Average Error: 29.5 → 0.5
Time: 3.8s
Precision: binary64
\[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -209.68920730807355 \lor \neg \left(-2 \cdot x \leq 1.2623611574830718 \cdot 10^{-16}\right):\\ \;\;\;\;\frac{\frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{2}{1 + e^{-2 \cdot x}} - 1 \cdot 1}{1 + \frac{2}{1 + e^{-2 \cdot x}}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1 - {x}^{3} \cdot \left(x \cdot 5.551115123125783 \cdot 10^{-17} + 0.33333333333333337\right)\\ \end{array}\]
\frac{2}{1 + e^{-2 \cdot x}} - 1
↓
\begin{array}{l}
\mathbf{if}\;-2 \cdot x \leq -209.68920730807355 \lor \neg \left(-2 \cdot x \leq 1.2623611574830718 \cdot 10^{-16}\right):\\
\;\;\;\;\frac{\frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{2}{1 + e^{-2 \cdot x}} - 1 \cdot 1}{1 + \frac{2}{1 + e^{-2 \cdot x}}}\\

\mathbf{else}:\\
\;\;\;\;x \cdot 1 - {x}^{3} \cdot \left(x \cdot 5.551115123125783 \cdot 10^{-17} + 0.33333333333333337\right)\\

\end{array}
(FPCore (x y) :precision binary64 (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))
↓
(FPCore (x y)
 :precision binary64
 (if (or (<= (* -2.0 x) -209.68920730807355)
         (not (<= (* -2.0 x) 1.2623611574830718e-16)))
   (/
    (-
     (* (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) (/ 2.0 (+ 1.0 (exp (* -2.0 x)))))
     (* 1.0 1.0))
    (+ 1.0 (/ 2.0 (+ 1.0 (exp (* -2.0 x))))))
   (-
    (* x 1.0)
    (* (pow x 3.0) (+ (* x 5.551115123125783e-17) 0.33333333333333337)))))
double code(double x, double y) {
	return ((double) ((2.0 / ((double) (1.0 + ((double) exp(((double) (-2.0 * x))))))) - 1.0));
}
↓
double code(double x, double y) {
	double tmp;
	if (((((double) (-2.0 * x)) <= -209.68920730807355) || !(((double) (-2.0 * x)) <= 1.2623611574830718e-16))) {
		tmp = (((double) (((double) ((2.0 / ((double) (1.0 + ((double) exp(((double) (-2.0 * x))))))) * (2.0 / ((double) (1.0 + ((double) exp(((double) (-2.0 * x))))))))) - ((double) (1.0 * 1.0)))) / ((double) (1.0 + (2.0 / ((double) (1.0 + ((double) exp(((double) (-2.0 * x))))))))));
	} else {
		tmp = ((double) (((double) (x * 1.0)) - ((double) (((double) pow(x, 3.0)) * ((double) (((double) (x * 5.551115123125783e-17)) + 0.33333333333333337))))));
	}
	return tmp;
}

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 (*.f64 -2.0 x) < -209.68920730807355 or 1.26236115748307184e-16 < (*.f64 -2.0 x)

    1. Initial program 0.6

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

    3. Applied flip--_binary640.6

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

    4. Simplified0.6

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

    if -209.68920730807355 < (*.f64 -2.0 x) < 1.26236115748307184e-16

    1. Initial program 59.2

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

    2. Taylor expanded around 0 0.4

      \[\leadsto \color{blue}{1 \cdot x - \left(5.551115123125783 \cdot 10^{-17} \cdot {x}^{4} + 0.33333333333333337 \cdot {x}^{3}\right)}\]

    3. Simplified0.4

      \[\leadsto \color{blue}{1 \cdot x - {x}^{3} \cdot \left(x \cdot 5.551115123125783 \cdot 10^{-17} + 0.33333333333333337\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -209.68920730807355 \lor \neg \left(-2 \cdot x \leq 1.2623611574830718 \cdot 10^{-16}\right):\\ \;\;\;\;\frac{\frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{2}{1 + e^{-2 \cdot x}} - 1 \cdot 1}{1 + \frac{2}{1 + e^{-2 \cdot x}}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1 - {x}^{3} \cdot \left(x \cdot 5.551115123125783 \cdot 10^{-17} + 0.33333333333333337\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020205 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))