Average Error: 28.7 → 0.2
Time: 3.9s
Precision: binary64
\[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -79.47395879001132:\\ \;\;\;\;\sqrt[3]{\frac{2}{1 + e^{-2 \cdot x}} - 1} \cdot \left(\sqrt[3]{\frac{2}{1 + e^{-2 \cdot x}} - 1} \cdot \sqrt[3]{\frac{2}{1 + e^{-2 \cdot x}} - 1}\right)\\ \mathbf{elif}\;-2 \cdot x \leq 0.0002704380990454027:\\ \;\;\;\;x - 0.3333333333333333 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\log \left(e^{\frac{2}{1 + e^{-2 \cdot x}} - 1}\right) \cdot \left(\log \left(e^{\frac{2}{1 + e^{-2 \cdot x}} - 1}\right) \cdot \log \left(e^{\frac{2}{1 + e^{-2 \cdot x}} - 1}\right)\right)}\\ \end{array}\]
\frac{2}{1 + e^{-2 \cdot x}} - 1
↓
\begin{array}{l}
\mathbf{if}\;-2 \cdot x \leq -79.47395879001132:\\
\;\;\;\;\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{elif}\;-2 \cdot x \leq 0.0002704380990454027:\\
\;\;\;\;x - 0.3333333333333333 \cdot {x}^{3}\\

\mathbf{else}:\\
\;\;\;\;\sqrt[3]{\log \left(e^{\frac{2}{1 + e^{-2 \cdot x}} - 1}\right) \cdot \left(\log \left(e^{\frac{2}{1 + e^{-2 \cdot x}} - 1}\right) \cdot \log \left(e^{\frac{2}{1 + e^{-2 \cdot x}} - 1}\right)\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) -79.47395879001132)
   (*
    (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))))
   (if (<= (* -2.0 x) 0.0002704380990454027)
     (- x (* 0.3333333333333333 (pow x 3.0)))
     (cbrt
      (*
       (log (exp (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0)))
       (*
        (log (exp (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0)))
        (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) <= -79.47395879001132) {
		tmp = 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));
	} else if ((-2.0 * x) <= 0.0002704380990454027) {
		tmp = x - (0.3333333333333333 * pow(x, 3.0));
	} else {
		tmp = cbrt(log(exp((2.0 / (1.0 + exp(-2.0 * x))) - 1.0)) * (log(exp((2.0 / (1.0 + exp(-2.0 * x))) - 1.0)) * 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) < -79.473958790011324

    1. Initial program 0

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

    3. Applied add-cube-cbrt_binary64_7950

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

    if -79.473958790011324 < (*.f64 -2 x) < 2.704380990454027e-4

    1. Initial program 58.9

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

    2. Taylor expanded around 0 0.4

      \[\leadsto \color{blue}{x - 0.3333333333333333 \cdot {x}^{3}}\]

    if 2.704380990454027e-4 < (*.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_binary64_7990.1

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

    4. Applied add-log-exp_binary64_7990.1

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

    5. Applied diff-log_binary64_8520.1

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

    6. Simplified0.1

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

    8. Applied add-cbrt-cube_binary64_7960.1

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -79.47395879001132:\\ \;\;\;\;\sqrt[3]{\frac{2}{1 + e^{-2 \cdot x}} - 1} \cdot \left(\sqrt[3]{\frac{2}{1 + e^{-2 \cdot x}} - 1} \cdot \sqrt[3]{\frac{2}{1 + e^{-2 \cdot x}} - 1}\right)\\ \mathbf{elif}\;-2 \cdot x \leq 0.0002704380990454027:\\ \;\;\;\;x - 0.3333333333333333 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\log \left(e^{\frac{2}{1 + e^{-2 \cdot x}} - 1}\right) \cdot \left(\log \left(e^{\frac{2}{1 + e^{-2 \cdot x}} - 1}\right) \cdot \log \left(e^{\frac{2}{1 + e^{-2 \cdot x}} - 1}\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2021058 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))