Average Error: 30.3 → 0.1
Time: 8.2s
Precision: binary64
\[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -8.271617323873846:\\ \;\;\;\;\sqrt{\frac{2}{1 + e^{-2 \cdot x}} - 1} \cdot \sqrt{\frac{2}{1 + e^{-2 \cdot x}} - 1}\\ \mathbf{elif}\;-2 \cdot x \leq 0.009839941204378123:\\ \;\;\;\;x + \left(0.13333333333333333 \cdot {x}^{5} - \left(0.3333333333333333 \cdot {x}^{3} + 0.05396825396825397 \cdot {x}^{7}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{\frac{2}{1 + e^{-2 \cdot x}} - 1}\right)\\ \end{array}\]
\frac{2}{1 + e^{-2 \cdot x}} - 1
↓
\begin{array}{l}
\mathbf{if}\;-2 \cdot x \leq -8.271617323873846:\\
\;\;\;\;\sqrt{\frac{2}{1 + e^{-2 \cdot x}} - 1} \cdot \sqrt{\frac{2}{1 + e^{-2 \cdot x}} - 1}\\

\mathbf{elif}\;-2 \cdot x \leq 0.009839941204378123:\\
\;\;\;\;x + \left(0.13333333333333333 \cdot {x}^{5} - \left(0.3333333333333333 \cdot {x}^{3} + 0.05396825396825397 \cdot {x}^{7}\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\log \left(e^{\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) -8.271617323873846)
   (*
    (sqrt (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))
    (sqrt (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0)))
   (if (<= (* -2.0 x) 0.009839941204378123)
     (+
      x
      (-
       (* 0.13333333333333333 (pow x 5.0))
       (+
        (* 0.3333333333333333 (pow x 3.0))
        (* 0.05396825396825397 (pow x 7.0)))))
     (log (exp (- (/ 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) <= -8.271617323873846) {
		tmp = sqrt((2.0 / (1.0 + exp(-2.0 * x))) - 1.0) * sqrt((2.0 / (1.0 + exp(-2.0 * x))) - 1.0);
	} else if ((-2.0 * x) <= 0.009839941204378123) {
		tmp = x + ((0.13333333333333333 * pow(x, 5.0)) - ((0.3333333333333333 * pow(x, 3.0)) + (0.05396825396825397 * pow(x, 7.0))));
	} else {
		tmp = log(exp((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) < -8.27161732387384596

    1. Initial program 0.0

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

    3. Applied add-sqr-sqrt_binary64_24870.0

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

    4. Simplified0.0

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

    5. Simplified0.0

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

    if -8.27161732387384596 < (*.f64 -2 x) < 0.0098399412043781234

    1. Initial program 58.8

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

    2. Taylor expanded around 0 0.1

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

    3. Simplified0.1

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

    5. Applied associate--l+_binary64_24020.1

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

    if 0.0098399412043781234 < (*.f64 -2 x)

    1. Initial program 0.0

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

    3. Applied add-log-exp_binary64_25040.0

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

    4. Applied add-log-exp_binary64_25040.0

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

    5. Applied diff-log_binary64_25570.0

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

    6. Simplified0.0

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -8.271617323873846:\\ \;\;\;\;\sqrt{\frac{2}{1 + e^{-2 \cdot x}} - 1} \cdot \sqrt{\frac{2}{1 + e^{-2 \cdot x}} - 1}\\ \mathbf{elif}\;-2 \cdot x \leq 0.009839941204378123:\\ \;\;\;\;x + \left(0.13333333333333333 \cdot {x}^{5} - \left(0.3333333333333333 \cdot {x}^{3} + 0.05396825396825397 \cdot {x}^{7}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{\frac{2}{1 + e^{-2 \cdot x}} - 1}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2021050 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))