Average Error: 29.6 → 0.9
Time: 24.3s
Precision: 64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 100.06482820348397:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{2}{3}, \left(x \cdot x\right) \cdot x, 2 - x \cdot x\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(e^{\left(\varepsilon + -1\right) \cdot x}, \frac{1}{\varepsilon}, e^{\left(\varepsilon + -1\right) \cdot x} - \frac{\frac{1}{\varepsilon} - 1}{e^{\mathsf{fma}\left(\varepsilon, x, x\right)}}\right)}{2}\\ \end{array}\]
\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 100.06482820348397:\\
\;\;\;\;\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^{\left(\varepsilon + -1\right) \cdot x}, \frac{1}{\varepsilon}, e^{\left(\varepsilon + -1\right) \cdot x} - \frac{\frac{1}{\varepsilon} - 1}{e^{\mathsf{fma}\left(\varepsilon, x, x\right)}}\right)}{2}\\

\end{array}
double f(double x, double eps) {
        double r767618 = 1.0;
        double r767619 = eps;
        double r767620 = r767618 / r767619;
        double r767621 = r767618 + r767620;
        double r767622 = r767618 - r767619;
        double r767623 = x;
        double r767624 = r767622 * r767623;
        double r767625 = -r767624;
        double r767626 = exp(r767625);
        double r767627 = r767621 * r767626;
        double r767628 = r767620 - r767618;
        double r767629 = r767618 + r767619;
        double r767630 = r767629 * r767623;
        double r767631 = -r767630;
        double r767632 = exp(r767631);
        double r767633 = r767628 * r767632;
        double r767634 = r767627 - r767633;
        double r767635 = 2.0;
        double r767636 = r767634 / r767635;
        return r767636;
}
↓
double f(double x, double eps) {
        double r767637 = x;
        double r767638 = 100.06482820348397;
        bool r767639 = r767637 <= r767638;
        double r767640 = 0.6666666666666666;
        double r767641 = r767637 * r767637;
        double r767642 = r767641 * r767637;
        double r767643 = 2.0;
        double r767644 = r767643 - r767641;
        double r767645 = fma(r767640, r767642, r767644);
        double r767646 = r767645 / r767643;
        double r767647 = eps;
        double r767648 = -1.0;
        double r767649 = r767647 + r767648;
        double r767650 = r767649 * r767637;
        double r767651 = exp(r767650);
        double r767652 = 1.0;
        double r767653 = r767652 / r767647;
        double r767654 = r767653 - r767652;
        double r767655 = fma(r767647, r767637, r767637);
        double r767656 = exp(r767655);
        double r767657 = r767654 / r767656;
        double r767658 = r767651 - r767657;
        double r767659 = fma(r767651, r767653, r767658);
        double r767660 = r767659 / r767643;
        double r767661 = r767639 ? r767646 : r767660;
        return r767661;
}

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 2 regimes

  2. if x < 100.06482820348397

    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

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

    if 100.06482820348397 < 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. Simplified0.3

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

    3. Taylor expanded around inf 0.3

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

    4. Simplified0.3

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

  4. Final simplification0.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 100.06482820348397:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{2}{3}, \left(x \cdot x\right) \cdot x, 2 - x \cdot x\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(e^{\left(\varepsilon + -1\right) \cdot x}, \frac{1}{\varepsilon}, e^{\left(\varepsilon + -1\right) \cdot x} - \frac{\frac{1}{\varepsilon} - 1}{e^{\mathsf{fma}\left(\varepsilon, x, x\right)}}\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019153 +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))