Average Error: 29.8 → 0.1
Time: 7.6s
Precision: binary64
\[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -108.47803473002091:\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} - 1\\ \mathbf{elif}\;-2 \cdot x \leq 9.884693008229484 \cdot 10^{-05}:\\ \;\;\;\;x + \left(0.13333333333333333 \cdot {x}^{5} - 0.3333333333333333 \cdot {x}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \sqrt{\frac{2}{1 + e^{-2 \cdot x}}}\right) \cdot \left(\sqrt{\frac{2}{1 + e^{-2 \cdot x}}} + -1\right)\\ \end{array}\]
\frac{2}{1 + e^{-2 \cdot x}} - 1
↓
\begin{array}{l}
\mathbf{if}\;-2 \cdot x \leq -108.47803473002091:\\
\;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} - 1\\

\mathbf{elif}\;-2 \cdot x \leq 9.884693008229484 \cdot 10^{-05}:\\
\;\;\;\;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) -108.47803473002091)
   (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0)
   (if (<= (* -2.0 x) 9.884693008229484e-05)
     (+
      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) <= -108.47803473002091) {
		tmp = (2.0 / (1.0 + exp(-2.0 * x))) - 1.0;
	} else if ((-2.0 * x) <= 9.884693008229484e-05) {
		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) < -108.47803473002091

    1. Initial program 0

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

    if -108.47803473002091 < (*.f64 -2 x) < 9.88469300822948405e-5

    1. Initial program 58.9

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

    2. Taylor expanded around 0 0.2

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

    4. Applied associate--l+_binary64_3560.2

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

    if 9.88469300822948405e-5 < (*.f64 -2 x)

    1. Initial program 0.1

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

    3. Applied add-sqr-sqrt_binary64_4410.1

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

    4. Applied difference-of-sqr-1_binary64_3890.1

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

    5. Simplified0.1

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

    6. Simplified0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -108.47803473002091:\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} - 1\\ \mathbf{elif}\;-2 \cdot x \leq 9.884693008229484 \cdot 10^{-05}:\\ \;\;\;\;x + \left(0.13333333333333333 \cdot {x}^{5} - 0.3333333333333333 \cdot {x}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \sqrt{\frac{2}{1 + e^{-2 \cdot x}}}\right) \cdot \left(\sqrt{\frac{2}{1 + e^{-2 \cdot x}}} + -1\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020308 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))