Average Error: 29.3 → 0.1
Time: 4.3s
Precision: binary64
\[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.012262932310461267:\\ \;\;\;\;\frac{{\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{3} - 1}{1 + \frac{2 + \frac{4}{1 + e^{-2 \cdot x}}}{1 + e^{-2 \cdot x}}}\\ \mathbf{elif}\;-2 \cdot x \leq 7.641716791549214 \cdot 10^{-09}:\\ \;\;\;\;\left(x + 0.13333333333333333 \cdot {x}^{5}\right) - 0.3333333333333333 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{4}{\left(1 + e^{-2 \cdot x}\right) \cdot \left(1 + e^{-2 \cdot x}\right)} + -1}{1 + \frac{2}{1 + e^{-2 \cdot x}}}\\ \end{array}\]
\frac{2}{1 + e^{-2 \cdot x}} - 1
↓
\begin{array}{l}
\mathbf{if}\;-2 \cdot x \leq -0.012262932310461267:\\
\;\;\;\;\frac{{\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{3} - 1}{1 + \frac{2 + \frac{4}{1 + e^{-2 \cdot x}}}{1 + e^{-2 \cdot x}}}\\

\mathbf{elif}\;-2 \cdot x \leq 7.641716791549214 \cdot 10^{-09}:\\
\;\;\;\;\left(x + 0.13333333333333333 \cdot {x}^{5}\right) - 0.3333333333333333 \cdot {x}^{3}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{4}{\left(1 + e^{-2 \cdot x}\right) \cdot \left(1 + e^{-2 \cdot x}\right)} + -1}{1 + \frac{2}{1 + e^{-2 \cdot x}}}\\

\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) -0.012262932310461267)
   (/
    (- (pow (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 3.0) 1.0)
    (+
     1.0
     (/ (+ 2.0 (/ 4.0 (+ 1.0 (exp (* -2.0 x))))) (+ 1.0 (exp (* -2.0 x))))))
   (if (<= (* -2.0 x) 7.641716791549214e-09)
     (-
      (+ x (* 0.13333333333333333 (pow x 5.0)))
      (* 0.3333333333333333 (pow x 3.0)))
     (/
      (+ (/ 4.0 (* (+ 1.0 (exp (* -2.0 x))) (+ 1.0 (exp (* -2.0 x))))) -1.0)
      (+ 1.0 (/ 2.0 (+ 1.0 (exp (* -2.0 x)))))))))
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) <= -0.012262932310461267) {
		tmp = (pow((2.0 / (1.0 + exp(-2.0 * x))), 3.0) - 1.0) / (1.0 + ((2.0 + (4.0 / (1.0 + exp(-2.0 * x)))) / (1.0 + exp(-2.0 * x))));
	} else if ((-2.0 * x) <= 7.641716791549214e-09) {
		tmp = (x + (0.13333333333333333 * pow(x, 5.0))) - (0.3333333333333333 * pow(x, 3.0));
	} else {
		tmp = ((4.0 / ((1.0 + exp(-2.0 * x)) * (1.0 + exp(-2.0 * x)))) + -1.0) / (1.0 + (2.0 / (1.0 + exp(-2.0 * x))));
	}
	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) < -0.0122629323104612668

    1. Initial program 0.0

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

    3. Applied flip3--_binary64_31510.0

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

    4. Simplified0.0

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

    5. Simplified0.0

      \[\leadsto \frac{{\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{3} - 1}{\color{blue}{1 + \frac{2 + \frac{4}{1 + e^{-2 \cdot x}}}{1 + e^{-2 \cdot x}}}}\]

    if -0.0122629323104612668 < (*.f64 -2 x) < 7.6417167915492141e-9

    1. Initial program 59.4

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

    2. Taylor expanded around 0 0.0

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

    if 7.6417167915492141e-9 < (*.f64 -2 x)

    1. Initial program 0.3

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

    3. Applied flip--_binary64_31220.3

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

    4. Simplified0.3

      \[\leadsto \frac{\color{blue}{\frac{4}{\left(1 + e^{-2 \cdot x}\right) \cdot \left(1 + e^{-2 \cdot x}\right)} + -1}}{\frac{2}{1 + e^{-2 \cdot x}} + 1}\]

    5. Simplified0.3

      \[\leadsto \frac{\frac{4}{\left(1 + e^{-2 \cdot x}\right) \cdot \left(1 + e^{-2 \cdot x}\right)} + -1}{\color{blue}{1 + \frac{2}{1 + e^{-2 \cdot x}}}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.012262932310461267:\\ \;\;\;\;\frac{{\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{3} - 1}{1 + \frac{2 + \frac{4}{1 + e^{-2 \cdot x}}}{1 + e^{-2 \cdot x}}}\\ \mathbf{elif}\;-2 \cdot x \leq 7.641716791549214 \cdot 10^{-09}:\\ \;\;\;\;\left(x + 0.13333333333333333 \cdot {x}^{5}\right) - 0.3333333333333333 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{4}{\left(1 + e^{-2 \cdot x}\right) \cdot \left(1 + e^{-2 \cdot x}\right)} + -1}{1 + \frac{2}{1 + e^{-2 \cdot x}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020356 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))