Average Error: 29.3 → 0.4
Time: 4.2s
Precision: binary64
\[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.287660782793311:\\ \;\;\;\;\frac{\log \left(e^{{\left(\frac{2}{1 + {\left(e^{-2}\right)}^{x}}\right)}^{2} + -1}\right)}{\frac{1 + {\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{3}}{1 + \frac{-2 + \frac{4}{1 + e^{-2 \cdot x}}}{1 + e^{-2 \cdot x}}}}\\ \mathbf{elif}\;-2 \cdot x \leq 2.922242698280268 \cdot 10^{-16}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 + \frac{4}{\left(1 + e^{-2 \cdot x}\right) \cdot \left(1 + e^{-2 \cdot x}\right)}}{1 + \frac{2}{1 + e^{-2 \cdot x}}}\\ \end{array}\]
\frac{2}{1 + e^{-2 \cdot x}} - 1
↓
\begin{array}{l}
\mathbf{if}\;-2 \cdot x \leq -0.287660782793311:\\
\;\;\;\;\frac{\log \left(e^{{\left(\frac{2}{1 + {\left(e^{-2}\right)}^{x}}\right)}^{2} + -1}\right)}{\frac{1 + {\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{3}}{1 + \frac{-2 + \frac{4}{1 + e^{-2 \cdot x}}}{1 + e^{-2 \cdot x}}}}\\

\mathbf{elif}\;-2 \cdot x \leq 2.922242698280268 \cdot 10^{-16}:\\
\;\;\;\;x\\

\mathbf{else}:\\
\;\;\;\;\frac{-1 + \frac{4}{\left(1 + e^{-2 \cdot x}\right) \cdot \left(1 + e^{-2 \cdot x}\right)}}{1 + \frac{2}{1 + e^{-2 \cdot x}}}\\

\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) -0.287660782793311)
   (/
    (log (exp (+ (pow (/ 2.0 (+ 1.0 (pow (exp -2.0) x))) 2.0) -1.0)))
    (/
     (+ 1.0 (pow (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 3.0))
     (+
      1.0
      (/ (+ -2.0 (/ 4.0 (+ 1.0 (exp (* -2.0 x))))) (+ 1.0 (exp (* -2.0 x)))))))
   (if (<= (* -2.0 x) 2.922242698280268e-16)
     x
     (/
      (+ -1.0 (/ 4.0 (* (+ 1.0 (exp (* -2.0 x))) (+ 1.0 (exp (* -2.0 x))))))
      (+ 1.0 (/ 2.0 (+ 1.0 (exp (* -2.0 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) <= -0.287660782793311) {
		tmp = log(exp(pow((2.0 / (1.0 + pow(exp(-2.0), x))), 2.0) + -1.0)) / ((1.0 + pow((2.0 / (1.0 + exp(-2.0 * x))), 3.0)) / (1.0 + ((-2.0 + (4.0 / (1.0 + exp(-2.0 * x)))) / (1.0 + exp(-2.0 * x)))));
	} else if ((-2.0 * x) <= 2.922242698280268e-16) {
		tmp = x;
	} else {
		tmp = (-1.0 + (4.0 / ((1.0 + exp(-2.0 * x)) * (1.0 + exp(-2.0 * x))))) / (1.0 + (2.0 / (1.0 + exp(-2.0 * 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 3 regimes

  2. if (*.f64 -2 x) < -0.28766078279331098

    1. Initial program 0.0

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

    3. Applied flip--_binary64_24400.0

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

    4. Simplified0.0

      \[\leadsto \frac{\color{blue}{\frac{4}{\left(1 + e^{-2 \cdot x}\right) \cdot \left(1 + e^{-2 \cdot x}\right)} + -1}}{\frac{2}{1 + e^{-2 \cdot x}} + 1}\]

    5. Simplified0.0

      \[\leadsto \frac{\frac{4}{\left(1 + e^{-2 \cdot x}\right) \cdot \left(1 + e^{-2 \cdot x}\right)} + -1}{\color{blue}{1 + \frac{2}{1 + e^{-2 \cdot x}}}}\]
    6. Using strategy rm

    7. Applied flip3-+_binary64_24680.0

      \[\leadsto \frac{\frac{4}{\left(1 + e^{-2 \cdot x}\right) \cdot \left(1 + e^{-2 \cdot x}\right)} + -1}{\color{blue}{\frac{{1}^{3} + {\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{3}}{1 \cdot 1 + \left(\frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{2}{1 + e^{-2 \cdot x}} - 1 \cdot \frac{2}{1 + e^{-2 \cdot x}}\right)}}}\]

