Average Error: 29.8 → 1.0
Time: 5.9s
Precision: 64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 192.56282467868999:\\ \;\;\;\;\frac{\left(0.66666666666666674 \cdot {x}^{3} + 2\right) - 1 \cdot {x}^{2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \sqrt[3]{{\left(\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}\right)}^{3}}}{2}\\ \end{array}\]
\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 192.56282467868999:\\
\;\;\;\;\frac{\left(0.66666666666666674 \cdot {x}^{3} + 2\right) - 1 \cdot {x}^{2}}{2}\\

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

\end{array}
double code(double x, double eps) {
	return ((double) (((double) (((double) (((double) (1.0 + ((double) (1.0 / eps)))) * ((double) exp(((double) -(((double) (((double) (1.0 - eps)) * x)))))))) - ((double) (((double) (((double) (1.0 / eps)) - 1.0)) * ((double) exp(((double) -(((double) (((double) (1.0 + eps)) * x)))))))))) / 2.0));
}
↓
double code(double x, double eps) {
	double VAR;
	if ((x <= 192.56282467869)) {
		VAR = ((double) (((double) (((double) (((double) (0.6666666666666667 * ((double) pow(x, 3.0)))) + 2.0)) - ((double) (1.0 * ((double) pow(x, 2.0)))))) / 2.0));
	} else {
		VAR = ((double) (((double) (((double) (((double) (1.0 + ((double) (1.0 / eps)))) * ((double) exp(((double) -(((double) (((double) (1.0 - eps)) * x)))))))) - ((double) cbrt(((double) pow(((double) (((double) (((double) (1.0 / eps)) - 1.0)) * ((double) exp(((double) -(((double) (((double) (1.0 + eps)) * x)))))))), 3.0)))))) / 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 < 192.56282467869

    1. Initial program 39.0

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

    2. Taylor expanded around 0 1.3

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

    if 192.56282467869 < x

    1. Initial program 0.1

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

    3. Applied add-cbrt-cube0.1

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

    4. Applied add-cbrt-cube41.4

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

    5. Applied cbrt-unprod41.4

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

    6. Simplified0.1

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

  4. Final simplification1.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 192.56282467868999:\\ \;\;\;\;\frac{\left(0.66666666666666674 \cdot {x}^{3} + 2\right) - 1 \cdot {x}^{2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \sqrt[3]{{\left(\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}\right)}^{3}}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020122 
(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))