Average Error: 29.0 → 5.5
Time: 8.4s
Precision: 64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 0.0250881002166713322:\\ \;\;\;\;\left(\left(\left(\sqrt[3]{\sqrt[3]{\mathsf{fma}\left(1.38778 \cdot 10^{-17}, \frac{{x}^{3}}{\varepsilon}, 1 - 0.5 \cdot {x}^{2}\right)}} \cdot \sqrt[3]{\sqrt[3]{\mathsf{fma}\left(1.38778 \cdot 10^{-17}, \frac{{x}^{3}}{\varepsilon}, 1 - 0.5 \cdot {x}^{2}\right)}}\right) \cdot \sqrt[3]{\sqrt[3]{\mathsf{fma}\left(1.38778 \cdot 10^{-17}, \frac{{x}^{3}}{\varepsilon}, 1 - 0.5 \cdot {x}^{2}\right)}}\right) \cdot \sqrt[3]{\mathsf{fma}\left(1.38778 \cdot 10^{-17}, \frac{{x}^{3}}{\varepsilon}, 1 - 0.5 \cdot {x}^{2}\right)}\right) \cdot \left({1}^{\frac{1}{3}} - 0.166666666666666657 \cdot \left({x}^{2} \cdot {1}^{\frac{1}{3}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\\ \end{array}\]
\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 0.0250881002166713322:\\
\;\;\;\;\left(\left(\left(\sqrt[3]{\sqrt[3]{\mathsf{fma}\left(1.38778 \cdot 10^{-17}, \frac{{x}^{3}}{\varepsilon}, 1 - 0.5 \cdot {x}^{2}\right)}} \cdot \sqrt[3]{\sqrt[3]{\mathsf{fma}\left(1.38778 \cdot 10^{-17}, \frac{{x}^{3}}{\varepsilon}, 1 - 0.5 \cdot {x}^{2}\right)}}\right) \cdot \sqrt[3]{\sqrt[3]{\mathsf{fma}\left(1.38778 \cdot 10^{-17}, \frac{{x}^{3}}{\varepsilon}, 1 - 0.5 \cdot {x}^{2}\right)}}\right) \cdot \sqrt[3]{\mathsf{fma}\left(1.38778 \cdot 10^{-17}, \frac{{x}^{3}}{\varepsilon}, 1 - 0.5 \cdot {x}^{2}\right)}\right) \cdot \left({1}^{\frac{1}{3}} - 0.166666666666666657 \cdot \left({x}^{2} \cdot {1}^{\frac{1}{3}}\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\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}\\

\end{array}
double f(double x, double eps) {
        double r60716 = 1.0;
        double r60717 = eps;
        double r60718 = r60716 / r60717;
        double r60719 = r60716 + r60718;
        double r60720 = r60716 - r60717;
        double r60721 = x;
        double r60722 = r60720 * r60721;
        double r60723 = -r60722;
        double r60724 = exp(r60723);
        double r60725 = r60719 * r60724;
        double r60726 = r60718 - r60716;
        double r60727 = r60716 + r60717;
        double r60728 = r60727 * r60721;
        double r60729 = -r60728;
        double r60730 = exp(r60729);
        double r60731 = r60726 * r60730;
        double r60732 = r60725 - r60731;
        double r60733 = 2.0;
        double r60734 = r60732 / r60733;
        return r60734;
}
↓
double f(double x, double eps) {
        double r60735 = x;
        double r60736 = 0.025088100216671332;
        bool r60737 = r60735 <= r60736;
        double r60738 = 1.3877787807814457e-17;
        double r60739 = 3.0;
        double r60740 = pow(r60735, r60739);
        double r60741 = eps;
        double r60742 = r60740 / r60741;
        double r60743 = 1.0;
        double r60744 = 0.5;
        double r60745 = 2.0;
        double r60746 = pow(r60735, r60745);
        double r60747 = r60744 * r60746;
        double r60748 = r60743 - r60747;
        double r60749 = fma(r60738, r60742, r60748);
        double r60750 = cbrt(r60749);
        double r60751 = cbrt(r60750);
        double r60752 = r60751 * r60751;
        double r60753 = r60752 * r60751;
        double r60754 = r60753 * r60750;
        double r60755 = 0.3333333333333333;
        double r60756 = pow(r60743, r60755);
        double r60757 = 0.16666666666666666;
        double r60758 = r60746 * r60756;
        double r60759 = r60757 * r60758;
        double r60760 = r60756 - r60759;
        double r60761 = r60754 * r60760;
        double r60762 = r60743 / r60741;
        double r60763 = r60743 + r60762;
        double r60764 = r60743 - r60741;
        double r60765 = r60764 * r60735;
        double r60766 = -r60765;
        double r60767 = exp(r60766);
        double r60768 = r60763 * r60767;
        double r60769 = r60762 - r60743;
        double r60770 = r60743 + r60741;
        double r60771 = r60770 * r60735;
        double r60772 = -r60771;
        double r60773 = exp(r60772);
        double r60774 = r60769 * r60773;
        double r60775 = r60768 - r60774;
        double r60776 = 2.0;
        double r60777 = r60775 / r60776;
        double r60778 = r60737 ? r60761 : r60777;
        return r60778;
}

