Average Error: 29.7 → 0.9
Time: 40.5s
Precision: 64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 120.3533821425713199460005853325128555298:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt[3]{\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)}, 0.6666666666666667406815349750104360282421, 2 - \left(x \cdot 1\right) \cdot x\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(e^{x \cdot \left(\varepsilon - 1\right)}, \frac{1}{\varepsilon} + 1, \frac{1 - \frac{1}{\varepsilon}}{e^{x \cdot \left(\varepsilon + 1\right)}}\right)}{2}\\ \end{array}\]
\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 r2511655 = 1.0;
        double r2511656 = eps;
        double r2511657 = r2511655 / r2511656;
        double r2511658 = r2511655 + r2511657;
        double r2511659 = r2511655 - r2511656;
        double r2511660 = x;
        double r2511661 = r2511659 * r2511660;
        double r2511662 = -r2511661;
        double r2511663 = exp(r2511662);
        double r2511664 = r2511658 * r2511663;
        double r2511665 = r2511657 - r2511655;
        double r2511666 = r2511655 + r2511656;
        double r2511667 = r2511666 * r2511660;
        double r2511668 = -r2511667;
        double r2511669 = exp(r2511668);
        double r2511670 = r2511665 * r2511669;
        double r2511671 = r2511664 - r2511670;
        double r2511672 = 2.0;
        double r2511673 = r2511671 / r2511672;
        return r2511673;
}
↓
double f(double x, double eps) {
        double r2511674 = x;
        double r2511675 = 120.35338214257132;
        bool r2511676 = r2511674 <= r2511675;
        double r2511677 = r2511674 * r2511674;
        double r2511678 = r2511677 * r2511674;
        double r2511679 = r2511678 * r2511678;
        double r2511680 = r2511679 * r2511678;
        double r2511681 = cbrt(r2511680);
        double r2511682 = 0.6666666666666667;
        double r2511683 = 2.0;
        double r2511684 = 1.0;
        double r2511685 = r2511674 * r2511684;
        double r2511686 = r2511685 * r2511674;
        double r2511687 = r2511683 - r2511686;
        double r2511688 = fma(r2511681, r2511682, r2511687);
        double r2511689 = r2511688 / r2511683;
        double r2511690 = eps;
        double r2511691 = r2511690 - r2511684;
        double r2511692 = r2511674 * r2511691;
        double r2511693 = exp(r2511692);
        double r2511694 = r2511684 / r2511690;
        double r2511695 = r2511694 + r2511684;
        double r2511696 = r2511684 - r2511694;
        double r2511697 = r2511690 + r2511684;
        double r2511698 = r2511674 * r2511697;
        double r2511699 = exp(r2511698);
        double r2511700 = r2511696 / r2511699;
        double r2511701 = fma(r2511693, r2511695, r2511700);
        double r2511702 = r2511701 / r2511683;
        double r2511703 = r2511676 ? r2511689 : r2511702;
        return r2511703;
}

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))