Average Error: 29.5 → 0.7
Time: 4.1s
Precision: binary64
\[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -2821070241.128962:\\ \;\;\;\;\sqrt[3]{\frac{2}{1 + e^{-2 \cdot x}} - 1} \cdot \left(\sqrt[3]{\frac{2}{1 + e^{-2 \cdot x}} - 1} \cdot \sqrt[3]{\frac{2}{1 + e^{-2 \cdot x}} - 1}\right)\\ \mathbf{elif}\;-2 \cdot x \leq 4.399426332164473 \cdot 10^{-13}:\\ \;\;\;\;\left(x + 0.13333333333333333 \cdot {x}^{5}\right) - 0.3333333333333333 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} - 1\\ \end{array}\]
\frac{2}{1 + e^{-2 \cdot x}} - 1
↓
\begin{array}{l}
\mathbf{if}\;-2 \cdot x \leq -2821070241.128962:\\
\;\;\;\;\sqrt[3]{\frac{2}{1 + 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)\\

\mathbf{elif}\;-2 \cdot x \leq 4.399426332164473 \cdot 10^{-13}:\\
\;\;\;\;\left(x + 0.13333333333333333 \cdot {x}^{5}\right) - 0.3333333333333333 \cdot {x}^{3}\\

\mathbf{else}:\\
\;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} - 1\\

\end{array}
(FPCore (x y) :precision binary64 (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))
↓
(FPCore (x y)
 :precision binary64
 (if (<= (* -2.0 x) -2821070241.128962)
   (*
    (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))))
   (if (<= (* -2.0 x) 4.399426332164473e-13)
     (-
      (+ x (* 0.13333333333333333 (pow x 5.0)))
      (* 0.3333333333333333 (pow x 3.0)))
     (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))))
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) <= -2821070241.128962) {
		tmp = 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));
	} else if ((-2.0 * x) <= 4.399426332164473e-13) {
		tmp = (x + (0.13333333333333333 * pow(x, 5.0))) - (0.3333333333333333 * pow(x, 3.0));
	} else {
		tmp = (2.0 / (1.0 + exp(-2.0 * x))) - 1.0;
	}
	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 3 regimes

  2. if (*.f64 -2 x) < -2821070241.12896204

    1. Initial program 0

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

    3. Applied add-cube-cbrt_binary64_31820

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

    if -2821070241.12896204 < (*.f64 -2 x) < 4.39942633216447318e-13

    1. Initial program 58.5

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

    2. Taylor expanded around 0 0.9

      \[\leadsto \color{blue}{\left(x + 0.13333333333333333 \cdot {x}^{5}\right) - 0.3333333333333333 \cdot {x}^{3}}\]

    if 4.39942633216447318e-13 < (*.f64 -2 x)

    1. Initial program 0.8

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -2821070241.128962:\\ \;\;\;\;\sqrt[3]{\frac{2}{1 + e^{-2 \cdot x}} - 1} \cdot \left(\sqrt[3]{\frac{2}{1 + e^{-2 \cdot x}} - 1} \cdot \sqrt[3]{\frac{2}{1 + e^{-2 \cdot x}} - 1}\right)\\ \mathbf{elif}\;-2 \cdot x \leq 4.399426332164473 \cdot 10^{-13}:\\ \;\;\;\;\left(x + 0.13333333333333333 \cdot {x}^{5}\right) - 0.3333333333333333 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} - 1\\ \end{array}\]

Reproduce

herbie shell --seed 2020346 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))