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 2 regimes

  2. if x < 0.025088100216671332

    1. Initial program 38.7

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

    2. Simplified38.7

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

    3. Taylor expanded around 0 7.3

      \[\leadsto \color{blue}{\left(1.38778 \cdot 10^{-17} \cdot \frac{{x}^{3}}{\varepsilon} + 1\right) - 0.5 \cdot {x}^{2}}\]

    4. Simplified7.3

      \[\leadsto \color{blue}{\mathsf{fma}\left(1.38778 \cdot 10^{-17}, \frac{{x}^{3}}{\varepsilon}, 1 - 0.5 \cdot {x}^{2}\right)}\]
    5. Using strategy rm

    6. Applied add-cube-cbrt7.3

      \[\leadsto \color{blue}{\left(\sqrt[3]{\mathsf{fma}\left(1.38778 \cdot 10^{-17}, \frac{{x}^{3}}{\varepsilon}, 1 - 0.5 \cdot {x}^{2}\right)} \cdot \sqrt[3]{\mathsf{fma}\left(1.38778 \cdot 10^{-17}, \frac{{x}^{3}}{\varepsilon}, 1 - 0.5 \cdot {x}^{2}\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(1.38778 \cdot 10^{-17}, \frac{{x}^{3}}{\varepsilon}, 1 - 0.5 \cdot {x}^{2}\right)}}\]

    7. Taylor expanded around 0 7.1

      \[\leadsto \left(\sqrt[3]{\mathsf{fma}\left(1.38778 \cdot 10^{-17}, \frac{{x}^{3}}{\varepsilon}, 1 - 0.5 \cdot {x}^{2}\right)} \cdot \sqrt[3]{\mathsf{fma}\left(1.38778 \cdot 10^{-17}, \frac{{x}^{3}}{\varepsilon}, 1 - 0.5 \cdot {x}^{2}\right)}\right) \cdot \color{blue}{\left({1}^{\frac{1}{3}} - 0.166666666666666657 \cdot \left({x}^{2} \cdot {1}^{\frac{1}{3}}\right)\right)}\]
    8. Using strategy rm

    9. Applied add-cube-cbrt7.1

      \[\leadsto \left(\color{blue}{\left(\left(\sqrt[3]{\sqrt[3]{\mathsf{fma}\left(1.38778 \cdot 10^{-17}, \frac{{x}^{3}}{\varepsilon}, 1 - 0.5 \cdot {x}^{2}\right)}} \cdot \sqrt[3]{\sqrt[3]{\mathsf{fma}\left(1.38778 \cdot 10^{-17}, \frac{{x}^{3}}{\varepsilon}, 1 - 0.5 \cdot {x}^{2}\right)}}\right) \cdot \sqrt[3]{\sqrt[3]{\mathsf{fma}\left(1.38778 \cdot 10^{-17}, \frac{{x}^{3}}{\varepsilon}, 1 - 0.5 \cdot {x}^{2}\right)}}\right)} \cdot \sqrt[3]{\mathsf{fma}\left(1.38778 \cdot 10^{-17}, \frac{{x}^{3}}{\varepsilon}, 1 - 0.5 \cdot {x}^{2}\right)}\right) \cdot \left({1}^{\frac{1}{3}} - 0.166666666666666657 \cdot \left({x}^{2} \cdot {1}^{\frac{1}{3}}\right)\right)\]

    if 0.025088100216671332 < x

    1. Initial program 0.9

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

  4. Final simplification5.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 0.0250881002166713322:\\ \;\;\;\;\left(\left(\left(\sqrt[3]{\sqrt[3]{\mathsf{fma}\left(1.38778 \cdot 10^{-17}, \frac{{x}^{3}}{\varepsilon}, 1 - 0.5 \cdot {x}^{2}\right)}} \cdot \sqrt[3]{\sqrt[3]{\mathsf{fma}\left(1.38778 \cdot 10^{-17}, \frac{{x}^{3}}{\varepsilon}, 1 - 0.5 \cdot {x}^{2}\right)}}\right) \cdot \sqrt[3]{\sqrt[3]{\mathsf{fma}\left(1.38778 \cdot 10^{-17}, \frac{{x}^{3}}{\varepsilon}, 1 - 0.5 \cdot {x}^{2}\right)}}\right) \cdot \sqrt[3]{\mathsf{fma}\left(1.38778 \cdot 10^{-17}, \frac{{x}^{3}}{\varepsilon}, 1 - 0.5 \cdot {x}^{2}\right)}\right) \cdot \left({1}^{\frac{1}{3}} - 0.166666666666666657 \cdot \left({x}^{2} \cdot {1}^{\frac{1}{3}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020049 +o rules:numerics
(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))