Average Error: 29.4 → 0.1
Time: 7.0s
Precision: 64
\[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \le -0.0037159352560609659:\\ \;\;\;\;\frac{{\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{3} - {1}^{3}}{\left(\frac{2}{e^{-2 \cdot x} + 1} \cdot \left(\sqrt[3]{\frac{2}{1 + e^{-2 \cdot x}} + 1} \cdot \sqrt[3]{\frac{2}{1 + e^{-2 \cdot x}} + 1}\right)\right) \cdot \sqrt[3]{\frac{2}{1 + e^{-2 \cdot x}} + 1} + 1 \cdot 1}\\ \mathbf{elif}\;-2 \cdot x \le 5.17445771563928202 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(1, x, -\mathsf{fma}\left(5.55112 \cdot 10^{-17}, {x}^{4}, 0.33333333333333337 \cdot {x}^{3}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\left(\frac{\sqrt{2}}{1}\right)}^{3}, {\left(\frac{\sqrt{2}}{1 + e^{-2 \cdot x}}\right)}^{3}, -{1}^{3}\right)}{\frac{2}{e^{-2 \cdot x} + 1} \cdot \left(\frac{2}{1 + e^{-2 \cdot x}} + 1\right) + 1 \cdot 1}\\ \end{array}\]
\frac{2}{1 + e^{-2 \cdot x}} - 1
↓
\begin{array}{l}
\mathbf{if}\;-2 \cdot x \le -0.0037159352560609659:\\
\;\;\;\;\frac{{\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{3} - {1}^{3}}{\left(\frac{2}{e^{-2 \cdot x} + 1} \cdot \left(\sqrt[3]{\frac{2}{1 + e^{-2 \cdot x}} + 1} \cdot \sqrt[3]{\frac{2}{1 + e^{-2 \cdot x}} + 1}\right)\right) \cdot \sqrt[3]{\frac{2}{1 + e^{-2 \cdot x}} + 1} + 1 \cdot 1}\\

\mathbf{elif}\;-2 \cdot x \le 5.17445771563928202 \cdot 10^{-7}:\\
\;\;\;\;\mathsf{fma}\left(1, x, -\mathsf{fma}\left(5.55112 \cdot 10^{-17}, {x}^{4}, 0.33333333333333337 \cdot {x}^{3}\right)\right)\\

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

\end{array}
double code(double x, double y) {
	return ((2.0 / (1.0 + exp((-2.0 * x)))) - 1.0);
}
↓
double code(double x, double y) {
	double VAR;
	if (((-2.0 * x) <= -0.003715935256060966)) {
		VAR = ((pow((2.0 / (1.0 + exp((-2.0 * x)))), 3.0) - pow(1.0, 3.0)) / ((((2.0 / (exp((-2.0 * x)) + 1.0)) * (cbrt(((2.0 / (1.0 + exp((-2.0 * x)))) + 1.0)) * cbrt(((2.0 / (1.0 + exp((-2.0 * x)))) + 1.0)))) * cbrt(((2.0 / (1.0 + exp((-2.0 * x)))) + 1.0))) + (1.0 * 1.0)));
	} else {
		double VAR_1;
		if (((-2.0 * x) <= 5.174457715639282e-07)) {
			VAR_1 = fma(1.0, x, -fma(5.551115123125783e-17, pow(x, 4.0), (0.33333333333333337 * pow(x, 3.0))));
		} else {
			VAR_1 = (fma(pow((sqrt(2.0) / 1.0), 3.0), pow((sqrt(2.0) / (1.0 + exp((-2.0 * x)))), 3.0), -pow(1.0, 3.0)) / (((2.0 / (exp((-2.0 * x)) + 1.0)) * ((2.0 / (1.0 + exp((-2.0 * x)))) + 1.0)) + (1.0 * 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) < -0.003715935256060966

    1. Initial program 0.0

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

    3. Applied flip3--0.0

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

    4. Simplified0.0

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

    6. Applied add-cube-cbrt0.0

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

    7. Applied associate-*r*0.0

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

    if -0.003715935256060966 < (* -2.0 x) < 5.174457715639282e-07

    1. Initial program 59.3

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

    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. Simplified0.0

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

    if 5.174457715639282e-07 < (* -2.0 x)

    1. Initial program 0.2

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

    3. Applied flip3--0.2

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

    4. Simplified0.2

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

    6. Applied *-un-lft-identity0.2

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

    7. Applied add-sqr-sqrt0.2

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

    8. Applied times-frac0.2

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

    9. Applied unpow-prod-down0.2

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

    10. Applied fma-neg0.2

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \le -0.0037159352560609659:\\ \;\;\;\;\frac{{\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{3} - {1}^{3}}{\left(\frac{2}{e^{-2 \cdot x} + 1} \cdot \left(\sqrt[3]{\frac{2}{1 + e^{-2 \cdot x}} + 1} \cdot \sqrt[3]{\frac{2}{1 + e^{-2 \cdot x}} + 1}\right)\right) \cdot \sqrt[3]{\frac{2}{1 + e^{-2 \cdot x}} + 1} + 1 \cdot 1}\\ \mathbf{elif}\;-2 \cdot x \le 5.17445771563928202 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(1, x, -\mathsf{fma}\left(5.55112 \cdot 10^{-17}, {x}^{4}, 0.33333333333333337 \cdot {x}^{3}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\left(\frac{\sqrt{2}}{1}\right)}^{3}, {\left(\frac{\sqrt{2}}{1 + e^{-2 \cdot x}}\right)}^{3}, -{1}^{3}\right)}{\frac{2}{e^{-2 \cdot x} + 1} \cdot \left(\frac{2}{1 + e^{-2 \cdot x}} + 1\right) + 1 \cdot 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020091 +o rules:numerics
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2 (+ 1 (exp (* -2 x)))) 1))