Average Error: 29.5 → 1.0
Time: 33.2s
Precision: 64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 226.33586080847886:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(x \cdot x\right) \cdot x, \frac{2}{3}, 2 - x \cdot x\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)} \cdot \left(\frac{1}{\varepsilon} + 1\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot \left(\sqrt{e^{\left(-x\right) \cdot \left(\varepsilon + 1\right)}} \cdot \sqrt{e^{\left(-x\right) \cdot \left(\varepsilon + 1\right)}}\right)}{2}\\ \end{array}\]
\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 226.33586080847886:\\
\;\;\;\;\frac{\mathsf{fma}\left(\left(x \cdot x\right) \cdot x, \frac{2}{3}, 2 - x \cdot x\right)}{2}\\

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

\end{array}
double f(double x, double eps) {
        double r1864556 = 1.0;
        double r1864557 = eps;
        double r1864558 = r1864556 / r1864557;
        double r1864559 = r1864556 + r1864558;
        double r1864560 = r1864556 - r1864557;
        double r1864561 = x;
        double r1864562 = r1864560 * r1864561;
        double r1864563 = -r1864562;
        double r1864564 = exp(r1864563);
        double r1864565 = r1864559 * r1864564;
        double r1864566 = r1864558 - r1864556;
        double r1864567 = r1864556 + r1864557;
        double r1864568 = r1864567 * r1864561;
        double r1864569 = -r1864568;
        double r1864570 = exp(r1864569);
        double r1864571 = r1864566 * r1864570;
        double r1864572 = r1864565 - r1864571;
        double r1864573 = 2.0;
        double r1864574 = r1864572 / r1864573;
        return r1864574;
}
↓
double f(double x, double eps) {
        double r1864575 = x;
        double r1864576 = 226.33586080847886;
        bool r1864577 = r1864575 <= r1864576;
        double r1864578 = r1864575 * r1864575;
        double r1864579 = r1864578 * r1864575;
        double r1864580 = 0.6666666666666666;
        double r1864581 = 2.0;
        double r1864582 = r1864581 - r1864578;
        double r1864583 = fma(r1864579, r1864580, r1864582);
        double r1864584 = r1864583 / r1864581;
        double r1864585 = -r1864575;
        double r1864586 = 1.0;
        double r1864587 = eps;
        double r1864588 = r1864586 - r1864587;
        double r1864589 = r1864585 * r1864588;
        double r1864590 = exp(r1864589);
        double r1864591 = r1864586 / r1864587;
        double r1864592 = r1864591 + r1864586;
        double r1864593 = r1864590 * r1864592;
        double r1864594 = r1864591 - r1864586;
        double r1864595 = r1864587 + r1864586;
        double r1864596 = r1864585 * r1864595;
        double r1864597 = exp(r1864596);
        double r1864598 = sqrt(r1864597);
        double r1864599 = r1864598 * r1864598;
        double r1864600 = r1864594 * r1864599;
        double r1864601 = r1864593 - r1864600;
        double r1864602 = r1864601 / r1864581;
        double r1864603 = r1864577 ? r1864584 : r1864602;
        return r1864603;
}

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 2 regimes

  2. if x < 226.33586080847886

    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. Taylor expanded around 0 1.3

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

    3. Simplified1.3

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

    if 226.33586080847886 < 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. Using strategy rm

    3. Applied add-sqr-sqrt0.1

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

  4. Final simplification1.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 226.33586080847886:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(x \cdot x\right) \cdot x, \frac{2}{3}, 2 - x \cdot x\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)} \cdot \left(\frac{1}{\varepsilon} + 1\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot \left(\sqrt{e^{\left(-x\right) \cdot \left(\varepsilon + 1\right)}} \cdot \sqrt{e^{\left(-x\right) \cdot \left(\varepsilon + 1\right)}}\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019143 +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))