Average Error: 29.4 → 0.9
Time: 15.5s
Precision: 64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 179.02536135484516:\\ \;\;\;\;\frac{2 + \left(x \cdot x\right) \cdot \left(0.66666666666666674 \cdot x - 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 \cdot \left(\frac{e^{-x \cdot \left(1 - \varepsilon\right)}}{\varepsilon} + e^{-x \cdot \left(1 - \varepsilon\right)}\right) - \frac{\frac{1}{\varepsilon} - 1}{e^{\left(1 + \varepsilon\right) \cdot x}}}{2}\\ \end{array}\]
\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 179.02536135484516:\\
\;\;\;\;\frac{2 + \left(x \cdot x\right) \cdot \left(0.66666666666666674 \cdot x - 1\right)}{2}\\

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

\end{array}
double f(double x, double eps) {
        double r44670 = 1.0;
        double r44671 = eps;
        double r44672 = r44670 / r44671;
        double r44673 = r44670 + r44672;
        double r44674 = r44670 - r44671;
        double r44675 = x;
        double r44676 = r44674 * r44675;
        double r44677 = -r44676;
        double r44678 = exp(r44677);
        double r44679 = r44673 * r44678;
        double r44680 = r44672 - r44670;
        double r44681 = r44670 + r44671;
        double r44682 = r44681 * r44675;
        double r44683 = -r44682;
        double r44684 = exp(r44683);
        double r44685 = r44680 * r44684;
        double r44686 = r44679 - r44685;
        double r44687 = 2.0;
        double r44688 = r44686 / r44687;
        return r44688;
}
↓
double f(double x, double eps) {
        double r44689 = x;
        double r44690 = 179.02536135484516;
        bool r44691 = r44689 <= r44690;
        double r44692 = 2.0;
        double r44693 = r44689 * r44689;
        double r44694 = 0.6666666666666667;
        double r44695 = r44694 * r44689;
        double r44696 = 1.0;
        double r44697 = r44695 - r44696;
        double r44698 = r44693 * r44697;
        double r44699 = r44692 + r44698;
        double r44700 = r44699 / r44692;
        double r44701 = eps;
        double r44702 = r44696 - r44701;
        double r44703 = r44689 * r44702;
        double r44704 = -r44703;
        double r44705 = exp(r44704);
        double r44706 = r44705 / r44701;
        double r44707 = r44706 + r44705;
        double r44708 = r44696 * r44707;
        double r44709 = r44696 / r44701;
        double r44710 = r44709 - r44696;
        double r44711 = r44696 + r44701;
        double r44712 = r44711 * r44689;
        double r44713 = exp(r44712);
        double r44714 = r44710 / r44713;
        double r44715 = r44708 - r44714;
        double r44716 = r44715 / r44692;
        double r44717 = r44691 ? r44700 : r44716;
        return r44717;
}

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 < 179.02536135484516

    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. Simplified38.9

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

    3. Taylor expanded around 0 1.2

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

    4. Simplified1.2

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

    if 179.02536135484516 < 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

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

    3. Taylor expanded around inf 0.1

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

    4. Simplified0.1

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

  4. Final simplification0.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 179.02536135484516:\\ \;\;\;\;\frac{2 + \left(x \cdot x\right) \cdot \left(0.66666666666666674 \cdot x - 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 \cdot \left(\frac{e^{-x \cdot \left(1 - \varepsilon\right)}}{\varepsilon} + e^{-x \cdot \left(1 - \varepsilon\right)}\right) - \frac{\frac{1}{\varepsilon} - 1}{e^{\left(1 + \varepsilon\right) \cdot x}}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020045 
(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))