Average Error: 29.5 → 0.9
Time: 19.0s
Precision: 64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 41.6575809018513:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{2}{3} \cdot x, x \cdot x, 2 - x \cdot x\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(e^{\left(\varepsilon + -1\right) \cdot x}, \frac{1}{\varepsilon}, e^{\left(\varepsilon + -1\right) \cdot x} - \frac{\frac{1}{\varepsilon} - 1}{{\left(e^{\sqrt[3]{\mathsf{fma}\left(\varepsilon, x, x\right)} \cdot \sqrt[3]{\mathsf{fma}\left(\varepsilon, x, x\right)}}\right)}^{\left(\sqrt[3]{\mathsf{fma}\left(\varepsilon, x, x\right)}\right)}}\right)}{2}\\ \end{array}\]
\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 41.6575809018513:\\
\;\;\;\;\frac{\mathsf{fma}\left(\frac{2}{3} \cdot x, x \cdot x, 2 - x \cdot x\right)}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\mathsf{fma}\left(e^{\left(\varepsilon + -1\right) \cdot x}, \frac{1}{\varepsilon}, e^{\left(\varepsilon + -1\right) \cdot x} - \frac{\frac{1}{\varepsilon} - 1}{{\left(e^{\sqrt[3]{\mathsf{fma}\left(\varepsilon, x, x\right)} \cdot \sqrt[3]{\mathsf{fma}\left(\varepsilon, x, x\right)}}\right)}^{\left(\sqrt[3]{\mathsf{fma}\left(\varepsilon, x, x\right)}\right)}}\right)}{2}\\

\end{array}
double f(double x, double eps) {
        double r854596 = 1.0;
        double r854597 = eps;
        double r854598 = r854596 / r854597;
        double r854599 = r854596 + r854598;
        double r854600 = r854596 - r854597;
        double r854601 = x;
        double r854602 = r854600 * r854601;
        double r854603 = -r854602;
        double r854604 = exp(r854603);
        double r854605 = r854599 * r854604;
        double r854606 = r854598 - r854596;
        double r854607 = r854596 + r854597;
        double r854608 = r854607 * r854601;
        double r854609 = -r854608;
        double r854610 = exp(r854609);
        double r854611 = r854606 * r854610;
        double r854612 = r854605 - r854611;
        double r854613 = 2.0;
        double r854614 = r854612 / r854613;
        return r854614;
}
↓
double f(double x, double eps) {
        double r854615 = x;
        double r854616 = 41.6575809018513;
        bool r854617 = r854615 <= r854616;
        double r854618 = 0.6666666666666666;
        double r854619 = r854618 * r854615;
        double r854620 = r854615 * r854615;
        double r854621 = 2.0;
        double r854622 = r854621 - r854620;
        double r854623 = fma(r854619, r854620, r854622);
        double r854624 = r854623 / r854621;
        double r854625 = eps;
        double r854626 = -1.0;
        double r854627 = r854625 + r854626;
        double r854628 = r854627 * r854615;
        double r854629 = exp(r854628);
        double r854630 = 1.0;
        double r854631 = r854630 / r854625;
        double r854632 = r854631 - r854630;
        double r854633 = fma(r854625, r854615, r854615);
        double r854634 = cbrt(r854633);
        double r854635 = r854634 * r854634;
        double r854636 = exp(r854635);
        double r854637 = pow(r854636, r854634);
        double r854638 = r854632 / r854637;
        double r854639 = r854629 - r854638;
        double r854640 = fma(r854629, r854631, r854639);
        double r854641 = r854640 / r854621;
        double r854642 = r854617 ? r854624 : r854641;
        return r854642;
}

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 2 regimes

  2. if x < 41.6575809018513

    1. Initial program 39.1

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

    2. Simplified39.1

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

    3. Taylor expanded around 0 1.1

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

    4. Simplified1.1

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

    if 41.6575809018513 < 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

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

    4. Applied add-cube-cbrt0.1

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

    5. Applied exp-prod0.1

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

  4. Final simplification0.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 41.6575809018513:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{2}{3} \cdot x, x \cdot x, 2 - x \cdot x\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(e^{\left(\varepsilon + -1\right) \cdot x}, \frac{1}{\varepsilon}, e^{\left(\varepsilon + -1\right) \cdot x} - \frac{\frac{1}{\varepsilon} - 1}{{\left(e^{\sqrt[3]{\mathsf{fma}\left(\varepsilon, x, x\right)} \cdot \sqrt[3]{\mathsf{fma}\left(\varepsilon, x, x\right)}}\right)}^{\left(\sqrt[3]{\mathsf{fma}\left(\varepsilon, x, x\right)}\right)}}\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019156 +o rules:numerics
(FPCore (x eps)
  :name "NMSE Section 6.1 mentioned, A"
  (/ (- (* (+ 1 (/ 1 eps)) (exp (- (* (- 1 eps) x)))) (* (- (/ 1 eps) 1) (exp (- (* (+ 1 eps) x))))) 2))