Average Error: 29.2 → 0.8
Time: 6.2s
Precision: 64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 1.84613317056484205:\\ \;\;\;\;\frac{\frac{{\left(0.66666666666666674 \cdot {x}^{3} + 2\right)}^{3} - {\left(1 \cdot {x}^{2}\right)}^{3}}{\left(0.66666666666666674 \cdot {x}^{3} + 2\right) \cdot \left(\left(0.66666666666666674 \cdot {x}^{3} + 2\right) + 1 \cdot {x}^{2}\right) + 1 \cdot \left(1 \cdot {x}^{4}\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot {\left(e^{\sqrt[3]{-\left(1 - \varepsilon\right) \cdot x} \cdot \sqrt[3]{-\left(1 - \varepsilon\right) \cdot x}}\right)}^{\left(\sqrt[3]{-\left(1 - \varepsilon\right) \cdot x}\right)} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\\ \end{array}\]
\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 1.84613317056484205:\\
\;\;\;\;\frac{\frac{{\left(0.66666666666666674 \cdot {x}^{3} + 2\right)}^{3} - {\left(1 \cdot {x}^{2}\right)}^{3}}{\left(0.66666666666666674 \cdot {x}^{3} + 2\right) \cdot \left(\left(0.66666666666666674 \cdot {x}^{3} + 2\right) + 1 \cdot {x}^{2}\right) + 1 \cdot \left(1 \cdot {x}^{4}\right)}}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot {\left(e^{\sqrt[3]{-\left(1 - \varepsilon\right) \cdot x} \cdot \sqrt[3]{-\left(1 - \varepsilon\right) \cdot x}}\right)}^{\left(\sqrt[3]{-\left(1 - \varepsilon\right) \cdot x}\right)} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{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 <= 1.846133170564842)) {
		VAR = (((pow(((0.6666666666666667 * pow(x, 3.0)) + 2.0), 3.0) - pow((1.0 * pow(x, 2.0)), 3.0)) / ((((0.6666666666666667 * pow(x, 3.0)) + 2.0) * (((0.6666666666666667 * pow(x, 3.0)) + 2.0) + (1.0 * pow(x, 2.0)))) + (1.0 * (1.0 * pow(x, 4.0))))) / 2.0);
	} else {
		VAR = ((((1.0 + (1.0 / eps)) * pow(exp((cbrt(-((1.0 - eps) * x)) * cbrt(-((1.0 - eps) * x)))), cbrt(-((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 < 1.846133170564842

    1. Initial program 38.9

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

    2. Taylor expanded around 0 1.0

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

    4. Applied flip3--1.0

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

    5. Simplified1.0

      \[\leadsto \frac{\frac{{\left(0.66666666666666674 \cdot {x}^{3} + 2\right)}^{3} - {\left(1 \cdot {x}^{2}\right)}^{3}}{\color{blue}{\left(0.66666666666666674 \cdot {x}^{3} + 2\right) \cdot \left(\left(0.66666666666666674 \cdot {x}^{3} + 2\right) + 1 \cdot {x}^{2}\right) + 1 \cdot \left(1 \cdot {x}^{4}\right)}}}{2}\]

    if 1.846133170564842 < x

    1. Initial program 0.4

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

    3. Applied add-cube-cbrt0.4

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

    4. Applied exp-prod0.4

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

  4. Final simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 1.84613317056484205:\\ \;\;\;\;\frac{\frac{{\left(0.66666666666666674 \cdot {x}^{3} + 2\right)}^{3} - {\left(1 \cdot {x}^{2}\right)}^{3}}{\left(0.66666666666666674 \cdot {x}^{3} + 2\right) \cdot \left(\left(0.66666666666666674 \cdot {x}^{3} + 2\right) + 1 \cdot {x}^{2}\right) + 1 \cdot \left(1 \cdot {x}^{4}\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot {\left(e^{\sqrt[3]{-\left(1 - \varepsilon\right) \cdot x} \cdot \sqrt[3]{-\left(1 - \varepsilon\right) \cdot x}}\right)}^{\left(\sqrt[3]{-\left(1 - \varepsilon\right) \cdot x}\right)} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020075 
(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))