Average Error: 29.9 → 0.1
Time: 2.4s
Precision: binary64
\[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.3910841835833815 \lor \neg \left(-2 \cdot x \leq 2.1678637251384385 \cdot 10^{-07}\right):\\ \;\;\;\;\frac{2}{\log \left(e \cdot e^{e^{-2 \cdot x}}\right)} - 1\\ \mathbf{else}:\\ \;\;\;\;\left(x + 0.13333333333333333 \cdot {x}^{5}\right) - 0.3333333333333333 \cdot {x}^{3}\\ \end{array}\]
\frac{2}{1 + e^{-2 \cdot x}} - 1
↓
\begin{array}{l}
\mathbf{if}\;-2 \cdot x \leq -0.3910841835833815 \lor \neg \left(-2 \cdot x \leq 2.1678637251384385 \cdot 10^{-07}\right):\\
\;\;\;\;\frac{2}{\log \left(e \cdot e^{e^{-2 \cdot x}}\right)} - 1\\

\mathbf{else}:\\
\;\;\;\;\left(x + 0.13333333333333333 \cdot {x}^{5}\right) - 0.3333333333333333 \cdot {x}^{3}\\

\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) -0.3910841835833815)
         (not (<= (* -2.0 x) 2.1678637251384385e-07)))
   (- (/ 2.0 (log (* E (exp (exp (* -2.0 x)))))) 1.0)
   (-
    (+ x (* 0.13333333333333333 (pow x 5.0)))
    (* 0.3333333333333333 (pow x 3.0)))))
double code(double x, double y) {
	return (2.0 / (1.0 + exp(-2.0 * x))) - 1.0;
}
↓
double code(double x, double y) {
	double tmp;
	if (((-2.0 * x) <= -0.3910841835833815) || !((-2.0 * x) <= 2.1678637251384385e-07)) {
		tmp = (2.0 / log(((double) M_E) * exp(exp(-2.0 * x)))) - 1.0;
	} else {
		tmp = (x + (0.13333333333333333 * pow(x, 5.0))) - (0.3333333333333333 * pow(x, 3.0));
	}
	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 x) < -0.39108418358338148 or 2.1678637251384385e-7 < (*.f64 -2 x)

    1. Initial program 0.1

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

    3. Applied add-log-exp_binary640.2

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

    4. Applied add-log-exp_binary640.2

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

    5. Applied sum-log_binary640.2

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

    6. Simplified0.2

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

    if -0.39108418358338148 < (*.f64 -2 x) < 2.1678637251384385e-7

    1. Initial program 59.1

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

    2. Taylor expanded around 0 0.1

      \[\leadsto \color{blue}{\left(x + 0.13333333333333333 \cdot {x}^{5}\right) - 0.3333333333333333 \cdot {x}^{3}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.3910841835833815 \lor \neg \left(-2 \cdot x \leq 2.1678637251384385 \cdot 10^{-07}\right):\\ \;\;\;\;\frac{2}{\log \left(e \cdot e^{e^{-2 \cdot x}}\right)} - 1\\ \mathbf{else}:\\ \;\;\;\;\left(x + 0.13333333333333333 \cdot {x}^{5}\right) - 0.3333333333333333 \cdot {x}^{3}\\ \end{array}\]

Reproduce

herbie shell --seed 2020220 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))