Average Error: 29.2 → 1.1
Time: 49.6s
Precision: 64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 245.97670246303022:\\ \;\;\;\;\frac{\left(\left(\mathsf{fma}\left(\frac{2}{3}, \left(x \cdot x\right) \cdot x, \left(\left(2 - x \cdot x\right)\right)\right)\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{1}{\varepsilon} + 1\right) \cdot \frac{1}{e^{x \cdot \left(1 - \varepsilon\right)}} - e^{\left(-\left(1 + \varepsilon\right)\right) \cdot x} \cdot \left(\frac{1}{\varepsilon} - 1\right)}{2}\\ \end{array}\]
\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 245.97670246303022:\\
\;\;\;\;\frac{\left(\left(\mathsf{fma}\left(\frac{2}{3}, \left(x \cdot x\right) \cdot x, \left(\left(2 - x \cdot x\right)\right)\right)\right)\right)}{2}\\

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

\end{array}
double f(double x, double eps) {
        double r1310596 = 1.0;
        double r1310597 = eps;
        double r1310598 = r1310596 / r1310597;
        double r1310599 = r1310596 + r1310598;
        double r1310600 = r1310596 - r1310597;
        double r1310601 = x;
        double r1310602 = r1310600 * r1310601;
        double r1310603 = -r1310602;
        double r1310604 = exp(r1310603);
        double r1310605 = r1310599 * r1310604;
        double r1310606 = r1310598 - r1310596;
        double r1310607 = r1310596 + r1310597;
        double r1310608 = r1310607 * r1310601;
        double r1310609 = -r1310608;
        double r1310610 = exp(r1310609);
        double r1310611 = r1310606 * r1310610;
        double r1310612 = r1310605 - r1310611;
        double r1310613 = 2.0;
        double r1310614 = r1310612 / r1310613;
        return r1310614;
}
↓
double f(double x, double eps) {
        double r1310615 = x;
        double r1310616 = 245.97670246303022;
        bool r1310617 = r1310615 <= r1310616;
        double r1310618 = 0.6666666666666666;
        double r1310619 = r1310615 * r1310615;
        double r1310620 = r1310619 * r1310615;
        double r1310621 = 2.0;
        double r1310622 = r1310621 - r1310619;
        double r1310623 = /* ERROR: no posit support in C */;
        double r1310624 = /* ERROR: no posit support in C */;
        double r1310625 = fma(r1310618, r1310620, r1310624);
        double r1310626 = /* ERROR: no posit support in C */;
        double r1310627 = /* ERROR: no posit support in C */;
        double r1310628 = r1310627 / r1310621;
        double r1310629 = 1.0;
        double r1310630 = eps;
        double r1310631 = r1310629 / r1310630;
        double r1310632 = r1310631 + r1310629;
        double r1310633 = r1310629 - r1310630;
        double r1310634 = r1310615 * r1310633;
        double r1310635 = exp(r1310634);
        double r1310636 = r1310629 / r1310635;
        double r1310637 = r1310632 * r1310636;
        double r1310638 = r1310629 + r1310630;
        double r1310639 = -r1310638;
        double r1310640 = r1310639 * r1310615;
        double r1310641 = exp(r1310640);
        double r1310642 = r1310631 - r1310629;
        double r1310643 = r1310641 * r1310642;
        double r1310644 = r1310637 - r1310643;
        double r1310645 = r1310644 / r1310621;
        double r1310646 = r1310617 ? r1310628 : r1310645;
        return r1310646;
}

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 2 regimes

  2. if x < 245.97670246303022

    1. Initial program 38.8

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

    2. Taylor expanded around 0 1.2

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

    3. Simplified1.2

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

    5. Applied insert-posit161.4

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

    7. Applied insert-posit161.4

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

    if 245.97670246303022 < 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 exp-neg0.1

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

  4. Final simplification1.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 245.97670246303022:\\ \;\;\;\;\frac{\left(\left(\mathsf{fma}\left(\frac{2}{3}, \left(x \cdot x\right) \cdot x, \left(\left(2 - x \cdot x\right)\right)\right)\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{1}{\varepsilon} + 1\right) \cdot \frac{1}{e^{x \cdot \left(1 - \varepsilon\right)}} - e^{\left(-\left(1 + \varepsilon\right)\right) \cdot x} \cdot \left(\frac{1}{\varepsilon} - 1\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019146 +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))