Average Error: 29.6 → 1.1
Time: 27.3s
Precision: 64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 121.4869422814261525900292326696217060089:\\ \;\;\;\;\frac{0.6666666666666667406815349750104360282421 \cdot \left(\left(x \cdot x\right) \cdot x\right) + \left(2 - \left(x \cdot 1\right) \cdot x\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 \cdot \left(e^{\left(\varepsilon - 1\right) \cdot x} + \frac{e^{\left(\varepsilon - 1\right) \cdot x}}{\varepsilon}\right) - e^{\left(-\left(1 + \varepsilon\right)\right) \cdot x} \cdot \left(\frac{1}{\varepsilon} - 1\right)}{2}\\ \end{array}\]
\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 \le 121.4869422814261525900292326696217060089:\\
\;\;\;\;\frac{0.6666666666666667406815349750104360282421 \cdot \left(\left(x \cdot x\right) \cdot x\right) + \left(2 - \left(x \cdot 1\right) \cdot x\right)}{2}\\

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

\end{array}
double f(double x, double eps) {
        double r2766613 = 1.0;
        double r2766614 = eps;
        double r2766615 = r2766613 / r2766614;
        double r2766616 = r2766613 + r2766615;
        double r2766617 = r2766613 - r2766614;
        double r2766618 = x;
        double r2766619 = r2766617 * r2766618;
        double r2766620 = -r2766619;
        double r2766621 = exp(r2766620);
        double r2766622 = r2766616 * r2766621;
        double r2766623 = r2766615 - r2766613;
        double r2766624 = r2766613 + r2766614;
        double r2766625 = r2766624 * r2766618;
        double r2766626 = -r2766625;
        double r2766627 = exp(r2766626);
        double r2766628 = r2766623 * r2766627;
        double r2766629 = r2766622 - r2766628;
        double r2766630 = 2.0;
        double r2766631 = r2766629 / r2766630;
        return r2766631;
}
↓
double f(double x, double eps) {
        double r2766632 = x;
        double r2766633 = 121.48694228142615;
        bool r2766634 = r2766632 <= r2766633;
        double r2766635 = 0.6666666666666667;
        double r2766636 = r2766632 * r2766632;
        double r2766637 = r2766636 * r2766632;
        double r2766638 = r2766635 * r2766637;
        double r2766639 = 2.0;
        double r2766640 = 1.0;
        double r2766641 = r2766632 * r2766640;
        double r2766642 = r2766641 * r2766632;
        double r2766643 = r2766639 - r2766642;
        double r2766644 = r2766638 + r2766643;
        double r2766645 = r2766644 / r2766639;
        double r2766646 = eps;
        double r2766647 = r2766646 - r2766640;
        double r2766648 = r2766647 * r2766632;
        double r2766649 = exp(r2766648);
        double r2766650 = r2766649 / r2766646;
        double r2766651 = r2766649 + r2766650;
        double r2766652 = r2766640 * r2766651;
        double r2766653 = r2766640 + r2766646;
        double r2766654 = -r2766653;
        double r2766655 = r2766654 * r2766632;
        double r2766656 = exp(r2766655);
        double r2766657 = r2766640 / r2766646;
        double r2766658 = r2766657 - r2766640;
        double r2766659 = r2766656 * r2766658;
        double r2766660 = r2766652 - r2766659;
        double r2766661 = r2766660 / r2766639;
        double r2766662 = r2766634 ? r2766645 : r2766661;
        return r2766662;
}

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 < 121.48694228142615

    1. Initial program 39.5

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

    2. Taylor expanded around 0 1.4

      \[\leadsto \frac{\color{blue}{\left(0.6666666666666667406815349750104360282421 \cdot {x}^{3} + 2\right) - 1 \cdot {x}^{2}}}{2}\]

    3. Simplified1.4

      \[\leadsto \frac{\color{blue}{0.6666666666666667406815349750104360282421 \cdot \left(\left(x \cdot x\right) \cdot x\right) + \left(2 - \left(1 \cdot x\right) \cdot x\right)}}{2}\]

    if 121.48694228142615 < x

    1. Initial program 0.2

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

    2. Taylor expanded around inf 0.2

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

    3. Simplified0.2

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

  4. Final simplification1.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 121.4869422814261525900292326696217060089:\\ \;\;\;\;\frac{0.6666666666666667406815349750104360282421 \cdot \left(\left(x \cdot x\right) \cdot x\right) + \left(2 - \left(x \cdot 1\right) \cdot x\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 \cdot \left(e^{\left(\varepsilon - 1\right) \cdot x} + \frac{e^{\left(\varepsilon - 1\right) \cdot x}}{\varepsilon}\right) - e^{\left(-\left(1 + \varepsilon\right)\right) \cdot x} \cdot \left(\frac{1}{\varepsilon} - 1\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019171 
(FPCore (x eps)
  :name "NMSE Section 6.1 mentioned, A"
  (/ (- (* (+ 1.0 (/ 1.0 eps)) (exp (- (* (- 1.0 eps) x)))) (* (- (/ 1.0 eps) 1.0) (exp (- (* (+ 1.0 eps) x))))) 2.0))