Average Error: 29.5 → 0.0
Time: 4.1s
Precision: binary64
\[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -0.0008260042363517994 \lor \neg \left(x \leq 0.0006575528516900341\right):\\ \;\;\;\;\sqrt[3]{\frac{2}{1 + {\left(e^{-2}\right)}^{x}} - 1} \cdot \left(\sqrt[3]{\frac{2}{1 + {\left(e^{-2}\right)}^{x}} - 1} \cdot \sqrt[3]{\frac{{\left(\frac{2}{1 + {\left(e^{-2}\right)}^{x}}\right)}^{3} - {1}^{3}}{1 \cdot 1 + 2 \cdot \frac{1 + \frac{2}{1 + {\left(e^{-2}\right)}^{x}}}{1 + {\left(e^{-2}\right)}^{x}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1 - {x}^{3} \cdot \left(x \cdot 5.551115123125783 \cdot 10^{-17} + 0.33333333333333337\right)\\ \end{array}\]
\frac{2}{1 + e^{-2 \cdot x}} - 1
↓
\begin{array}{l}
\mathbf{if}\;x \leq -0.0008260042363517994 \lor \neg \left(x \leq 0.0006575528516900341\right):\\
\;\;\;\;\sqrt[3]{\frac{2}{1 + {\left(e^{-2}\right)}^{x}} - 1} \cdot \left(\sqrt[3]{\frac{2}{1 + {\left(e^{-2}\right)}^{x}} - 1} \cdot \sqrt[3]{\frac{{\left(\frac{2}{1 + {\left(e^{-2}\right)}^{x}}\right)}^{3} - {1}^{3}}{1 \cdot 1 + 2 \cdot \frac{1 + \frac{2}{1 + {\left(e^{-2}\right)}^{x}}}{1 + {\left(e^{-2}\right)}^{x}}}}\right)\\

\mathbf{else}:\\
\;\;\;\;x \cdot 1 - {x}^{3} \cdot \left(x \cdot 5.551115123125783 \cdot 10^{-17} + 0.33333333333333337\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 (<= x -0.0008260042363517994) (not (<= x 0.0006575528516900341)))
   (*
    (cbrt (- (/ 2.0 (+ 1.0 (pow (exp -2.0) x))) 1.0))
    (*
     (cbrt (- (/ 2.0 (+ 1.0 (pow (exp -2.0) x))) 1.0))
     (cbrt
      (/
       (- (pow (/ 2.0 (+ 1.0 (pow (exp -2.0) x))) 3.0) (pow 1.0 3.0))
       (+
        (* 1.0 1.0)
        (*
         2.0
         (/
          (+ 1.0 (/ 2.0 (+ 1.0 (pow (exp -2.0) x))))
          (+ 1.0 (pow (exp -2.0) x)))))))))
   (-
    (* x 1.0)
    (* (pow x 3.0) (+ (* x 5.551115123125783e-17) 0.33333333333333337)))))
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 VAR;
	if (((x <= -0.0008260042363517994) || !(x <= 0.0006575528516900341))) {
		VAR = ((double) (((double) cbrt(((double) ((2.0 / ((double) (1.0 + ((double) pow(((double) exp(-2.0)), x))))) - 1.0)))) * ((double) (((double) cbrt(((double) ((2.0 / ((double) (1.0 + ((double) pow(((double) exp(-2.0)), x))))) - 1.0)))) * ((double) cbrt((((double) (((double) pow((2.0 / ((double) (1.0 + ((double) pow(((double) exp(-2.0)), x))))), 3.0)) - ((double) pow(1.0, 3.0)))) / ((double) (((double) (1.0 * 1.0)) + ((double) (2.0 * (((double) (1.0 + (2.0 / ((double) (1.0 + ((double) pow(((double) exp(-2.0)), x))))))) / ((double) (1.0 + ((double) pow(((double) exp(-2.0)), x))))))))))))))));
	} else {
		VAR = ((double) (((double) (x * 1.0)) - ((double) (((double) pow(x, 3.0)) * ((double) (((double) (x * 5.551115123125783e-17)) + 0.33333333333333337))))));
	}
	return VAR;
}

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 x < -8.2600423635179943e-4 or 6.5755285169003409e-4 < x

    1. Initial program 0.0

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

    3. Applied add-cube-cbrt0.1

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

    4. Simplified0.1

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

    5. Simplified0.1

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

    7. Applied flip3--0.1

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

    8. Simplified0.1

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

    if -8.2600423635179943e-4 < x < 6.5755285169003409e-4

    1. Initial program 59.2

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

    2. Taylor expanded around 0 0.0

      \[\leadsto \color{blue}{1 \cdot x - \left(5.551115123125783 \cdot 10^{-17} \cdot {x}^{4} + 0.33333333333333337 \cdot {x}^{3}\right)}\]

    3. Simplified0.0

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0008260042363517994 \lor \neg \left(x \leq 0.0006575528516900341\right):\\ \;\;\;\;\sqrt[3]{\frac{2}{1 + {\left(e^{-2}\right)}^{x}} - 1} \cdot \left(\sqrt[3]{\frac{2}{1 + {\left(e^{-2}\right)}^{x}} - 1} \cdot \sqrt[3]{\frac{{\left(\frac{2}{1 + {\left(e^{-2}\right)}^{x}}\right)}^{3} - {1}^{3}}{1 \cdot 1 + 2 \cdot \frac{1 + \frac{2}{1 + {\left(e^{-2}\right)}^{x}}}{1 + {\left(e^{-2}\right)}^{x}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1 - {x}^{3} \cdot \left(x \cdot 5.551115123125783 \cdot 10^{-17} + 0.33333333333333337\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020198 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))