Average Error: 29.5 → 1.0
Time: 4.6s
Precision: binary64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq 104.67319938439192:\\ \;\;\;\;\frac{\sqrt[3]{{\left(2 + x \cdot \left(\sqrt[3]{x \cdot 0.6666666666666667 - 1} \cdot \left(x \cdot \left(\sqrt[3]{x \cdot 0.6666666666666667 - 1} \cdot \sqrt[3]{x \cdot 0.6666666666666667 - 1}\right)\right)\right)\right)}^{3}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{x \cdot \left(\varepsilon - 1\right)} + {\left(e^{\left(-\varepsilon\right) - 1}\right)}^{x} \cdot \left(1 - \frac{1}{\varepsilon}\right)}{2}\\ \end{array}\]
\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 104.67319938439192:\\
\;\;\;\;\frac{\sqrt[3]{{\left(2 + x \cdot \left(\sqrt[3]{x \cdot 0.6666666666666667 - 1} \cdot \left(x \cdot \left(\sqrt[3]{x \cdot 0.6666666666666667 - 1} \cdot \sqrt[3]{x \cdot 0.6666666666666667 - 1}\right)\right)\right)\right)}^{3}}}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{x \cdot \left(\varepsilon - 1\right)} + {\left(e^{\left(-\varepsilon\right) - 1}\right)}^{x} \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 <= 104.67319938439192)) {
		VAR = (((double) cbrt(((double) pow(((double) (2.0 + ((double) (x * ((double) (((double) cbrt(((double) (((double) (x * 0.6666666666666667)) - 1.0)))) * ((double) (x * ((double) (((double) cbrt(((double) (((double) (x * 0.6666666666666667)) - 1.0)))) * ((double) cbrt(((double) (((double) (x * 0.6666666666666667)) - 1.0)))))))))))))), 3.0)))) / 2.0);
	} else {
		VAR = (((double) (((double) (((double) (1.0 + (1.0 / eps))) * ((double) exp(((double) (x * ((double) (eps - 1.0)))))))) + ((double) (((double) pow(((double) exp(((double) (((double) -(eps)) - 1.0)))), x)) * ((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 < 104.67319938439192

    1. Initial program 38.8

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

    2. Taylor expanded around 0 1.3

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

    3. Simplified1.3

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

    5. Applied add-cbrt-cube1.3

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

    6. Simplified1.3

      \[\leadsto \frac{\sqrt[3]{\color{blue}{{\left(2 + x \cdot \left(x \cdot \left(x \cdot 0.6666666666666667 - 1\right)\right)\right)}^{3}}}}{2}\]
    7. Using strategy rm

    8. Applied add-cube-cbrt1.3

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

    9. Applied associate-*r*1.3

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

    if 104.67319938439192 < 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 distribute-lft-neg-in0.1

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

    4. Applied exp-prod0.1

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

    5. Simplified0.1

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

  4. Final simplification1.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 104.67319938439192:\\ \;\;\;\;\frac{\sqrt[3]{{\left(2 + x \cdot \left(\sqrt[3]{x \cdot 0.6666666666666667 - 1} \cdot \left(x \cdot \left(\sqrt[3]{x \cdot 0.6666666666666667 - 1} \cdot \sqrt[3]{x \cdot 0.6666666666666667 - 1}\right)\right)\right)\right)}^{3}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{x \cdot \left(\varepsilon - 1\right)} + {\left(e^{\left(-\varepsilon\right) - 1}\right)}^{x} \cdot \left(1 - \frac{1}{\varepsilon}\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020196 
(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))