Average Error: 29.7 → 1.1
Time: 27.3s
Precision: 64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 138.7487461970468:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{2}{3}, \left(x \cdot x\right) \cdot x, 2 - x \cdot x\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(e^{x \cdot \varepsilon - x}, \frac{1}{\varepsilon}, e^{x \cdot \varepsilon - x} - \frac{\frac{1}{\varepsilon} - 1}{e^{\mathsf{fma}\left(x, \varepsilon, x\right)}}\right)}{2}\\ \end{array}\]
\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 138.7487461970468:\\
\;\;\;\;\frac{\mathsf{fma}\left(\frac{2}{3}, \left(x \cdot x\right) \cdot x, 2 - x \cdot x\right)}{2}\\

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

\end{array}
double f(double x, double eps) {
        double r1218617 = 1.0;
        double r1218618 = eps;
        double r1218619 = r1218617 / r1218618;
        double r1218620 = r1218617 + r1218619;
        double r1218621 = r1218617 - r1218618;
        double r1218622 = x;
        double r1218623 = r1218621 * r1218622;
        double r1218624 = -r1218623;
        double r1218625 = exp(r1218624);
        double r1218626 = r1218620 * r1218625;
        double r1218627 = r1218619 - r1218617;
        double r1218628 = r1218617 + r1218618;
        double r1218629 = r1218628 * r1218622;
        double r1218630 = -r1218629;
        double r1218631 = exp(r1218630);
        double r1218632 = r1218627 * r1218631;
        double r1218633 = r1218626 - r1218632;
        double r1218634 = 2.0;
        double r1218635 = r1218633 / r1218634;
        return r1218635;
}
↓
double f(double x, double eps) {
        double r1218636 = x;
        double r1218637 = 138.7487461970468;
        bool r1218638 = r1218636 <= r1218637;
        double r1218639 = 0.6666666666666666;
        double r1218640 = r1218636 * r1218636;
        double r1218641 = r1218640 * r1218636;
        double r1218642 = 2.0;
        double r1218643 = r1218642 - r1218640;
        double r1218644 = fma(r1218639, r1218641, r1218643);
        double r1218645 = r1218644 / r1218642;
        double r1218646 = eps;
        double r1218647 = r1218636 * r1218646;
        double r1218648 = r1218647 - r1218636;
        double r1218649 = exp(r1218648);
        double r1218650 = 1.0;
        double r1218651 = r1218650 / r1218646;
        double r1218652 = r1218651 - r1218650;
        double r1218653 = fma(r1218636, r1218646, r1218636);
        double r1218654 = exp(r1218653);
        double r1218655 = r1218652 / r1218654;
        double r1218656 = r1218649 - r1218655;
        double r1218657 = fma(r1218649, r1218651, r1218656);
        double r1218658 = r1218657 / r1218642;
        double r1218659 = r1218638 ? r1218645 : r1218658;
        return r1218659;
}

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 2 regimes

  2. if x < 138.7487461970468

    1. Initial program 38.9

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

    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

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

    if 138.7487461970468 < x

    1. Initial program 0.3

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

    2. Simplified0.4

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

  4. Final simplification1.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 138.7487461970468:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{2}{3}, \left(x \cdot x\right) \cdot x, 2 - x \cdot x\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(e^{x \cdot \varepsilon - x}, \frac{1}{\varepsilon}, e^{x \cdot \varepsilon - x} - \frac{\frac{1}{\varepsilon} - 1}{e^{\mathsf{fma}\left(x, \varepsilon, x\right)}}\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019134 +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))