Average Error: 29.4 → 0.2
Time: 3.1s
Precision: binary64
\[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -40.51106863404356:\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} - 1\\ \mathbf{elif}\;-2 \cdot x \leq 8.169908299674417 \cdot 10^{-09}:\\ \;\;\;\;x + \left(0.13333333333333333 \cdot {x}^{5} - 0.3333333333333333 \cdot {x}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \sqrt{\frac{2}{1 + e^{-2 \cdot x}}}\right) \cdot \left(\sqrt{\frac{2}{1 + e^{-2 \cdot x}}} - 1\right)\\ \end{array}\]
\frac{2}{1 + e^{-2 \cdot x}} - 1
↓
\begin{array}{l}
\mathbf{if}\;-2 \cdot x \leq -40.51106863404356:\\
\;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} - 1\\

\mathbf{elif}\;-2 \cdot x \leq 8.169908299674417 \cdot 10^{-09}:\\
\;\;\;\;x + \left(0.13333333333333333 \cdot {x}^{5} - 0.3333333333333333 \cdot {x}^{3}\right)\\

\mathbf{else}:\\
\;\;\;\;\left(1 + \sqrt{\frac{2}{1 + e^{-2 \cdot x}}}\right) \cdot \left(\sqrt{\frac{2}{1 + e^{-2 \cdot x}}} - 1\right)\\

\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) -40.51106863404356)
   (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0)
   (if (<= (* -2.0 x) 8.169908299674417e-09)
     (+
      x
      (-
       (* 0.13333333333333333 (pow x 5.0))
       (* 0.3333333333333333 (pow x 3.0))))
     (*
      (+ 1.0 (sqrt (/ 2.0 (+ 1.0 (exp (* -2.0 x))))))
      (- (sqrt (/ 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) <= -40.51106863404356) {
		tmp = (2.0 / (1.0 + exp(-2.0 * x))) - 1.0;
	} else if ((-2.0 * x) <= 8.169908299674417e-09) {
		tmp = x + ((0.13333333333333333 * pow(x, 5.0)) - (0.3333333333333333 * pow(x, 3.0)));
	} else {
		tmp = (1.0 + sqrt(2.0 / (1.0 + exp(-2.0 * x)))) * (sqrt(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) < -40.5110686340435606

    1. Initial program 0

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

    3. Applied *-un-lft-identity_binary640

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

    4. Applied associate-/r*_binary640

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

    if -40.5110686340435606 < (*.f64 -2 x) < 8.169908299674417e-9

    1. Initial program 59.1

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

    2. Taylor expanded around 0 0.2

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

    3. Simplified0.2

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

    5. Applied sub-neg_binary640.2

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

    6. Applied associate-+l+_binary640.2

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

    7. Simplified0.2

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

    if 8.169908299674417e-9 < (*.f64 -2 x)

    1. Initial program 0.3

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

    3. Applied add-sqr-sqrt_binary640.3

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

    4. Applied difference-of-sqr-1_binary640.3

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -40.51106863404356:\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} - 1\\ \mathbf{elif}\;-2 \cdot x \leq 8.169908299674417 \cdot 10^{-09}:\\ \;\;\;\;x + \left(0.13333333333333333 \cdot {x}^{5} - 0.3333333333333333 \cdot {x}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \sqrt{\frac{2}{1 + e^{-2 \cdot x}}}\right) \cdot \left(\sqrt{\frac{2}{1 + e^{-2 \cdot x}}} - 1\right)\\ \end{array}\]

Alternatives

Reproduce

herbie shell --seed 2021118 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))