < (* -2.0 x)

    1. Initial program 1.0

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

    3. Applied add-cbrt-cube1.0

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

    4. Simplified1.0

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \le -0.03841085747037655:\\ \;\;\;\;e^{\log \left(\mathsf{fma}\left(\frac{2}{{1}^{3} + {\left(e^{-2 \cdot x}\right)}^{3}}, 1 \cdot 1 + \left(e^{-2 \cdot x} \cdot e^{-2 \cdot x} - 1 \cdot e^{-2 \cdot x}\right), -1\right)\right)}\\ \mathbf{elif}\;-2 \cdot x \le 3.94349320920764624 \cdot 10^{-16}:\\ \;\;\;\;\mathsf{fma}\left(1, x, -\mathsf{fma}\left(5.55112 \cdot 10^{-17}, {x}^{4}, 0.33333333333333337 \cdot {x}^{3}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(\frac{2}{1 + e^{-2 \cdot x}} - 1\right)}^{3}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020122 +o rules:numerics
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2 (+ 1 (exp (* -2 x)))) 1))