Average Error: 29.2 → 0.5
Time: 3.3s
Precision: binary64
\[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -37501469.614610635:\\ \;\;\;\;\log \left(\frac{e^{\frac{2}{1 + e^{-2 \cdot x}}}}{e}\right)\\ \mathbf{elif}\;-2 \cdot x \leq 0.01698418759074461:\\ \;\;\;\;\left(x + 0.13333333333333333 \cdot {x}^{5}\right) - \left(0.05396825396825397 \cdot {x}^{7} + 0.3333333333333333 \cdot {x}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} - 1\\ \end{array}\]
\frac{2}{1 + e^{-2 \cdot x}} - 1
↓
\begin{array}{l}
\mathbf{if}\;-2 \cdot x \leq -37501469.614610635:\\
\;\;\;\;\log \left(\frac{e^{\frac{2}{1 + e^{-2 \cdot x}}}}{e}\right)\\

\mathbf{elif}\;-2 \cdot x \leq 0.01698418759074461:\\
\;\;\;\;\left(x + 0.13333333333333333 \cdot {x}^{5}\right) - \left(0.05396825396825397 \cdot {x}^{7} + 0.3333333333333333 \cdot {x}^{3}\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{2}{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) -37501469.614610635)
   (log (/ (exp (/ 2.0 (+ 1.0 (exp (* -2.0 x))))) E))
   (if (<= (* -2.0 x) 0.01698418759074461)
     (-
      (+ x (* 0.13333333333333333 (pow x 5.0)))
      (+
       (* 0.05396825396825397 (pow x 7.0))
       (* 0.3333333333333333 (pow x 3.0))))
     (- (/ 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) <= -37501469.614610635) {
		tmp = log(exp(2.0 / (1.0 + exp(-2.0 * x))) / ((double) M_E));
	} else if ((-2.0 * x) <= 0.01698418759074461) {
		tmp = (x + (0.13333333333333333 * pow(x, 5.0))) - ((0.05396825396825397 * pow(x, 7.0)) + (0.3333333333333333 * pow(x, 3.0)));
	} else {
		tmp = (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) < -37501469.6146106347

    1. Initial program 0

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

    3. Applied add-log-exp_binary640

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

    4. Applied add-log-exp_binary640

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

    5. Applied diff-log_binary640

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

    if -37501469.6146106347 < (*.f64 -2 x) < 0.016984187590744611

    1. Initial program 58.0

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

    2. Taylor expanded around 0 1.1

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

    if 0.016984187590744611 < (*.f64 -2 x)

    1. Initial program 0.0

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

    3. Applied *-un-lft-identity_binary640.0

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -37501469.614610635:\\ \;\;\;\;\log \left(\frac{e^{\frac{2}{1 + e^{-2 \cdot x}}}}{e}\right)\\ \mathbf{elif}\;-2 \cdot x \leq 0.01698418759074461:\\ \;\;\;\;\left(x + 0.13333333333333333 \cdot {x}^{5}\right) - \left(0.05396825396825397 \cdot {x}^{7} + 0.3333333333333333 \cdot {x}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} - 1\\ \end{array}\]

Alternatives

Reproduce

herbie shell --seed 2021126 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))