Average Error: 29.3 → 0.6
Time: 2.7s
Precision: binary64
\[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -35.212264247415106 \lor \neg \left(-2 \cdot x \leq 2.2895961090980154 \cdot 10^{-18}\right):\\ \;\;\;\;\sqrt[3]{\frac{2}{1 + e^{-2 \cdot x}} - 1} \cdot \left(\sqrt[3]{\frac{2}{1 + e^{-2 \cdot x}} - 1} \cdot \sqrt[3]{\frac{2}{1 + e^{-2 \cdot x}} - 1}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1 - \left(5.551115123125783 \cdot 10^{-17} \cdot {x}^{4} + 0.33333333333333337 \cdot {x}^{3}\right)\\ \end{array}\]
\frac{2}{1 + e^{-2 \cdot x}} - 1
↓
\begin{array}{l}
\mathbf{if}\;-2 \cdot x \leq -35.212264247415106 \lor \neg \left(-2 \cdot x \leq 2.2895961090980154 \cdot 10^{-18}\right):\\
\;\;\;\;\sqrt[3]{\frac{2}{1 + e^{-2 \cdot x}} - 1} \cdot \left(\sqrt[3]{\frac{2}{1 + e^{-2 \cdot x}} - 1} \cdot \sqrt[3]{\frac{2}{1 + e^{-2 \cdot x}} - 1}\right)\\

\mathbf{else}:\\
\;\;\;\;x \cdot 1 - \left(5.551115123125783 \cdot 10^{-17} \cdot {x}^{4} + 0.33333333333333337 \cdot {x}^{3}\right)\\

\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) -35.212264247415106)
         (not (<= (* -2.0 x) 2.2895961090980154e-18)))
   (*
    (cbrt (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))
    (*
     (cbrt (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))
     (cbrt (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))))
   (-
    (* x 1.0)
    (+
     (* 5.551115123125783e-17 (pow x 4.0))
     (* 0.33333333333333337 (pow x 3.0))))))
double code(double x, double y) {
	return ((double) ((2.0 / ((double) (1.0 + ((double) exp(((double) (-2.0 * x))))))) - 1.0));
}
↓
double code(double x, double y) {
	double tmp;
	if (((((double) (-2.0 * x)) <= -35.212264247415106) || !(((double) (-2.0 * x)) <= 2.2895961090980154e-18))) {
		tmp = ((double) (((double) cbrt(((double) ((2.0 / ((double) (1.0 + ((double) exp(((double) (-2.0 * x))))))) - 1.0)))) * ((double) (((double) cbrt(((double) ((2.0 / ((double) (1.0 + ((double) exp(((double) (-2.0 * x))))))) - 1.0)))) * ((double) cbrt(((double) ((2.0 / ((double) (1.0 + ((double) exp(((double) (-2.0 * x))))))) - 1.0))))))));
	} else {
		tmp = ((double) (((double) (x * 1.0)) - ((double) (((double) (5.551115123125783e-17 * ((double) pow(x, 4.0)))) + ((double) (0.33333333333333337 * ((double) 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 -2.0 x) < -35.212264247415106 or 2.28959610909801543e-18 < (*.f64 -2.0 x)

    1. Initial program 1.0

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

    3. Applied add-cube-cbrt_binary641.0

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

    if -35.212264247415106 < (*.f64 -2.0 x) < 2.28959610909801543e-18

    1. Initial program 59.5

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

    2. Taylor expanded around 0 0.2

      \[\leadsto \color{blue}{1 \cdot x - \left(5.551115123125783 \cdot 10^{-17} \cdot {x}^{4} + 0.33333333333333337 \cdot {x}^{3}\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -35.212264247415106 \lor \neg \left(-2 \cdot x \leq 2.2895961090980154 \cdot 10^{-18}\right):\\ \;\;\;\;\sqrt[3]{\frac{2}{1 + e^{-2 \cdot x}} - 1} \cdot \left(\sqrt[3]{\frac{2}{1 + e^{-2 \cdot x}} - 1} \cdot \sqrt[3]{\frac{2}{1 + e^{-2 \cdot x}} - 1}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1 - \left(5.551115123125783 \cdot 10^{-17} \cdot {x}^{4} + 0.33333333333333337 \cdot {x}^{3}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020210 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))