Average Error: 29.2 → 0.5
Time: 3.8s
Precision: binary64
\[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -109316.8462524794 \lor \neg \left(-2 \cdot x \leq 4.223574591298496 \cdot 10^{-06}\right):\\ \;\;\;\;\sqrt[3]{\frac{2}{1 + e^{-2 \cdot x}} \cdot \left(\frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{2}{1 + e^{-2 \cdot x}}\right)} - 1\\ \mathbf{else}:\\ \;\;\;\;x - 0.3333333333333333 \cdot {x}^{3}\\ \end{array}\]
\frac{2}{1 + e^{-2 \cdot x}} - 1
↓
\begin{array}{l}
\mathbf{if}\;-2 \cdot x \leq -109316.8462524794 \lor \neg \left(-2 \cdot x \leq 4.223574591298496 \cdot 10^{-06}\right):\\
\;\;\;\;\sqrt[3]{\frac{2}{1 + e^{-2 \cdot x}} \cdot \left(\frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{2}{1 + e^{-2 \cdot x}}\right)} - 1\\

\mathbf{else}:\\
\;\;\;\;x - 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) -109316.8462524794)
         (not (<= (* -2.0 x) 4.223574591298496e-06)))
   (-
    (cbrt
     (*
      (/ 2.0 (+ 1.0 (exp (* -2.0 x))))
      (* (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) (/ 2.0 (+ 1.0 (exp (* -2.0 x)))))))
    1.0)
   (- x (* 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) <= -109316.8462524794) || !((-2.0 * x) <= 4.223574591298496e-06)) {
		tmp = cbrt((2.0 / (1.0 + exp(-2.0 * x))) * ((2.0 / (1.0 + exp(-2.0 * x))) * (2.0 / (1.0 + exp(-2.0 * x))))) - 1.0;
	} else {
		tmp = x - (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) < -109316.846252479401 or 4.22357459129849562e-6 < (*.f64 -2 x)

    1. Initial program 0.1

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

    3. Applied add-cbrt-cube_binary64_7960.1

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

    if -109316.846252479401 < (*.f64 -2 x) < 4.22357459129849562e-6

    1. Initial program 58.4

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

    2. Taylor expanded around 0 0.8

      \[\leadsto \color{blue}{x - 0.3333333333333333 \cdot {x}^{3}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -109316.8462524794 \lor \neg \left(-2 \cdot x \leq 4.223574591298496 \cdot 10^{-06}\right):\\ \;\;\;\;\sqrt[3]{\frac{2}{1 + e^{-2 \cdot x}} \cdot \left(\frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{2}{1 + e^{-2 \cdot x}}\right)} - 1\\ \mathbf{else}:\\ \;\;\;\;x - 0.3333333333333333 \cdot {x}^{3}\\ \end{array}\]

Reproduce

herbie shell --seed 2021044 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))