Average Error: 29.5 → 0.4
Time: 6.1s
Precision: 64
\[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \le -0.0107417321816840241:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\sqrt{\sqrt[3]{1 + e^{-2 \cdot x}} \cdot \sqrt[3]{1 + e^{-2 \cdot x}}}}, \frac{\frac{2}{\sqrt{1 + e^{-2 \cdot x}}}}{\sqrt{\sqrt[3]{1 + e^{-2 \cdot x}}}}, -1\right)\\ \mathbf{elif}\;-2 \cdot x \le 1.14760911155323326 \cdot 10^{-17}:\\ \;\;\;\;\mathsf{fma}\left(1, x, -\mathsf{fma}\left(5.55112 \cdot 10^{-17}, {x}^{4}, 0.33333333333333337 \cdot {x}^{3}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\sqrt{\frac{2}{\sqrt{1 + e^{-2 \cdot x}}}}}{\sqrt{\sqrt{1 + e^{-2 \cdot x}}}}, \frac{\sqrt{\frac{2}{\sqrt{1 + e^{-2 \cdot x}}}}}{\sqrt{\sqrt{1 + e^{-2 \cdot x}}}}, -1\right)\\ \end{array}\]
\frac{2}{1 + e^{-2 \cdot x}} - 1
↓
\begin{array}{l}
\mathbf{if}\;-2 \cdot x \le -0.0107417321816840241:\\
\;\;\;\;\mathsf{fma}\left(\frac{1}{\sqrt{\sqrt[3]{1 + e^{-2 \cdot x}} \cdot \sqrt[3]{1 + e^{-2 \cdot x}}}}, \frac{\frac{2}{\sqrt{1 + e^{-2 \cdot x}}}}{\sqrt{\sqrt[3]{1 + e^{-2 \cdot x}}}}, -1\right)\\

\mathbf{elif}\;-2 \cdot x \le 1.14760911155323326 \cdot 10^{-17}:\\
\;\;\;\;\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{\sqrt{\frac{2}{\sqrt{1 + e^{-2 \cdot x}}}}}{\sqrt{\sqrt{1 + e^{-2 \cdot x}}}}, \frac{\sqrt{\frac{2}{\sqrt{1 + e^{-2 \cdot x}}}}}{\sqrt{\sqrt{1 + e^{-2 \cdot x}}}}, -1\right)\\

\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) <= -0.010741732181684024)) {
		VAR = fma((1.0 / sqrt((cbrt((1.0 + exp((-2.0 * x)))) * cbrt((1.0 + exp((-2.0 * x))))))), ((2.0 / sqrt((1.0 + exp((-2.0 * x))))) / sqrt(cbrt((1.0 + exp((-2.0 * x)))))), -1.0);
	} else {
		double VAR_1;
		if (((-2.0 * x) <= 1.1476091115532333e-17)) {
			VAR_1 = fma(1.0, x, -fma(5.551115123125783e-17, pow(x, 4.0), (0.33333333333333337 * pow(x, 3.0))));
		} else {
			VAR_1 = fma((sqrt((2.0 / sqrt((1.0 + exp((-2.0 * x)))))) / sqrt(sqrt((1.0 + exp((-2.0 * x)))))), (sqrt((2.0 / sqrt((1.0 + exp((-2.0 * x)))))) / sqrt(sqrt((1.0 + exp((-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) < -0.010741732181684024

    1. Initial program 0.0

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

    3. Applied add-sqr-sqrt0.0

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

    4. Applied associate-/r*0.1

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

    6. Applied add-cube-cbrt0.1

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

    7. Applied sqrt-prod0.1

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

    8. Applied *-un-lft-identity0.1

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

    9. Applied times-frac0.1

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

    10. Applied fma-neg0.1

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

    if -0.010741732181684024 < (* -2.0 x) < 1.1476091115532333e-17

    1. Initial program 59.7

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

    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 1.1476091115532333e-17 < (* -2.0 x)

    1. Initial program 1.2

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

    3. Applied add-sqr-sqrt1.2

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

    4. Applied associate-/r*1.2

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

    6. Applied add-sqr-sqrt1.2

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

    7. Applied sqrt-prod1.3

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

    8. Applied add-sqr-sqrt1.3

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

    9. Applied times-frac1.3

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

    10. Applied fma-neg1.3

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \le -0.0107417321816840241:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\sqrt{\sqrt[3]{1 + e^{-2 \cdot x}} \cdot \sqrt[3]{1 + e^{-2 \cdot x}}}}, \frac{\frac{2}{\sqrt{1 + e^{-2 \cdot x}}}}{\sqrt{\sqrt[3]{1 + e^{-2 \cdot x}}}}, -1\right)\\ \mathbf{elif}\;-2 \cdot x \le 1.14760911155323326 \cdot 10^{-17}:\\ \;\;\;\;\mathsf{fma}\left(1, x, -\mathsf{fma}\left(5.55112 \cdot 10^{-17}, {x}^{4}, 0.33333333333333337 \cdot {x}^{3}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\sqrt{\frac{2}{\sqrt{1 + e^{-2 \cdot x}}}}}{\sqrt{\sqrt{1 + e^{-2 \cdot x}}}}, \frac{\sqrt{\frac{2}{\sqrt{1 + e^{-2 \cdot x}}}}}{\sqrt{\sqrt{1 + e^{-2 \cdot x}}}}, -1\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020106 +o rules:numerics
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2 (+ 1 (exp (* -2 x)))) 1))