Average Error: 29.1 → 0.4
Time: 2.9s
Precision: binary64
\[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
↓
\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.8699315969460777 \lor \neg \left(-2 \cdot x \leq 4.4707392111573346 \cdot 10^{-15}\right):\\ \;\;\;\;\log \left(e^{\frac{2}{1 + e^{-2 \cdot x}} - 1}\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
\frac{2}{1 + e^{-2 \cdot x}} - 1
↓
\begin{array}{l}
\mathbf{if}\;-2 \cdot x \leq -0.8699315969460777 \lor \neg \left(-2 \cdot x \leq 4.4707392111573346 \cdot 10^{-15}\right):\\
\;\;\;\;\log \left(e^{\frac{2}{1 + e^{-2 \cdot x}} - 1}\right)\\

\mathbf{else}:\\
\;\;\;\;x\\


\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) -0.8699315969460777)
         (not (<= (* -2.0 x) 4.4707392111573346e-15)))
   (log (exp (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0)))
   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) <= -0.8699315969460777) || !((-2.0 * x) <= 4.4707392111573346e-15)) {
		tmp = log(exp((2.0 / (1.0 + exp(-2.0 * x))) - 1.0));
	} else {
		tmp = 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 2 regimes

  2. if (*.f64 -2 x) < -0.8699315969460777 or 4.4707392111573346e-15 < (*.f64 -2 x)

    1. Initial program 0.6

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

    2. Applied add-log-exp_binary640.6

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

    3. Applied add-log-exp_binary640.6

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

    4. Applied diff-log_binary640.6

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

    5. Simplified0.6

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

    if -0.8699315969460777 < (*.f64 -2 x) < 4.4707392111573346e-15

    1. Initial program 59.7

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

    2. Taylor expanded in x around 0 0.2

      \[\leadsto \color{blue}{x} \]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.8699315969460777 \lor \neg \left(-2 \cdot x \leq 4.4707392111573346 \cdot 10^{-15}\right):\\ \;\;\;\;\log \left(e^{\frac{2}{1 + e^{-2 \cdot x}} - 1}\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Reproduce

herbie shell --seed 2022081 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))