Average Error: 29.3 → 0.3
Time: 4.0s
Precision: binary64
\[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -29080703.03703688:\\ \;\;\;\;e^{\log \left(\frac{2}{1 + e^{-2 \cdot x}} - 1\right)}\\ \mathbf{elif}\;-2 \cdot x \leq 8.059635556623151 \cdot 10^{-05}:\\ \;\;\;\;\left(x + 0.13333333333333333 \cdot {x}^{5}\right) - 0.3333333333333333 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{2} \cdot \sqrt[3]{2}}{\sqrt{1 + e^{-2 \cdot x}}} \cdot \frac{\sqrt[3]{2}}{\sqrt{1 + e^{-2 \cdot x}}} - 1\\ \end{array}\]
\frac{2}{1 + e^{-2 \cdot x}} - 1
↓
\begin{array}{l}
\mathbf{if}\;-2 \cdot x \leq -29080703.03703688:\\
\;\;\;\;e^{\log \left(\frac{2}{1 + e^{-2 \cdot x}} - 1\right)}\\

\mathbf{elif}\;-2 \cdot x \leq 8.059635556623151 \cdot 10^{-05}:\\
\;\;\;\;\left(x + 0.13333333333333333 \cdot {x}^{5}\right) - 0.3333333333333333 \cdot {x}^{3}\\

\mathbf{else}:\\
\;\;\;\;\frac{\sqrt[3]{2} \cdot \sqrt[3]{2}}{\sqrt{1 + e^{-2 \cdot x}}} \cdot \frac{\sqrt[3]{2}}{\sqrt{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) -29080703.03703688)
   (exp (log (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0)))
   (if (<= (* -2.0 x) 8.059635556623151e-05)
     (-
      (+ x (* 0.13333333333333333 (pow x 5.0)))
      (* 0.3333333333333333 (pow x 3.0)))
     (-
      (*
       (/ (* (cbrt 2.0) (cbrt 2.0)) (sqrt (+ 1.0 (exp (* -2.0 x)))))
       (/ (cbrt 2.0) (sqrt (+ 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) <= -29080703.03703688) {
		tmp = exp(log((2.0 / (1.0 + exp(-2.0 * x))) - 1.0));
	} else if ((-2.0 * x) <= 8.059635556623151e-05) {
		tmp = (x + (0.13333333333333333 * pow(x, 5.0))) - (0.3333333333333333 * pow(x, 3.0));
	} else {
		tmp = (((cbrt(2.0) * cbrt(2.0)) / sqrt(1.0 + exp(-2.0 * x))) * (cbrt(2.0) / sqrt(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) < -29080703.037036881

    1. Initial program 0

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

    3. Applied add-exp-log_binary64_1160

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

    if -29080703.037036881 < (*.f64 -2 x) < 8.0596355566231513e-5

    1. Initial program 58.6

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

    2. Taylor expanded around 0 0.5

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

    if 8.0596355566231513e-5 < (*.f64 -2 x)

    1. Initial program 0.1

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

    3. Applied add-sqr-sqrt_binary64_1000.1

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

    4. Applied add-cube-cbrt_binary64_1130.1

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

    5. Applied times-frac_binary64_840.1

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -29080703.03703688:\\ \;\;\;\;e^{\log \left(\frac{2}{1 + e^{-2 \cdot x}} - 1\right)}\\ \mathbf{elif}\;-2 \cdot x \leq 8.059635556623151 \cdot 10^{-05}:\\ \;\;\;\;\left(x + 0.13333333333333333 \cdot {x}^{5}\right) - 0.3333333333333333 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{2} \cdot \sqrt[3]{2}}{\sqrt{1 + e^{-2 \cdot x}}} \cdot \frac{\sqrt[3]{2}}{\sqrt{1 + e^{-2 \cdot x}}} - 1\\ \end{array}\]

Reproduce

herbie shell --seed 2020343 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))