Average Error: 29.5 → 0.1
Time: 4.0s
Precision: 64
\[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \le -0.028220417364492139:\\ \;\;\;\;\left(\sqrt[3]{\mathsf{fma}\left(\frac{2}{1 \cdot 1 - e^{-2 \cdot x} \cdot e^{-2 \cdot x}}, 1 - e^{-2 \cdot x}, -1\right)} \cdot \sqrt[3]{\mathsf{fma}\left(\frac{2}{1 \cdot 1 - e^{-2 \cdot x} \cdot e^{-2 \cdot x}}, 1 - e^{-2 \cdot x}, -1\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(\frac{2}{1 \cdot 1 - e^{-2 \cdot x} \cdot e^{-2 \cdot x}}, 1 - e^{-2 \cdot x}, -1\right)}\\ \mathbf{elif}\;-2 \cdot x \le 4.939188832806807 \cdot 10^{-4}:\\ \;\;\;\;\mathsf{fma}\left(1, x, -\mathsf{fma}\left(5.55112 \cdot 10^{-17}, {x}^{4}, 0.33333333333333337 \cdot {x}^{3}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\sqrt{1 + e^{-2 \cdot x}}}, \frac{2}{\sqrt{1 + e^{-2 \cdot x}}}, -1\right)\\ \end{array}\]
\frac{2}{1 + e^{-2 \cdot x}} - 1
↓
\begin{array}{l}
\mathbf{if}\;-2 \cdot x \le -0.028220417364492139:\\
\;\;\;\;\left(\sqrt[3]{\mathsf{fma}\left(\frac{2}{1 \cdot 1 - e^{-2 \cdot x} \cdot e^{-2 \cdot x}}, 1 - e^{-2 \cdot x}, -1\right)} \cdot \sqrt[3]{\mathsf{fma}\left(\frac{2}{1 \cdot 1 - e^{-2 \cdot x} \cdot e^{-2 \cdot x}}, 1 - e^{-2 \cdot x}, -1\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(\frac{2}{1 \cdot 1 - e^{-2 \cdot x} \cdot e^{-2 \cdot x}}, 1 - e^{-2 \cdot x}, -1\right)}\\

\mathbf{elif}\;-2 \cdot x \le 4.939188832806807 \cdot 10^{-4}:\\
\;\;\;\;\mathsf{fma}\left(1, x, -\mathsf{fma}\left(5.55112 \cdot 10^{-17}, {x}^{4}, 0.33333333333333337 \cdot {x}^{3}\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(\frac{1}{\sqrt{1 + e^{-2 \cdot x}}}, \frac{2}{\sqrt{1 + e^{-2 \cdot x}}}, -1\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)) <= -0.02822041736449214)) {
		VAR = ((double) (((double) (((double) cbrt(((double) fma(((double) (2.0 / ((double) (((double) (1.0 * 1.0)) - ((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)))))) * ((double) cbrt(((double) fma(((double) (2.0 / ((double) (((double) (1.0 * 1.0)) - ((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)))))))) * ((double) cbrt(((double) fma(((double) (2.0 / ((double) (((double) (1.0 * 1.0)) - ((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)) <= 0.0004939188832806807)) {
			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) fma(((double) (1.0 / ((double) sqrt(((double) (1.0 + ((double) exp(((double) (-2.0 * x)))))))))), ((double) (2.0 / ((double) sqrt(((double) (1.0 + ((double) exp(((double) (-2.0 * x)))))))))), ((double) -(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) < -0.02822041736449214

    1. Initial program 0.0

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

    3. Applied flip-+0.0

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

    4. Applied associate-/r/0.0

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

    5. Applied fma-neg0.0

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

    7. Applied add-cube-cbrt0.0

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

    if -0.02822041736449214 < (* -2.0 x) < 0.0004939188832806807

    1. Initial program 59.0

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

    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 0.0004939188832806807 < (* -2.0 x)

    1. Initial program 0.1

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
    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 *-un-lft-identity0.1

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

    5. Applied times-frac0.1

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

    6. Applied fma-neg0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \le -0.028220417364492139:\\ \;\;\;\;\left(\sqrt[3]{\mathsf{fma}\left(\frac{2}{1 \cdot 1 - e^{-2 \cdot x} \cdot e^{-2 \cdot x}}, 1 - e^{-2 \cdot x}, -1\right)} \cdot \sqrt[3]{\mathsf{fma}\left(\frac{2}{1 \cdot 1 - e^{-2 \cdot x} \cdot e^{-2 \cdot x}}, 1 - e^{-2 \cdot x}, -1\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(\frac{2}{1 \cdot 1 - e^{-2 \cdot x} \cdot e^{-2 \cdot x}}, 1 - e^{-2 \cdot x}, -1\right)}\\ \mathbf{elif}\;-2 \cdot x \le 4.939188832806807 \cdot 10^{-4}:\\ \;\;\;\;\mathsf{fma}\left(1, x, -\mathsf{fma}\left(5.55112 \cdot 10^{-17}, {x}^{4}, 0.33333333333333337 \cdot {x}^{3}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\sqrt{1 + e^{-2 \cdot x}}}, \frac{2}{\sqrt{1 + e^{-2 \cdot x}}}, -1\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020113 +o rules:numerics
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2 (+ 1 (exp (* -2 x)))) 1))