Average Error: 29.4 → 1.1
Time: 7.5s
Precision: 64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 12.312429890800345:\\ \;\;\;\;\frac{\sqrt[3]{{\left(\left(0.66666666666666674 \cdot {x}^{3} + 2\right) - 1 \cdot {x}^{2}\right)}^{3}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{\left(\left(\left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\sqrt[3]{e^{-\left(1 - \varepsilon\right) \cdot x}} \cdot \sqrt[3]{e^{-\left(1 - \varepsilon\right) \cdot x}}\right)\right) \cdot \sqrt[3]{e^{-\left(1 - \varepsilon\right) \cdot x}} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}\right) \cdot \left(\left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\sqrt[3]{e^{-\left(1 - \varepsilon\right) \cdot x}} \cdot \sqrt[3]{e^{-\left(1 - \varepsilon\right) \cdot x}}\right)\right) \cdot \sqrt[3]{e^{-\left(1 - \varepsilon\right) \cdot x}} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}\right)\right) \cdot \left(\left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\sqrt[3]{e^{-\left(1 - \varepsilon\right) \cdot x}} \cdot \sqrt[3]{e^{-\left(1 - \varepsilon\right) \cdot x}}\right)\right) \cdot \sqrt[3]{e^{-\left(1 - \varepsilon\right) \cdot x}} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}\right)}}{2}\\ \end{array}\]
\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}
↓
\begin{array}{l}
\mathbf{if}\;x \le 12.312429890800345:\\
\;\;\;\;\frac{\sqrt[3]{{\left(\left(0.66666666666666674 \cdot {x}^{3} + 2\right) - 1 \cdot {x}^{2}\right)}^{3}}}{2}\\

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

\end{array}
double code(double x, double eps) {
	return ((((1.0 + (1.0 / eps)) * exp(-((1.0 - eps) * x))) - (((1.0 / eps) - 1.0) * exp(-((1.0 + eps) * x)))) / 2.0);
}
↓
double code(double x, double eps) {
	double VAR;
	if ((x <= 12.312429890800345)) {
		VAR = (cbrt(pow((((0.6666666666666667 * pow(x, 3.0)) + 2.0) - (1.0 * pow(x, 2.0))), 3.0)) / 2.0);
	} else {
		VAR = (cbrt(((((((1.0 + (1.0 / eps)) * (cbrt(exp(-((1.0 - eps) * x))) * cbrt(exp(-((1.0 - eps) * x))))) * cbrt(exp(-((1.0 - eps) * x)))) - (((1.0 / eps) - 1.0) * exp(-((1.0 + eps) * x)))) * ((((1.0 + (1.0 / eps)) * (cbrt(exp(-((1.0 - eps) * x))) * cbrt(exp(-((1.0 - eps) * x))))) * cbrt(exp(-((1.0 - eps) * x)))) - (((1.0 / eps) - 1.0) * exp(-((1.0 + eps) * x))))) * ((((1.0 + (1.0 / eps)) * (cbrt(exp(-((1.0 - eps) * x))) * cbrt(exp(-((1.0 - eps) * x))))) * cbrt(exp(-((1.0 - eps) * x)))) - (((1.0 / eps) - 1.0) * exp(-((1.0 + eps) * x)))))) / 2.0);
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if x < 12.312429890800345

    1. Initial program 39.2

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]

    2. Taylor expanded around 0 1.3

      \[\leadsto \frac{\color{blue}{\left(0.66666666666666674 \cdot {x}^{3} + 2\right) - 1 \cdot {x}^{2}}}{2}\]
    3. Using strategy rm

    4. Applied add-cbrt-cube1.3

      \[\leadsto \frac{\color{blue}{\sqrt[3]{\left(\left(\left(0.66666666666666674 \cdot {x}^{3} + 2\right) - 1 \cdot {x}^{2}\right) \cdot \left(\left(0.66666666666666674 \cdot {x}^{3} + 2\right) - 1 \cdot {x}^{2}\right)\right) \cdot \left(\left(0.66666666666666674 \cdot {x}^{3} + 2\right) - 1 \cdot {x}^{2}\right)}}}{2}\]

    5. Simplified1.3

      \[\leadsto \frac{\sqrt[3]{\color{blue}{{\left(\left(0.66666666666666674 \cdot {x}^{3} + 2\right) - 1 \cdot {x}^{2}\right)}^{3}}}}{2}\]

    if 12.312429890800345 < x

    1. Initial program 0.3

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
    2. Using strategy rm

    3. Applied add-cube-cbrt0.3

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

    4. Applied associate-*r*0.3

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

    6. Applied add-cbrt-cube0.3

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

  4. Final simplification1.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 12.312429890800345:\\ \;\;\;\;\frac{\sqrt[3]{{\left(\left(0.66666666666666674 \cdot {x}^{3} + 2\right) - 1 \cdot {x}^{2}\right)}^{3}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{\left(\left(\left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\sqrt[3]{e^{-\left(1 - \varepsilon\right) \cdot x}} \cdot \sqrt[3]{e^{-\left(1 - \varepsilon\right) \cdot x}}\right)\right) \cdot \sqrt[3]{e^{-\left(1 - \varepsilon\right) \cdot x}} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}\right) \cdot \left(\left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\sqrt[3]{e^{-\left(1 - \varepsilon\right) \cdot x}} \cdot \sqrt[3]{e^{-\left(1 - \varepsilon\right) \cdot x}}\right)\right) \cdot \sqrt[3]{e^{-\left(1 - \varepsilon\right) \cdot x}} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}\right)\right) \cdot \left(\left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\sqrt[3]{e^{-\left(1 - \varepsilon\right) \cdot x}} \cdot \sqrt[3]{e^{-\left(1 - \varepsilon\right) \cdot x}}\right)\right) \cdot \sqrt[3]{e^{-\left(1 - \varepsilon\right) \cdot x}} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}\right)}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020078 
(FPCore (x eps)
  :name "NMSE Section 6.1 mentioned, A"
  :precision binary64
  (/ (- (* (+ 1 (/ 1 eps)) (exp (- (* (- 1 eps) x)))) (* (- (/ 1 eps) 1) (exp (- (* (+ 1 eps) x))))) 2))