Average Error: 28.9 → 1.0
Time: 4.8s
Precision: binary64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq 481.9795405026713:\\ \;\;\;\;\frac{{x}^{3} \cdot 0.6666666666666667 + \left(2 - 1 \cdot \left(x \cdot x\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\sqrt[3]{x} \cdot \left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\varepsilon - 1\right)\right)} + e^{x \cdot \left(\left(-\varepsilon\right) - 1\right)} \cdot \left(1 - \frac{1}{\varepsilon}\right)}{2}\\ \end{array}\]
\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 \leq 481.9795405026713:\\
\;\;\;\;\frac{{x}^{3} \cdot 0.6666666666666667 + \left(2 - 1 \cdot \left(x \cdot x\right)\right)}{2}\\

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

\end{array}
double code(double x, double eps) {
	return (((double) (((double) (((double) (1.0 + (1.0 / eps))) * ((double) exp(((double) -(((double) (((double) (1.0 - eps)) * x)))))))) - ((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 <= 481.9795405026713)) {
		VAR = (((double) (((double) (((double) pow(x, 3.0)) * 0.6666666666666667)) + ((double) (2.0 - ((double) (1.0 * ((double) (x * x)))))))) / 2.0);
	} else {
		VAR = (((double) (((double) (((double) (1.0 + (1.0 / eps))) * ((double) exp(((double) (((double) cbrt(x)) * ((double) (((double) (((double) cbrt(x)) * ((double) cbrt(x)))) * ((double) (eps - 1.0)))))))))) + ((double) (((double) exp(((double) (x * ((double) (((double) -(eps)) - 1.0)))))) * ((double) (1.0 - (1.0 / eps))))))) / 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 < 481.97954050267128

    1. Initial program Error: 38.3 bits

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

    2. Taylor expanded around 0 Error: 1.3 bits

      \[\leadsto \frac{\color{blue}{\left(0.6666666666666667 \cdot {x}^{3} + 2\right) - 1 \cdot {x}^{2}}}{2}\]

    3. SimplifiedError: 1.3 bits

      \[\leadsto \frac{\color{blue}{{x}^{3} \cdot 0.6666666666666667 + \left(2 - 1 \cdot \left(x \cdot x\right)\right)}}{2}\]

    if 481.97954050267128 < x

    1. Initial program Error: 0.0 bits

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

    3. Applied add-cube-cbrtError: 0.0 bits

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

    4. Applied associate-*r*Error: 0.0 bits

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

  4. Final simplificationError: 1.0 bits

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 481.9795405026713:\\ \;\;\;\;\frac{{x}^{3} \cdot 0.6666666666666667 + \left(2 - 1 \cdot \left(x \cdot x\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\sqrt[3]{x} \cdot \left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\varepsilon - 1\right)\right)} + e^{x \cdot \left(\left(-\varepsilon\right) - 1\right)} \cdot \left(1 - \frac{1}{\varepsilon}\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020200 
(FPCore (x eps)
  :name "NMSE Section 6.1 mentioned, A"
  :precision binary64
  (/ (- (* (+ 1.0 (/ 1.0 eps)) (exp (- (* (- 1.0 eps) x)))) (* (- (/ 1.0 eps) 1.0) (exp (- (* (+ 1.0 eps) x))))) 2.0))