Average Error: 29.7 → 1.0
Time: 25.9s
Precision: 64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 100.79533204800275:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{2}{3} \cdot x, x \cdot x, 2 - x \cdot x\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(e^{-\left(x + x \cdot \varepsilon\right)} + e^{x \cdot \varepsilon - x}\right) + \frac{e^{x \cdot \varepsilon - x}}{\varepsilon}\right) - \frac{e^{-\left(x + x \cdot \varepsilon\right)}}{\varepsilon}}{2}\\ \end{array}\]
\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 100.79533204800275:\\
\;\;\;\;\frac{\mathsf{fma}\left(\frac{2}{3} \cdot x, x \cdot x, 2 - x \cdot x\right)}{2}\\

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

\end{array}
double f(double x, double eps) {
        double r620585 = 1.0;
        double r620586 = eps;
        double r620587 = r620585 / r620586;
        double r620588 = r620585 + r620587;
        double r620589 = r620585 - r620586;
        double r620590 = x;
        double r620591 = r620589 * r620590;
        double r620592 = -r620591;
        double r620593 = exp(r620592);
        double r620594 = r620588 * r620593;
        double r620595 = r620587 - r620585;
        double r620596 = r620585 + r620586;
        double r620597 = r620596 * r620590;
        double r620598 = -r620597;
        double r620599 = exp(r620598);
        double r620600 = r620595 * r620599;
        double r620601 = r620594 - r620600;
        double r620602 = 2.0;
        double r620603 = r620601 / r620602;
        return r620603;
}
↓
double f(double x, double eps) {
        double r620604 = x;
        double r620605 = 100.79533204800275;
        bool r620606 = r620604 <= r620605;
        double r620607 = 0.6666666666666666;
        double r620608 = r620607 * r620604;
        double r620609 = r620604 * r620604;
        double r620610 = 2.0;
        double r620611 = r620610 - r620609;
        double r620612 = fma(r620608, r620609, r620611);
        double r620613 = r620612 / r620610;
        double r620614 = eps;
        double r620615 = r620604 * r620614;
        double r620616 = r620604 + r620615;
        double r620617 = -r620616;
        double r620618 = exp(r620617);
        double r620619 = r620615 - r620604;
        double r620620 = exp(r620619);
        double r620621 = r620618 + r620620;
        double r620622 = r620620 / r620614;
        double r620623 = r620621 + r620622;
        double r620624 = r620618 / r620614;
        double r620625 = r620623 - r620624;
        double r620626 = r620625 / r620610;
        double r620627 = r620606 ? r620613 : r620626;
        return r620627;
}

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 2 regimes

  2. if x < 100.79533204800275

    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

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

    3. Simplified1.3

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

    4. Taylor expanded around -inf 1.3

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

    5. Simplified1.3

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

    if 100.79533204800275 < 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. Taylor expanded around -inf 0.1

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

  4. Final simplification1.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 100.79533204800275:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{2}{3} \cdot x, x \cdot x, 2 - x \cdot x\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(e^{-\left(x + x \cdot \varepsilon\right)} + e^{x \cdot \varepsilon - x}\right) + \frac{e^{x \cdot \varepsilon - x}}{\varepsilon}\right) - \frac{e^{-\left(x + x \cdot \varepsilon\right)}}{\varepsilon}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019151 +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))