Average Error: 29.3 → 0.7
Time: 2.5s
Precision: binary64
\[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -15543813.620502109:\\ \;\;\;\;\sqrt[3]{\frac{2}{1 + e^{-2 \cdot x}} - 1} \cdot \left(\sqrt[3]{\frac{2}{1 + e^{-2 \cdot x}} - 1} \cdot \sqrt[3]{\frac{2}{1 + e^{-2 \cdot x}} - 1}\right)\\ \mathbf{elif}\;-2 \cdot x \leq 1.509110211603142 \cdot 10^{-12}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{1 + e^{-2 \cdot x}}} \cdot \frac{2}{\sqrt{1 + e^{-2 \cdot x}}} - 1\\ \end{array}\]
\frac{2}{1 + e^{-2 \cdot x}} - 1
↓
\begin{array}{l}
\mathbf{if}\;-2 \cdot x \leq -15543813.620502109:\\
\;\;\;\;\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 1.509110211603142 \cdot 10^{-12}:\\
\;\;\;\;x\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{\sqrt{1 + e^{-2 \cdot x}}} \cdot \frac{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) -15543813.620502109)
   (*
    (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) 1.509110211603142e-12)
     x
     (-
      (*
       (/ 1.0 (sqrt (+ 1.0 (exp (* -2.0 x)))))
       (/ 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) <= -15543813.620502109) {
		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) <= 1.509110211603142e-12) {
		tmp = x;
	} else {
		tmp = ((1.0 / sqrt(1.0 + exp(-2.0 * x))) * (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) < -15543813.6205021087

    1. Initial program 0

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

    3. Applied add-cube-cbrt_binary64_28410

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

    if -15543813.6205021087 < (*.f64 -2 x) < 1.50911021160314208e-12

    1. Initial program 58.7

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

    2. Taylor expanded around 0 1.0

      \[\leadsto \color{blue}{x}\]

    if 1.50911021160314208e-12 < (*.f64 -2 x)

    1. Initial program 0.8

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

    3. Applied add-sqr-sqrt_binary64_28280.9

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

    4. Applied *-un-lft-identity_binary64_28060.9

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

    5. Applied times-frac_binary64_28120.9

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

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -15543813.620502109:\\ \;\;\;\;\sqrt[3]{\frac{2}{1 + e^{-2 \cdot x}} - 1} \cdot \left(\sqrt[3]{\frac{2}{1 + e^{-2 \cdot x}} - 1} \cdot \sqrt[3]{\frac{2}{1 + e^{-2 \cdot x}} - 1}\right)\\ \mathbf{elif}\;-2 \cdot x \leq 1.509110211603142 \cdot 10^{-12}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{1 + e^{-2 \cdot x}}} \cdot \frac{2}{\sqrt{1 + e^{-2 \cdot x}}} - 1\\ \end{array}\]

Reproduce

herbie shell --seed 2021032 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))