Average Error: 29.3 → 0.0
Time: 6.2s
Precision: binary64
\[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -3.219252166599685:\\ \;\;\;\;\frac{\frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{2}{1 + e^{-2 \cdot x}} - 1}{1 + \frac{2}{1 + e^{-2 \cdot x}}}\\ \mathbf{elif}\;-2 \cdot x \leq 0.014856573568863609:\\ \;\;\;\;x + \left(0.13333333333333333 \cdot {x}^{5} + \left({x}^{7} \cdot -0.05396825396825397 - 0.3333333333333333 \cdot {x}^{3}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\sqrt{1 + e^{-2 \cdot x}}}}{\sqrt{1 + e^{-2 \cdot x}}} - 1\\ \end{array}\]
\frac{2}{1 + e^{-2 \cdot x}} - 1
↓
\begin{array}{l}
\mathbf{if}\;-2 \cdot x \leq -3.219252166599685:\\
\;\;\;\;\frac{\frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{2}{1 + e^{-2 \cdot x}} - 1}{1 + \frac{2}{1 + e^{-2 \cdot x}}}\\

\mathbf{elif}\;-2 \cdot x \leq 0.014856573568863609:\\
\;\;\;\;x + \left(0.13333333333333333 \cdot {x}^{5} + \left({x}^{7} \cdot -0.05396825396825397 - 0.3333333333333333 \cdot {x}^{3}\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{2}{\sqrt{1 + e^{-2 \cdot x}}}}{\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) -3.219252166599685)
   (/
    (-
     (* (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) (/ 2.0 (+ 1.0 (exp (* -2.0 x)))))
     1.0)
    (+ 1.0 (/ 2.0 (+ 1.0 (exp (* -2.0 x))))))
   (if (<= (* -2.0 x) 0.014856573568863609)
     (+
      x
      (+
       (* 0.13333333333333333 (pow x 5.0))
       (-
        (* (pow x 7.0) -0.05396825396825397)
        (* 0.3333333333333333 (pow x 3.0)))))
     (-
      (/
       (/ 2.0 (sqrt (+ 1.0 (exp (* -2.0 x)))))
       (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) <= -3.219252166599685) {
		tmp = (((2.0 / (1.0 + exp(-2.0 * x))) * (2.0 / (1.0 + exp(-2.0 * x)))) - 1.0) / (1.0 + (2.0 / (1.0 + exp(-2.0 * x))));
	} else if ((-2.0 * x) <= 0.014856573568863609) {
		tmp = x + ((0.13333333333333333 * pow(x, 5.0)) + ((pow(x, 7.0) * -0.05396825396825397) - (0.3333333333333333 * pow(x, 3.0))));
	} else {
		tmp = ((2.0 / sqrt(1.0 + exp(-2.0 * x))) / 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) < -3.2192521665996852

    1. Initial program 0.0

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

    3. Applied flip--_binary64_7350.0

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

    if -3.2192521665996852 < (*.f64 -2 x) < 0.0148565735688636087

    1. Initial program 58.9

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

    2. Taylor expanded around 0 0.1

      \[\leadsto \color{blue}{\left(x + 0.13333333333333333 \cdot {x}^{5}\right) - \left(0.05396825396825397 \cdot {x}^{7} + 0.3333333333333333 \cdot {x}^{3}\right)}\]

    3. Simplified0.1

      \[\leadsto \color{blue}{\left(x + 0.13333333333333333 \cdot {x}^{5}\right) - \left(0.3333333333333333 \cdot {x}^{3} + 0.05396825396825397 \cdot {x}^{7}\right)}\]
    4. Using strategy rm

    5. Applied associate--l+_binary64_6970.1

      \[\leadsto \color{blue}{x + \left(0.13333333333333333 \cdot {x}^{5} - \left(0.3333333333333333 \cdot {x}^{3} + 0.05396825396825397 \cdot {x}^{7}\right)\right)}\]

    6. Simplified0.1

      \[\leadsto x + \color{blue}{\left(0.13333333333333333 \cdot {x}^{5} + \left({x}^{7} \cdot -0.05396825396825397 - 0.3333333333333333 \cdot {x}^{3}\right)\right)}\]

    if 0.0148565735688636087 < (*.f64 -2 x)

    1. Initial program 0.0

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

    3. Applied add-sqr-sqrt_binary64_7820.0

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

    4. Applied associate-/r*_binary64_7040.0

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -3.219252166599685:\\ \;\;\;\;\frac{\frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{2}{1 + e^{-2 \cdot x}} - 1}{1 + \frac{2}{1 + e^{-2 \cdot x}}}\\ \mathbf{elif}\;-2 \cdot x \leq 0.014856573568863609:\\ \;\;\;\;x + \left(0.13333333333333333 \cdot {x}^{5} + \left({x}^{7} \cdot -0.05396825396825397 - 0.3333333333333333 \cdot {x}^{3}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\sqrt{1 + e^{-2 \cdot x}}}}{\sqrt{1 + e^{-2 \cdot x}}} - 1\\ \end{array}\]

Reproduce

herbie shell --seed 2021093 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))