Average Error: 30.1 → 0.9
Time: 23.6s
Precision: 64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 31.67008507871663525179428688716143369675:\\ \;\;\;\;\frac{2 + {x}^{2} \cdot \left(x \cdot 0.6666666666666667406815349750104360282421 - 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{\varepsilon} + 1}{\left(\sqrt[3]{e^{\left(1 - \varepsilon\right) \cdot x}} \cdot \sqrt[3]{e^{\left(1 - \varepsilon\right) \cdot x}}\right) \cdot \sqrt[3]{e^{\left(1 - \varepsilon\right) \cdot x}}} - \frac{\frac{1}{\varepsilon} - 1}{e^{\left(1 + \varepsilon\right) \cdot x}}}{2}\\ \end{array}\]
\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 31.67008507871663525179428688716143369675:\\
\;\;\;\;\frac{2 + {x}^{2} \cdot \left(x \cdot 0.6666666666666667406815349750104360282421 - 1\right)}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{\frac{1}{\varepsilon} + 1}{\left(\sqrt[3]{e^{\left(1 - \varepsilon\right) \cdot x}} \cdot \sqrt[3]{e^{\left(1 - \varepsilon\right) \cdot x}}\right) \cdot \sqrt[3]{e^{\left(1 - \varepsilon\right) \cdot x}}} - \frac{\frac{1}{\varepsilon} - 1}{e^{\left(1 + \varepsilon\right) \cdot x}}}{2}\\

\end{array}
double f(double x, double eps) {
        double r30640 = 1.0;
        double r30641 = eps;
        double r30642 = r30640 / r30641;
        double r30643 = r30640 + r30642;
        double r30644 = r30640 - r30641;
        double r30645 = x;
        double r30646 = r30644 * r30645;
        double r30647 = -r30646;
        double r30648 = exp(r30647);
        double r30649 = r30643 * r30648;
        double r30650 = r30642 - r30640;
        double r30651 = r30640 + r30641;
        double r30652 = r30651 * r30645;
        double r30653 = -r30652;
        double r30654 = exp(r30653);
        double r30655 = r30650 * r30654;
        double r30656 = r30649 - r30655;
        double r30657 = 2.0;
        double r30658 = r30656 / r30657;
        return r30658;
}
↓
double f(double x, double eps) {
        double r30659 = x;
        double r30660 = 31.670085078716635;
        bool r30661 = r30659 <= r30660;
        double r30662 = 2.0;
        double r30663 = 2.0;
        double r30664 = pow(r30659, r30663);
        double r30665 = 0.6666666666666667;
        double r30666 = r30659 * r30665;
        double r30667 = 1.0;
        double r30668 = r30666 - r30667;
        double r30669 = r30664 * r30668;
        double r30670 = r30662 + r30669;
        double r30671 = r30670 / r30662;
        double r30672 = eps;
        double r30673 = r30667 / r30672;
        double r30674 = r30673 + r30667;
        double r30675 = r30667 - r30672;
        double r30676 = r30675 * r30659;
        double r30677 = exp(r30676);
        double r30678 = cbrt(r30677);
        double r30679 = r30678 * r30678;
        double r30680 = r30679 * r30678;
        double r30681 = r30674 / r30680;
        double r30682 = r30673 - r30667;
        double r30683 = r30667 + r30672;
        double r30684 = r30683 * r30659;
        double r30685 = exp(r30684);
        double r30686 = r30682 / r30685;
        double r30687 = r30681 - r30686;
        double r30688 = r30687 / r30662;
        double r30689 = r30661 ? r30671 : r30688;
        return r30689;
}

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 < 31.670085078716635

    1. Initial program 39.8

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

    2. Simplified39.8

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

    3. Taylor expanded around 0 1.0

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

    4. Simplified1.0

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

    if 31.670085078716635 < x

    1. Initial program 0.3

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

    2. Simplified0.3

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

    4. Applied add-cube-cbrt0.3

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

  4. Final simplification0.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 31.67008507871663525179428688716143369675:\\ \;\;\;\;\frac{2 + {x}^{2} \cdot \left(x \cdot 0.6666666666666667406815349750104360282421 - 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{\varepsilon} + 1}{\left(\sqrt[3]{e^{\left(1 - \varepsilon\right) \cdot x}} \cdot \sqrt[3]{e^{\left(1 - \varepsilon\right) \cdot x}}\right) \cdot \sqrt[3]{e^{\left(1 - \varepsilon\right) \cdot x}}} - \frac{\frac{1}{\varepsilon} - 1}{e^{\left(1 + \varepsilon\right) \cdot x}}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019326 
(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))