Average Error: 29.5 → 0.0
Time: 3.1s
Precision: binary64
\[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.3204611975901199:\\ \;\;\;\;\frac{\frac{4}{\left(1 + e^{-2 \cdot x}\right) \cdot \left(1 + e^{-2 \cdot x}\right)} + -1}{1 + \frac{2}{1 + e^{-2 \cdot x}}}\\ \mathbf{elif}\;-2 \cdot x \leq 0.0010295493322116723:\\ \;\;\;\;x + \left(0.13333333333333333 \cdot {x}^{5} - 0.3333333333333333 \cdot {x}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \sqrt{\frac{2}{1 + e^{-2 \cdot x}}}\right) \cdot \left(-1 + \sqrt{\frac{2}{1 + e^{-2 \cdot x}}}\right)\\ \end{array}\]
\frac{2}{1 + e^{-2 \cdot x}} - 1
↓
\begin{array}{l}
\mathbf{if}\;-2 \cdot x \leq -0.3204611975901199:\\
\;\;\;\;\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}}}\\

\mathbf{elif}\;-2 \cdot x \leq 0.0010295493322116723:\\
\;\;\;\;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(-1 + \sqrt{\frac{2}{1 + e^{-2 \cdot x}}}\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) -0.3204611975901199)
   (/
    (+ (/ 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))))))
   (if (<= (* -2.0 x) 0.0010295493322116723)
     (+
      x
      (-
       (* 0.13333333333333333 (pow x 5.0))
       (* 0.3333333333333333 (pow x 3.0))))
     (*
      (+ 1.0 (sqrt (/ 2.0 (+ 1.0 (exp (* -2.0 x))))))
      (+ -1.0 (sqrt (/ 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) <= -0.3204611975901199) {
		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))));
	} else if ((-2.0 * x) <= 0.0010295493322116723) {
		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)))) * (-1.0 + sqrt(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) < -0.320461197590119873

    1. Initial program 0.0

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

    3. Applied flip--_binary640.0

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

    4. Simplified0.0

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

    5. Simplified0.0

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

    if -0.320461197590119873 < (*.f64 -2 x) < 0.00102954933221167228

    1. Initial program 59.2

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

    2. Taylor expanded around 0 0.0

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

    4. Applied associate--l+_binary640.0

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

    if 0.00102954933221167228 < (*.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_binary640.0

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

    4. Applied add-sqr-sqrt_binary640.0

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

    5. Applied difference-of-squares_binary640.0

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

    6. Simplified0.0

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

    7. Simplified0.0

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.3204611975901199:\\ \;\;\;\;\frac{\frac{4}{\left(1 + e^{-2 \cdot x}\right) \cdot \left(1 + e^{-2 \cdot x}\right)} + -1}{1 + \frac{2}{1 + e^{-2 \cdot x}}}\\ \mathbf{elif}\;-2 \cdot x \leq 0.0010295493322116723:\\ \;\;\;\;x + \left(0.13333333333333333 \cdot {x}^{5} - 0.3333333333333333 \cdot {x}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \sqrt{\frac{2}{1 + e^{-2 \cdot x}}}\right) \cdot \left(-1 + \sqrt{\frac{2}{1 + e^{-2 \cdot x}}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020231 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))