Average Error: 29.1 → 0.3
Time: 2.3s
Precision: binary64
\[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -244757.89631741017 \lor \neg \left(-2 \cdot x \leq 5.209272421169758 \cdot 10^{-05}\right):\\ \;\;\;\;\log \left(\frac{e^{\frac{2}{1 + e^{-2 \cdot x}}}}{e}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x + 0.13333333333333333 \cdot {x}^{5}\right) - 0.3333333333333333 \cdot {x}^{3}\\ \end{array}\]
\frac{2}{1 + e^{-2 \cdot x}} - 1
↓
\begin{array}{l}
\mathbf{if}\;-2 \cdot x \leq -244757.89631741017 \lor \neg \left(-2 \cdot x \leq 5.209272421169758 \cdot 10^{-05}\right):\\
\;\;\;\;\log \left(\frac{e^{\frac{2}{1 + e^{-2 \cdot x}}}}{e}\right)\\

\mathbf{else}:\\
\;\;\;\;\left(x + 0.13333333333333333 \cdot {x}^{5}\right) - 0.3333333333333333 \cdot {x}^{3}\\

\end{array}
(FPCore (x y) :precision binary64 (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))
↓
(FPCore (x y)
 :precision binary64
 (if (or (<= (* -2.0 x) -244757.89631741017)
         (not (<= (* -2.0 x) 5.209272421169758e-05)))
   (log (/ (exp (/ 2.0 (+ 1.0 (exp (* -2.0 x))))) E))
   (-
    (+ x (* 0.13333333333333333 (pow x 5.0)))
    (* 0.3333333333333333 (pow x 3.0)))))
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) <= -244757.89631741017) || !((-2.0 * x) <= 5.209272421169758e-05)) {
		tmp = log(exp(2.0 / (1.0 + exp(-2.0 * x))) / ((double) M_E));
	} else {
		tmp = (x + (0.13333333333333333 * pow(x, 5.0))) - (0.3333333333333333 * pow(x, 3.0));
	}
	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 2 regimes

  2. if (*.f64 -2 x) < -244757.89631741017 or 5.20927242116975782e-5 < (*.f64 -2 x)

    1. Initial program 0.1

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

    3. Applied add-log-exp_binary640.1

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

    4. Applied add-log-exp_binary640.1

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

    5. Applied diff-log_binary640.1

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

    6. Simplified0.1

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

    if -244757.89631741017 < (*.f64 -2 x) < 5.20927242116975782e-5

    1. Initial program 58.6

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

    2. Taylor expanded around 0 0.6

      \[\leadsto \color{blue}{\left(x + 0.13333333333333333 \cdot {x}^{5}\right) - 0.3333333333333333 \cdot {x}^{3}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -244757.89631741017 \lor \neg \left(-2 \cdot x \leq 5.209272421169758 \cdot 10^{-05}\right):\\ \;\;\;\;\log \left(\frac{e^{\frac{2}{1 + e^{-2 \cdot x}}}}{e}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x + 0.13333333333333333 \cdot {x}^{5}\right) - 0.3333333333333333 \cdot {x}^{3}\\ \end{array}\]

Reproduce

herbie shell --seed 2020219 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))