    8. Simplified0.0

      \[\leadsto \frac{\frac{4}{\left(1 + e^{-2 \cdot x}\right) \cdot \left(1 + e^{-2 \cdot x}\right)} + -1}{\frac{\color{blue}{1 + {\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{3}}}{1 \cdot 1 + \left(\frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{2}{1 + e^{-2 \cdot x}} - 1 \cdot \frac{2}{1 + e^{-2 \cdot x}}\right)}}\]

    9. Simplified0.0

      \[\leadsto \frac{\frac{4}{\left(1 + e^{-2 \cdot x}\right) \cdot \left(1 + e^{-2 \cdot x}\right)} + -1}{\frac{1 + {\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{3}}{\color{blue}{1 + \frac{-2 + \frac{4}{1 + e^{-2 \cdot x}}}{1 + e^{-2 \cdot x}}}}}\]
    10. Using strategy rm

    11. Applied add-log-exp_binary64_25040.0

      \[\leadsto \frac{\frac{4}{\left(1 + e^{-2 \cdot x}\right) \cdot \left(1 + e^{-2 \cdot x}\right)} + \color{blue}{\log \left(e^{-1}\right)}}{\frac{1 + {\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{3}}{1 + \frac{-2 + \frac{4}{1 + e^{-2 \cdot x}}}{1 + e^{-2 \cdot x}}}}\]

    12. Applied add-log-exp_binary64_25040.0

      \[\leadsto \frac{\color{blue}{\log \left(e^{\frac{4}{\left(1 + e^{-2 \cdot x}\right) \cdot \left(1 + e^{-2 \cdot x}\right)}}\right)} + \log \left(e^{-1}\right)}{\frac{1 + {\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{3}}{1 + \frac{-2 + \frac{4}{1 + e^{-2 \cdot x}}}{1 + e^{-2 \cdot x}}}}\]

    13. Applied sum-log_binary64_25560.0

      \[\leadsto \frac{\color{blue}{\log \left(e^{\frac{4}{\left(1 + e^{-2 \cdot x}\right) \cdot \left(1 + e^{-2 \cdot x}\right)}} \cdot e^{-1}\right)}}{\frac{1 + {\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{3}}{1 + \frac{-2 + \frac{4}{1 + e^{-2 \cdot x}}}{1 + e^{-2 \cdot x}}}}\]

    14. Simplified0.0

      \[\leadsto \frac{\log \color{blue}{\left(e^{{\left(\frac{2}{1 + {\left(e^{-2}\right)}^{x}}\right)}^{2} + -1}\right)}}{\frac{1 + {\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{3}}{1 + \frac{-2 + \frac{4}{1 + e^{-2 \cdot x}}}{1 + e^{-2 \cdot x}}}}\]

    if -0.28766078279331098 < (*.f64 -2 x) < 2.9222426982802681e-16

    1. Initial program 59.8

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

    2. Taylor expanded around 0 0.2

      \[\leadsto \color{blue}{x}\]

    if 2.9222426982802681e-16 < (*.f64 -2 x)

    1. Initial program 1.1

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

    3. Applied flip--_binary64_24401.1

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

    4. Simplified1.1

      \[\leadsto \frac{\color{blue}{\frac{4}{\left(1 + e^{-2 \cdot x}\right) \cdot \left(1 + e^{-2 \cdot x}\right)} + -1}}{\frac{2}{1 + e^{-2 \cdot x}} + 1}\]

    5. Simplified1.1

      \[\leadsto \frac{\frac{4}{\left(1 + e^{-2 \cdot x}\right) \cdot \left(1 + e^{-2 \cdot x}\right)} + -1}{\color{blue}{1 + \frac{2}{1 + e^{-2 \cdot x}}}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.287660782793311:\\ \;\;\;\;\frac{\log \left(e^{{\left(\frac{2}{1 + {\left(e^{-2}\right)}^{x}}\right)}^{2} + -1}\right)}{\frac{1 + {\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{3}}{1 + \frac{-2 + \frac{4}{1 + e^{-2 \cdot x}}}{1 + e^{-2 \cdot x}}}}\\ \mathbf{elif}\;-2 \cdot x \leq 2.922242698280268 \cdot 10^{-16}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 + \frac{4}{\left(1 + e^{-2 \cdot x}\right) \cdot \left(1 + e^{-2 \cdot x}\right)}}{1 + \frac{2}{1 + e^{-2 \cdot x}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020353 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))