Average Error: 30.1 → 1.1
Time: 7.0s
Precision: 64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 282.804764533081823:\\ \;\;\;\;1 + \left({\left(\sqrt[3]{\sqrt[3]{{x}^{2}}}\right)}^{5} \cdot \sqrt[3]{\sqrt[3]{{x}^{2}}}\right) \cdot \left(\sqrt[3]{{x}^{2}} \cdot \left(x \cdot 0.33333333333333337 - 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(\frac{\frac{1 + \frac{1}{\varepsilon}}{e^{\left(1 - \varepsilon\right) \cdot x}}}{2} - \frac{\frac{\frac{1}{\varepsilon} - 1}{e^{\left(1 + \varepsilon\right) \cdot x}}}{2}\right)}\\ \end{array}\]
\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 282.804764533081823:\\
\;\;\;\;1 + \left({\left(\sqrt[3]{\sqrt[3]{{x}^{2}}}\right)}^{5} \cdot \sqrt[3]{\sqrt[3]{{x}^{2}}}\right) \cdot \left(\sqrt[3]{{x}^{2}} \cdot \left(x \cdot 0.33333333333333337 - 0.5\right)\right)\\

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

\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 temp;
	if ((x <= 282.8047645330818)) {
		temp = (1.0 + ((pow(cbrt(cbrt(pow(x, 2.0))), 5.0) * cbrt(cbrt(pow(x, 2.0)))) * (cbrt(pow(x, 2.0)) * ((x * 0.33333333333333337) - 0.5))));
	} else {
		temp = exp(log(((((1.0 + (1.0 / eps)) / exp(((1.0 - eps) * x))) / 2.0) - ((((1.0 / eps) - 1.0) / exp(((1.0 + eps) * x))) / 2.0))));
	}
	return temp;
}

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 < 282.8047645330818

    1. Initial program 39.6

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

    2. Simplified39.6

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

    3. Taylor expanded around 0 1.4

      \[\leadsto \color{blue}{\left(0.33333333333333337 \cdot {x}^{3} + 1\right) - 0.5 \cdot {x}^{2}}\]

    4. Simplified1.4

      \[\leadsto \color{blue}{1 + {x}^{2} \cdot \left(x \cdot 0.33333333333333337 - 0.5\right)}\]
    5. Using strategy rm

    6. Applied add-cube-cbrt1.4

      \[\leadsto 1 + \color{blue}{\left(\left(\sqrt[3]{{x}^{2}} \cdot \sqrt[3]{{x}^{2}}\right) \cdot \sqrt[3]{{x}^{2}}\right)} \cdot \left(x \cdot 0.33333333333333337 - 0.5\right)\]

    7. Applied associate-*l*1.4

      \[\leadsto 1 + \color{blue}{\left(\sqrt[3]{{x}^{2}} \cdot \sqrt[3]{{x}^{2}}\right) \cdot \left(\sqrt[3]{{x}^{2}} \cdot \left(x \cdot 0.33333333333333337 - 0.5\right)\right)}\]
    8. Using strategy rm

    9. Applied add-cube-cbrt1.4

      \[\leadsto 1 + \left(\sqrt[3]{{x}^{2}} \cdot \color{blue}{\left(\left(\sqrt[3]{\sqrt[3]{{x}^{2}}} \cdot \sqrt[3]{\sqrt[3]{{x}^{2}}}\right) \cdot \sqrt[3]{\sqrt[3]{{x}^{2}}}\right)}\right) \cdot \left(\sqrt[3]{{x}^{2}} \cdot \left(x \cdot 0.33333333333333337 - 0.5\right)\right)\]

    10. Applied associate-*r*1.4

      \[\leadsto 1 + \color{blue}{\left(\left(\sqrt[3]{{x}^{2}} \cdot \left(\sqrt[3]{\sqrt[3]{{x}^{2}}} \cdot \sqrt[3]{\sqrt[3]{{x}^{2}}}\right)\right) \cdot \sqrt[3]{\sqrt[3]{{x}^{2}}}\right)} \cdot \left(\sqrt[3]{{x}^{2}} \cdot \left(x \cdot 0.33333333333333337 - 0.5\right)\right)\]

    11. Simplified1.4

      \[\leadsto 1 + \left(\color{blue}{{\left(\sqrt[3]{\sqrt[3]{{x}^{2}}}\right)}^{5}} \cdot \sqrt[3]{\sqrt[3]{{x}^{2}}}\right) \cdot \left(\sqrt[3]{{x}^{2}} \cdot \left(x \cdot 0.33333333333333337 - 0.5\right)\right)\]

    if 282.8047645330818 < 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. Simplified0.1

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

    4. Applied add-exp-log0.1

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

  4. Final simplification1.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 282.804764533081823:\\ \;\;\;\;1 + \left({\left(\sqrt[3]{\sqrt[3]{{x}^{2}}}\right)}^{5} \cdot \sqrt[3]{\sqrt[3]{{x}^{2}}}\right) \cdot \left(\sqrt[3]{{x}^{2}} \cdot \left(x \cdot 0.33333333333333337 - 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(\frac{\frac{1 + \frac{1}{\varepsilon}}{e^{\left(1 - \varepsilon\right) \cdot x}}}{2} - \frac{\frac{\frac{1}{\varepsilon} - 1}{e^{\left(1 + \varepsilon\right) \cdot x}}}{2}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020066 
(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))