Average Error: 29.2 → 0.0
Time: 4.1s
Precision: binary64
\[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.013230893820878172:\\ \;\;\;\;\sqrt[3]{{\left(\frac{2}{1 + e^{-2 \cdot x}} - 1\right)}^{2}} \cdot \sqrt[3]{\left(1 + \sqrt{\frac{2}{1 + e^{-2 \cdot x}}}\right) \cdot \left(\sqrt{\frac{2}{1 + e^{-2 \cdot x}}} + -1\right)}\\ \mathbf{elif}\;-2 \cdot x \leq 6.7885598968249405 \cdot 10^{-06}:\\ \;\;\;\;\left(x + 0.13333333333333333 \cdot {x}^{5}\right) - 0.3333333333333333 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(\frac{2}{1 + e^{-2 \cdot x}} - 1\right)}^{3}}\\ \end{array}\]
\frac{2}{1 + e^{-2 \cdot x}} - 1
↓
\begin{array}{l}
\mathbf{if}\;-2 \cdot x \leq -0.013230893820878172:\\
\;\;\;\;\sqrt[3]{{\left(\frac{2}{1 + e^{-2 \cdot x}} - 1\right)}^{2}} \cdot \sqrt[3]{\left(1 + \sqrt{\frac{2}{1 + e^{-2 \cdot x}}}\right) \cdot \left(\sqrt{\frac{2}{1 + e^{-2 \cdot x}}} + -1\right)}\\

\mathbf{elif}\;-2 \cdot x \leq 6.7885598968249405 \cdot 10^{-06}:\\
\;\;\;\;\left(x + 0.13333333333333333 \cdot {x}^{5}\right) - 0.3333333333333333 \cdot {x}^{3}\\

\mathbf{else}:\\
\;\;\;\;\sqrt[3]{{\left(\frac{2}{1 + e^{-2 \cdot x}} - 1\right)}^{3}}\\

\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) -0.013230893820878172)
   (*
    (cbrt (pow (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0) 2.0))
    (cbrt
     (*
      (+ 1.0 (sqrt (/ 2.0 (+ 1.0 (exp (* -2.0 x))))))
      (+ (sqrt (/ 2.0 (+ 1.0 (exp (* -2.0 x))))) -1.0))))
   (if (<= (* -2.0 x) 6.7885598968249405e-06)
     (-
      (+ x (* 0.13333333333333333 (pow x 5.0)))
      (* 0.3333333333333333 (pow x 3.0)))
     (cbrt (pow (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0) 3.0)))))
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) <= -0.013230893820878172) {
		tmp = cbrt(pow(((2.0 / (1.0 + exp(-2.0 * x))) - 1.0), 2.0)) * cbrt((1.0 + sqrt(2.0 / (1.0 + exp(-2.0 * x)))) * (sqrt(2.0 / (1.0 + exp(-2.0 * x))) + -1.0));
	} else if ((-2.0 * x) <= 6.7885598968249405e-06) {
		tmp = (x + (0.13333333333333333 * pow(x, 5.0))) - (0.3333333333333333 * pow(x, 3.0));
	} else {
		tmp = cbrt(pow(((2.0 / (1.0 + exp(-2.0 * x))) - 1.0), 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 3 regimes

  2. if (*.f64 -2 x) < -0.013230893820878172

    1. Initial program 0.0

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

    3. Applied add-cbrt-cube_binary640.0

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

    4. Simplified0.0

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

    6. Applied add-cube-cbrt_binary640.0

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

    7. Simplified0.0

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

    8. Simplified0.0

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

    10. Applied add-sqr-sqrt_binary640.0

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

    11. Applied difference-of-sqr-1_binary640.0

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

    12. Simplified0.0

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

    13. Simplified0.0

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

    if -0.013230893820878172 < (*.f64 -2 x) < 6.78855989682494053e-6

    1. Initial program 59.2

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

    2. Taylor expanded around 0 0.0

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

    if 6.78855989682494053e-6 < (*.f64 -2 x)

    1. Initial program 0.1

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

    3. Applied add-cbrt-cube_binary640.1

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

    4. Simplified0.1

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.013230893820878172:\\ \;\;\;\;\sqrt[3]{{\left(\frac{2}{1 + e^{-2 \cdot x}} - 1\right)}^{2}} \cdot \sqrt[3]{\left(1 + \sqrt{\frac{2}{1 + e^{-2 \cdot x}}}\right) \cdot \left(\sqrt{\frac{2}{1 + e^{-2 \cdot x}}} + -1\right)}\\ \mathbf{elif}\;-2 \cdot x \leq 6.7885598968249405 \cdot 10^{-06}:\\ \;\;\;\;\left(x + 0.13333333333333333 \cdot {x}^{5}\right) - 0.3333333333333333 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(\frac{2}{1 + e^{-2 \cdot x}} - 1\right)}^{3}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020253 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))