Average Error: 29.7 → 1.0
Time: 32.9s
Precision: 64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 2.1846861365271546:\\ \;\;\;\;\frac{\left(2 + \left(\frac{2}{3} \cdot x\right) \cdot \left(x \cdot x\right)\right) - x \cdot x}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(e^{\varepsilon \cdot x + \left(-x\right)} + \frac{e^{\varepsilon \cdot x + \left(-x\right)}}{\varepsilon}\right) - \frac{e^{\left(-x\right) - \varepsilon \cdot x}}{\varepsilon}\right) + e^{\left(-x\right) - \varepsilon \cdot x}}{2}\\ \end{array}\]
\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 2.1846861365271546:\\
\;\;\;\;\frac{\left(2 + \left(\frac{2}{3} \cdot x\right) \cdot \left(x \cdot x\right)\right) - x \cdot x}{2}\\

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

\end{array}
double f(double x, double eps) {
        double r1406582 = 1.0;
        double r1406583 = eps;
        double r1406584 = r1406582 / r1406583;
        double r1406585 = r1406582 + r1406584;
        double r1406586 = r1406582 - r1406583;
        double r1406587 = x;
        double r1406588 = r1406586 * r1406587;
        double r1406589 = -r1406588;
        double r1406590 = exp(r1406589);
        double r1406591 = r1406585 * r1406590;
        double r1406592 = r1406584 - r1406582;
        double r1406593 = r1406582 + r1406583;
        double r1406594 = r1406593 * r1406587;
        double r1406595 = -r1406594;
        double r1406596 = exp(r1406595);
        double r1406597 = r1406592 * r1406596;
        double r1406598 = r1406591 - r1406597;
        double r1406599 = 2.0;
        double r1406600 = r1406598 / r1406599;
        return r1406600;
}
↓
double f(double x, double eps) {
        double r1406601 = x;
        double r1406602 = 2.1846861365271546;
        bool r1406603 = r1406601 <= r1406602;
        double r1406604 = 2.0;
        double r1406605 = 0.6666666666666666;
        double r1406606 = r1406605 * r1406601;
        double r1406607 = r1406601 * r1406601;
        double r1406608 = r1406606 * r1406607;
        double r1406609 = r1406604 + r1406608;
        double r1406610 = r1406609 - r1406607;
        double r1406611 = r1406610 / r1406604;
        double r1406612 = eps;
        double r1406613 = r1406612 * r1406601;
        double r1406614 = -r1406601;
        double r1406615 = r1406613 + r1406614;
        double r1406616 = exp(r1406615);
        double r1406617 = r1406616 / r1406612;
        double r1406618 = r1406616 + r1406617;
        double r1406619 = r1406614 - r1406613;
        double r1406620 = exp(r1406619);
        double r1406621 = r1406620 / r1406612;
        double r1406622 = r1406618 - r1406621;
        double r1406623 = r1406622 + r1406620;
        double r1406624 = r1406623 / r1406604;
        double r1406625 = r1406603 ? r1406611 : r1406624;
        return r1406625;
}

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 < 2.1846861365271546

    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

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

    3. Taylor expanded around 0 1.3

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

    4. Simplified1.3

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

    5. Taylor expanded around 0 1.3

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

    6. Simplified1.3

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

    if 2.1846861365271546 < 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. Simplified0.4

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

    4. Applied associate--r-0.3

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

  4. Final simplification1.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 2.1846861365271546:\\ \;\;\;\;\frac{\left(2 + \left(\frac{2}{3} \cdot x\right) \cdot \left(x \cdot x\right)\right) - x \cdot x}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(e^{\varepsilon \cdot x + \left(-x\right)} + \frac{e^{\varepsilon \cdot x + \left(-x\right)}}{\varepsilon}\right) - \frac{e^{\left(-x\right) - \varepsilon \cdot x}}{\varepsilon}\right) + e^{\left(-x\right) - \varepsilon \cdot x}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019151 
(FPCore (x eps)
  :name "NMSE Section 6.1 mentioned, A"
  (/ (- (* (+ 1 (/ 1 eps)) (exp (- (* (- 1 eps) x)))) (* (- (/ 1 eps) 1) (exp (- (* (+ 1 eps) x))))) 2))