Average Error: 30.2 → 1.1
Time: 32.3s
Precision: 64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 10.85972191525816654689151619095355272293:\\ \;\;\;\;\frac{\log \left(e^{\mathsf{fma}\left({x}^{2}, 0.6666666666666667406815349750104360282421 \cdot x - 1, 2\right)}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, e^{-\left(1 - \varepsilon\right) \cdot x}, -\frac{\frac{1}{\varepsilon} - 1}{e^{\left(1 + \varepsilon\right) \cdot x}}\right)}{2}\\ \end{array}\]
\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 10.85972191525816654689151619095355272293:\\
\;\;\;\;\frac{\log \left(e^{\mathsf{fma}\left({x}^{2}, 0.6666666666666667406815349750104360282421 \cdot x - 1, 2\right)}\right)}{2}\\

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

\end{array}
double f(double x, double eps) {
        double r46708 = 1.0;
        double r46709 = eps;
        double r46710 = r46708 / r46709;
        double r46711 = r46708 + r46710;
        double r46712 = r46708 - r46709;
        double r46713 = x;
        double r46714 = r46712 * r46713;
        double r46715 = -r46714;
        double r46716 = exp(r46715);
        double r46717 = r46711 * r46716;
        double r46718 = r46710 - r46708;
        double r46719 = r46708 + r46709;
        double r46720 = r46719 * r46713;
        double r46721 = -r46720;
        double r46722 = exp(r46721);
        double r46723 = r46718 * r46722;
        double r46724 = r46717 - r46723;
        double r46725 = 2.0;
        double r46726 = r46724 / r46725;
        return r46726;
}
↓
double f(double x, double eps) {
        double r46727 = x;
        double r46728 = 10.859721915258167;
        bool r46729 = r46727 <= r46728;
        double r46730 = 2.0;
        double r46731 = pow(r46727, r46730);
        double r46732 = 0.6666666666666667;
        double r46733 = r46732 * r46727;
        double r46734 = 1.0;
        double r46735 = r46733 - r46734;
        double r46736 = 2.0;
        double r46737 = fma(r46731, r46735, r46736);
        double r46738 = exp(r46737);
        double r46739 = log(r46738);
        double r46740 = r46739 / r46736;
        double r46741 = eps;
        double r46742 = r46734 / r46741;
        double r46743 = r46734 + r46742;
        double r46744 = r46734 - r46741;
        double r46745 = r46744 * r46727;
        double r46746 = -r46745;
        double r46747 = exp(r46746);
        double r46748 = r46742 - r46734;
        double r46749 = r46734 + r46741;
        double r46750 = r46749 * r46727;
        double r46751 = exp(r46750);
        double r46752 = r46748 / r46751;
        double r46753 = -r46752;
        double r46754 = fma(r46743, r46747, r46753);
        double r46755 = r46754 / r46736;
        double r46756 = r46729 ? r46740 : r46755;
        return r46756;
}

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 2 regimes

  2. if x < 10.859721915258167

    1. Initial program 39.6

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

    2. Taylor expanded around 0 1.3

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

    3. Simplified1.3

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(0.6666666666666667406815349750104360282421, {x}^{3}, 2\right) - 1 \cdot {x}^{2}}}{2}\]
    4. Using strategy rm

    5. Applied add-log-exp1.3

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

    6. Applied add-log-exp1.3

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

    7. Applied diff-log1.3

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

    8. Simplified1.3

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

    if 10.859721915258167 < x

    1. Initial program 0.5

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

    3. Applied fma-neg0.5

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

    4. Simplified0.5

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

  4. Final simplification1.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 10.85972191525816654689151619095355272293:\\ \;\;\;\;\frac{\log \left(e^{\mathsf{fma}\left({x}^{2}, 0.6666666666666667406815349750104360282421 \cdot x - 1, 2\right)}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, e^{-\left(1 - \varepsilon\right) \cdot x}, -\frac{\frac{1}{\varepsilon} - 1}{e^{\left(1 + \varepsilon\right) \cdot x}}\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019303 +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))