Average Error: 29.7 → 0.9
Time: 40.7s
Precision: 64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 120.3533821425713199460005853325128555298:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt[3]{\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)}, 0.6666666666666667406815349750104360282421, 2 - \left(x \cdot 1\right) \cdot x\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(e^{x \cdot \left(\varepsilon - 1\right)}, \frac{1}{\varepsilon} + 1, \frac{1 - \frac{1}{\varepsilon}}{e^{x \cdot \left(\varepsilon + 1\right)}}\right)}{2}\\ \end{array}\]
\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 120.3533821425713199460005853325128555298:\\
\;\;\;\;\frac{\mathsf{fma}\left(\sqrt[3]{\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)}, 0.6666666666666667406815349750104360282421, 2 - \left(x \cdot 1\right) \cdot x\right)}{2}\\

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

\end{array}
double f(double x, double eps) {
        double r2498592 = 1.0;
        double r2498593 = eps;
        double r2498594 = r2498592 / r2498593;
        double r2498595 = r2498592 + r2498594;
        double r2498596 = r2498592 - r2498593;
        double r2498597 = x;
        double r2498598 = r2498596 * r2498597;
        double r2498599 = -r2498598;
        double r2498600 = exp(r2498599);
        double r2498601 = r2498595 * r2498600;
        double r2498602 = r2498594 - r2498592;
        double r2498603 = r2498592 + r2498593;
        double r2498604 = r2498603 * r2498597;
        double r2498605 = -r2498604;
        double r2498606 = exp(r2498605);
        double r2498607 = r2498602 * r2498606;
        double r2498608 = r2498601 - r2498607;
        double r2498609 = 2.0;
        double r2498610 = r2498608 / r2498609;
        return r2498610;
}
↓
double f(double x, double eps) {
        double r2498611 = x;
        double r2498612 = 120.35338214257132;
        bool r2498613 = r2498611 <= r2498612;
        double r2498614 = r2498611 * r2498611;
        double r2498615 = r2498614 * r2498611;
        double r2498616 = r2498615 * r2498615;
        double r2498617 = r2498616 * r2498615;
        double r2498618 = cbrt(r2498617);
        double r2498619 = 0.6666666666666667;
        double r2498620 = 2.0;
        double r2498621 = 1.0;
        double r2498622 = r2498611 * r2498621;
        double r2498623 = r2498622 * r2498611;
        double r2498624 = r2498620 - r2498623;
        double r2498625 = fma(r2498618, r2498619, r2498624);
        double r2498626 = r2498625 / r2498620;
        double r2498627 = eps;
        double r2498628 = r2498627 - r2498621;
        double r2498629 = r2498611 * r2498628;
        double r2498630 = exp(r2498629);
        double r2498631 = r2498621 / r2498627;
        double r2498632 = r2498631 + r2498621;
        double r2498633 = r2498621 - r2498631;
        double r2498634 = r2498627 + r2498621;
        double r2498635 = r2498611 * r2498634;
        double r2498636 = exp(r2498635);
        double r2498637 = r2498633 / r2498636;
        double r2498638 = fma(r2498630, r2498632, r2498637);
        double r2498639 = r2498638 / r2498620;
        double r2498640 = r2498613 ? r2498626 : r2498639;
        return r2498640;
}

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 2 regimes

  2. if x < 120.35338214257132

    1. Initial program 39.3

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

    2. Taylor expanded around 0 1.1

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

    3. Simplified1.1

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(x \cdot x\right) \cdot x, 0.6666666666666667406815349750104360282421, 2 - \left(1 \cdot x\right) \cdot x\right)}}{2}\]
    4. Using strategy rm

    5. Applied add-cbrt-cube1.1

      \[\leadsto \frac{\mathsf{fma}\left(\left(x \cdot x\right) \cdot \color{blue}{\sqrt[3]{\left(x \cdot x\right) \cdot x}}, 0.6666666666666667406815349750104360282421, 2 - \left(1 \cdot x\right) \cdot x\right)}{2}\]

    6. Applied add-cbrt-cube1.1

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sqrt[3]{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)}} \cdot \sqrt[3]{\left(x \cdot x\right) \cdot x}, 0.6666666666666667406815349750104360282421, 2 - \left(1 \cdot x\right) \cdot x\right)}{2}\]

    7. Applied cbrt-unprod1.1

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sqrt[3]{\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)}}, 0.6666666666666667406815349750104360282421, 2 - \left(1 \cdot x\right) \cdot x\right)}{2}\]

    8. Simplified1.1

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt[3]{\color{blue}{\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right)}}, 0.6666666666666667406815349750104360282421, 2 - \left(1 \cdot x\right) \cdot x\right)}{2}\]

    if 120.35338214257132 < 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

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

  4. Final simplification0.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 120.3533821425713199460005853325128555298:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt[3]{\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)}, 0.6666666666666667406815349750104360282421, 2 - \left(x \cdot 1\right) \cdot x\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(e^{x \cdot \left(\varepsilon - 1\right)}, \frac{1}{\varepsilon} + 1, \frac{1 - \frac{1}{\varepsilon}}{e^{x \cdot \left(\varepsilon + 1\right)}}\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019200 +o rules:numerics
(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))