Average Error: 30.1 → 0.9
Time: 1.5m
Precision: 64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 240.6932575900809:\\ \;\;\;\;\frac{\left(2 - x \cdot x\right) - \frac{-2}{3} \cdot \left(\left(x \cdot x\right) \cdot x\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(e^{\left(-x\right) \cdot \varepsilon + \left(-x\right)} + e^{\varepsilon \cdot x - x}\right) + \frac{e^{\varepsilon \cdot x - x}}{\varepsilon}\right) - \frac{e^{\left(-x\right) \cdot \varepsilon + \left(-x\right)}}{\varepsilon}}{2}\\ \end{array}\]
double f(double x, double eps) {
        double r9055661 = 1.0;
        double r9055662 = eps;
        double r9055663 = r9055661 / r9055662;
        double r9055664 = r9055661 + r9055663;
        double r9055665 = r9055661 - r9055662;
        double r9055666 = x;
        double r9055667 = r9055665 * r9055666;
        double r9055668 = -r9055667;
        double r9055669 = exp(r9055668);
        double r9055670 = r9055664 * r9055669;
        double r9055671 = r9055663 - r9055661;
        double r9055672 = r9055661 + r9055662;
        double r9055673 = r9055672 * r9055666;
        double r9055674 = -r9055673;
        double r9055675 = exp(r9055674);
        double r9055676 = r9055671 * r9055675;
        double r9055677 = r9055670 - r9055676;
        double r9055678 = 2.0;
        double r9055679 = r9055677 / r9055678;
        return r9055679;
}
↓
double f(double x, double eps) {
        double r9055680 = x;
        double r9055681 = 240.6932575900809;
        bool r9055682 = r9055680 <= r9055681;
        double r9055683 = 2.0;
        double r9055684 = r9055680 * r9055680;
        double r9055685 = r9055683 - r9055684;
        double r9055686 = -0.6666666666666666;
        double r9055687 = r9055684 * r9055680;
        double r9055688 = r9055686 * r9055687;
        double r9055689 = r9055685 - r9055688;
        double r9055690 = r9055689 / r9055683;
        double r9055691 = -r9055680;
        double r9055692 = eps;
        double r9055693 = r9055691 * r9055692;
        double r9055694 = r9055693 + r9055691;
        double r9055695 = exp(r9055694);
        double r9055696 = r9055692 * r9055680;
        double r9055697 = r9055696 - r9055680;
        double r9055698 = exp(r9055697);
        double r9055699 = r9055695 + r9055698;
        double r9055700 = r9055698 / r9055692;
        double r9055701 = r9055699 + r9055700;
        double r9055702 = r9055695 / r9055692;
        double r9055703 = r9055701 - r9055702;
        double r9055704 = r9055703 / r9055683;
        double r9055705 = r9055682 ? r9055690 : r9055704;
        return r9055705;
}


\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 240.6932575900809:\\
\;\;\;\;\frac{\left(2 - x \cdot x\right) - \frac{-2}{3} \cdot \left(\left(x \cdot x\right) \cdot x\right)}{2}\\

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

\end{array}

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 2 regimes

  2. if x < 240.6932575900809

    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 1.1

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

    3. Simplified1.1

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

    4. Taylor expanded around inf 1.1

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

    5. Simplified1.1

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

    if 240.6932575900809 < 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 simplification0.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 240.6932575900809:\\ \;\;\;\;\frac{\left(2 - x \cdot x\right) - \frac{-2}{3} \cdot \left(\left(x \cdot x\right) \cdot x\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(e^{\left(-x\right) \cdot \varepsilon + \left(-x\right)} + e^{\varepsilon \cdot x - x}\right) + \frac{e^{\varepsilon \cdot x - x}}{\varepsilon}\right) - \frac{e^{\left(-x\right) \cdot \varepsilon + \left(-x\right)}}{\varepsilon}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019102 
(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))