Average Error: 29.6 → 1.0
Time: 1.4m
Precision: 64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 1.2131256673255104:\\ \;\;\;\;\frac{x \cdot \left(\left(\frac{2}{3} \cdot x\right) \cdot x - x\right) + 2}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{e^{\left(-1 + \varepsilon\right) \cdot x}} \cdot \left(\sqrt[3]{e^{\left(-1 + \varepsilon\right) \cdot x}} \cdot \sqrt[3]{e^{\left(-1 + \varepsilon\right) \cdot x}}\right) + \left(\frac{e^{\left(-1 + \varepsilon\right) \cdot x}}{\varepsilon} - \left(\frac{e^{x \cdot \left(-1 - \varepsilon\right)}}{\varepsilon} - e^{x \cdot \left(-1 - \varepsilon\right)}\right)\right)}{2}\\ \end{array}\]
\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 1.2131256673255104:\\
\;\;\;\;\frac{x \cdot \left(\left(\frac{2}{3} \cdot x\right) \cdot x - x\right) + 2}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\sqrt[3]{e^{\left(-1 + \varepsilon\right) \cdot x}} \cdot \left(\sqrt[3]{e^{\left(-1 + \varepsilon\right) \cdot x}} \cdot \sqrt[3]{e^{\left(-1 + \varepsilon\right) \cdot x}}\right) + \left(\frac{e^{\left(-1 + \varepsilon\right) \cdot x}}{\varepsilon} - \left(\frac{e^{x \cdot \left(-1 - \varepsilon\right)}}{\varepsilon} - e^{x \cdot \left(-1 - \varepsilon\right)}\right)\right)}{2}\\

\end{array}
double f(double x, double eps) {
        double r9199109 = 1.0;
        double r9199110 = eps;
        double r9199111 = r9199109 / r9199110;
        double r9199112 = r9199109 + r9199111;
        double r9199113 = r9199109 - r9199110;
        double r9199114 = x;
        double r9199115 = r9199113 * r9199114;
        double r9199116 = -r9199115;
        double r9199117 = exp(r9199116);
        double r9199118 = r9199112 * r9199117;
        double r9199119 = r9199111 - r9199109;
        double r9199120 = r9199109 + r9199110;
        double r9199121 = r9199120 * r9199114;
        double r9199122 = -r9199121;
        double r9199123 = exp(r9199122);
        double r9199124 = r9199119 * r9199123;
        double r9199125 = r9199118 - r9199124;
        double r9199126 = 2.0;
        double r9199127 = r9199125 / r9199126;
        return r9199127;
}
↓
double f(double x, double eps) {
        double r9199128 = x;
        double r9199129 = 1.2131256673255104;
        bool r9199130 = r9199128 <= r9199129;
        double r9199131 = 0.6666666666666666;
        double r9199132 = r9199131 * r9199128;
        double r9199133 = r9199132 * r9199128;
        double r9199134 = r9199133 - r9199128;
        double r9199135 = r9199128 * r9199134;
        double r9199136 = 2.0;
        double r9199137 = r9199135 + r9199136;
        double r9199138 = r9199137 / r9199136;
        double r9199139 = -1.0;
        double r9199140 = eps;
        double r9199141 = r9199139 + r9199140;
        double r9199142 = r9199141 * r9199128;
        double r9199143 = exp(r9199142);
        double r9199144 = cbrt(r9199143);
        double r9199145 = r9199144 * r9199144;
        double r9199146 = r9199144 * r9199145;
        double r9199147 = r9199143 / r9199140;
        double r9199148 = r9199139 - r9199140;
        double r9199149 = r9199128 * r9199148;
        double r9199150 = exp(r9199149);
        double r9199151 = r9199150 / r9199140;
        double r9199152 = r9199151 - r9199150;
        double r9199153 = r9199147 - r9199152;
        double r9199154 = r9199146 + r9199153;
        double r9199155 = r9199154 / r9199136;
        double r9199156 = r9199130 ? r9199138 : r9199155;
        return r9199156;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if x < 1.2131256673255104

    1. Initial program 39.4

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

    2. Simplified39.4

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

    3. Taylor expanded around 0 1.2

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

    4. Simplified1.2

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

    if 1.2131256673255104 < x

    1. Initial program 0.6

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

    2. Simplified0.5

      \[\leadsto \color{blue}{\frac{\left(e^{x \cdot \left(\varepsilon + -1\right)} + \frac{e^{x \cdot \left(\varepsilon + -1\right)}}{\varepsilon}\right) - \left(\frac{e^{x \cdot \left(-1 - \varepsilon\right)}}{\varepsilon} - e^{x \cdot \left(-1 - \varepsilon\right)}\right)}{2}}\]
    3. Using strategy rm

    4. Applied associate--l+0.5

      \[\leadsto \frac{\color{blue}{e^{x \cdot \left(\varepsilon + -1\right)} + \left(\frac{e^{x \cdot \left(\varepsilon + -1\right)}}{\varepsilon} - \left(\frac{e^{x \cdot \left(-1 - \varepsilon\right)}}{\varepsilon} - e^{x \cdot \left(-1 - \varepsilon\right)}\right)\right)}}{2}\]
    5. Using strategy rm

    6. Applied add-cube-cbrt0.5

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

  4. Final simplification1.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 1.2131256673255104:\\ \;\;\;\;\frac{x \cdot \left(\left(\frac{2}{3} \cdot x\right) \cdot x - x\right) + 2}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{e^{\left(-1 + \varepsilon\right) \cdot x}} \cdot \left(\sqrt[3]{e^{\left(-1 + \varepsilon\right) \cdot x}} \cdot \sqrt[3]{e^{\left(-1 + \varepsilon\right) \cdot x}}\right) + \left(\frac{e^{\left(-1 + \varepsilon\right) \cdot x}}{\varepsilon} - \left(\frac{e^{x \cdot \left(-1 - \varepsilon\right)}}{\varepsilon} - e^{x \cdot \left(-1 - \varepsilon\right)}\right)\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019125 
(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))