Average Error: 28.9 → 0.8
Time: 10.9s
Precision: 64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 210.239463704909753:\\ \;\;\;\;1 \cdot \left({x}^{2} \cdot \left(0.33333333333333337 \cdot x - 0.5\right) + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \frac{1}{\varepsilon}}{\sqrt{e^{\left(1 - \varepsilon\right) \cdot x}} \cdot \sqrt{e^{\left(1 - \varepsilon\right) \cdot x}}}}{2} - \frac{\frac{\frac{1}{\varepsilon} - 1}{e^{\left(1 + \varepsilon\right) \cdot x}}}{2}\\ \end{array}\]
\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 210.239463704909753:\\
\;\;\;\;1 \cdot \left({x}^{2} \cdot \left(0.33333333333333337 \cdot x - 0.5\right) + 1\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{1 + \frac{1}{\varepsilon}}{\sqrt{e^{\left(1 - \varepsilon\right) \cdot x}} \cdot \sqrt{e^{\left(1 - \varepsilon\right) \cdot x}}}}{2} - \frac{\frac{\frac{1}{\varepsilon} - 1}{e^{\left(1 + \varepsilon\right) \cdot x}}}{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 <= 210.23946370490975)) {
		VAR = ((double) (1.0 * ((double) (((double) (((double) pow(x, 2.0)) * ((double) (((double) (0.33333333333333337 * x)) - 0.5)))) + 1.0))));
	} else {
		VAR = ((double) (((double) (((double) (((double) (1.0 + ((double) (1.0 / eps)))) / ((double) (((double) sqrt(((double) exp(((double) (((double) (1.0 - eps)) * x)))))) * ((double) sqrt(((double) exp(((double) (((double) (1.0 - eps)) * x)))))))))) / 2.0)) - ((double) (((double) (((double) (((double) (1.0 / eps)) - 1.0)) / ((double) exp(((double) (((double) (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 < 210.23946370490975

    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. Simplified38.8

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

    3. Taylor expanded around 0 1.1

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

    5. Applied *-un-lft-identity1.1

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

    6. Applied *-un-lft-identity1.1

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

    7. Applied distribute-lft-out--1.1

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

    8. Simplified1.1

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

    if 210.23946370490975 < 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-sqr-sqrt0.1

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

  4. Final simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 210.239463704909753:\\ \;\;\;\;1 \cdot \left({x}^{2} \cdot \left(0.33333333333333337 \cdot x - 0.5\right) + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \frac{1}{\varepsilon}}{\sqrt{e^{\left(1 - \varepsilon\right) \cdot x}} \cdot \sqrt{e^{\left(1 - \varepsilon\right) \cdot x}}}}{2} - \frac{\frac{\frac{1}{\varepsilon} - 1}{e^{\left(1 + \varepsilon\right) \cdot x}}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020114 
(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))