Average Error: 29.0 → 0.3
Time: 3.0s
Precision: 64
\[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \le -3165.5921330015635 \lor \neg \left(-2 \cdot x \le 1.4205456714418501 \cdot 10^{-8}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\sqrt{1 + e^{-2 \cdot x}}}, \frac{2}{\sqrt{1 + e^{-2 \cdot x}}}, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, x, -\mathsf{fma}\left(5.55112 \cdot 10^{-17}, {x}^{4}, 0.33333333333333337 \cdot {x}^{3}\right)\right)\\ \end{array}\]
\frac{2}{1 + e^{-2 \cdot x}} - 1
↓
\begin{array}{l}
\mathbf{if}\;-2 \cdot x \le -3165.5921330015635 \lor \neg \left(-2 \cdot x \le 1.4205456714418501 \cdot 10^{-8}\right):\\
\;\;\;\;\mathsf{fma}\left(\frac{1}{\sqrt{1 + e^{-2 \cdot x}}}, \frac{2}{\sqrt{1 + e^{-2 \cdot x}}}, -1\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(1, x, -\mathsf{fma}\left(5.55112 \cdot 10^{-17}, {x}^{4}, 0.33333333333333337 \cdot {x}^{3}\right)\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)) <= -3165.5921330015635) || !(((double) (-2.0 * x)) <= 1.4205456714418501e-08))) {
		VAR = ((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))));
	} else {
		VAR = ((double) fma(1.0, x, ((double) -(((double) fma(5.551115123125783e-17, ((double) pow(x, 4.0)), ((double) (0.33333333333333337 * ((double) pow(x, 3.0))))))))));
	}
	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 2 regimes

  2. if (* -2.0 x) < -3165.5921330015635 or 1.4205456714418501e-08 < (* -2.0 x)

    1. Initial program 0.2

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

    3. Applied add-sqr-sqrt0.2

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

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

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

    5. Applied times-frac0.2

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

    6. Applied fma-neg0.2

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

    if -3165.5921330015635 < (* -2.0 x) < 1.4205456714418501e-08

    1. Initial program 58.9

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

    2. Taylor expanded around 0 0.4

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

    3. Simplified0.4

      \[\leadsto \color{blue}{\mathsf{fma}\left(1, x, -\mathsf{fma}\left(5.55112 \cdot 10^{-17}, {x}^{4}, 0.33333333333333337 \cdot {x}^{3}\right)\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \le -3165.5921330015635 \lor \neg \left(-2 \cdot x \le 1.4205456714418501 \cdot 10^{-8}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\sqrt{1 + e^{-2 \cdot x}}}, \frac{2}{\sqrt{1 + e^{-2 \cdot x}}}, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, x, -\mathsf{fma}\left(5.55112 \cdot 10^{-17}, {x}^{4}, 0.33333333333333337 \cdot {x}^{3}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020121 +o rules:numerics
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2 (+ 1 (exp (* -2 x)))) 1))