Average Error: 30.2 → 1.1
Time: 22.6s
Precision: 64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le \frac{5944341917089697}{70368744177664}:\\ \;\;\;\;\frac{{x}^{2} \cdot \left(x \cdot \frac{3002399751580331}{4503599627370496} - 1\right) + 2}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - 1 \cdot \left(\frac{e^{-\left(x \cdot \varepsilon + 1 \cdot x\right)}}{\varepsilon} - e^{-\left(x \cdot \varepsilon + 1 \cdot x\right)}\right)}{2}\\ \end{array}\]
\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 \frac{5944341917089697}{70368744177664}:\\
\;\;\;\;\frac{{x}^{2} \cdot \left(x \cdot \frac{3002399751580331}{4503599627370496} - 1\right) + 2}{2}\\

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

\end{array}
double f(double x, double eps) {
        double r47084 = 1.0;
        double r47085 = eps;
        double r47086 = r47084 / r47085;
        double r47087 = r47084 + r47086;
        double r47088 = r47084 - r47085;
        double r47089 = x;
        double r47090 = r47088 * r47089;
        double r47091 = -r47090;
        double r47092 = exp(r47091);
        double r47093 = r47087 * r47092;
        double r47094 = r47086 - r47084;
        double r47095 = r47084 + r47085;
        double r47096 = r47095 * r47089;
        double r47097 = -r47096;
        double r47098 = exp(r47097);
        double r47099 = r47094 * r47098;
        double r47100 = r47093 - r47099;
        double r47101 = 2.0;
        double r47102 = r47100 / r47101;
        return r47102;
}
↓
double f(double x, double eps) {
        double r47103 = x;
        double r47104 = 5944341917089697.0;
        double r47105 = 70368744177664.0;
        double r47106 = r47104 / r47105;
        bool r47107 = r47103 <= r47106;
        double r47108 = 2.0;
        double r47109 = pow(r47103, r47108);
        double r47110 = 3002399751580331.0;
        double r47111 = 4503599627370496.0;
        double r47112 = r47110 / r47111;
        double r47113 = r47103 * r47112;
        double r47114 = 1.0;
        double r47115 = r47113 - r47114;
        double r47116 = r47109 * r47115;
        double r47117 = 2.0;
        double r47118 = r47116 + r47117;
        double r47119 = r47118 / r47117;
        double r47120 = eps;
        double r47121 = r47114 / r47120;
        double r47122 = r47114 + r47121;
        double r47123 = r47114 - r47120;
        double r47124 = r47123 * r47103;
        double r47125 = -r47124;
        double r47126 = exp(r47125);
        double r47127 = r47122 * r47126;
        double r47128 = r47103 * r47120;
        double r47129 = r47114 * r47103;
        double r47130 = r47128 + r47129;
        double r47131 = -r47130;
        double r47132 = exp(r47131);
        double r47133 = r47132 / r47120;
        double r47134 = r47133 - r47132;
        double r47135 = r47114 * r47134;
        double r47136 = r47127 - r47135;
        double r47137 = r47136 / r47117;
        double r47138 = r47107 ? r47119 : r47137;
        return r47138;
}

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 < 84.47417936124704

    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

      \[\leadsto \frac{\color{blue}{\left(0.6666666666666667406815349750104360282421 \cdot {x}^{3} + 2\right) - 1 \cdot {x}^{2}}}{2}\]

    3. Simplified1.3

      \[\leadsto \frac{\color{blue}{\left(\frac{3002399751580331}{4503599627370496} \cdot {x}^{3} + 2\right) - 1 \cdot {x}^{2}}}{2}\]
    4. Using strategy rm

    5. Applied add-cube-cbrt1.3

      \[\leadsto \frac{\left(\color{blue}{\left(\sqrt[3]{\frac{3002399751580331}{4503599627370496} \cdot {x}^{3}} \cdot \sqrt[3]{\frac{3002399751580331}{4503599627370496} \cdot {x}^{3}}\right) \cdot \sqrt[3]{\frac{3002399751580331}{4503599627370496} \cdot {x}^{3}}} + 2\right) - 1 \cdot {x}^{2}}{2}\]

    6. Taylor expanded around 0 1.3

      \[\leadsto \frac{\color{blue}{\left(0.6666666666666667406815349750104360282421 \cdot {x}^{3} + 2\right) - 1 \cdot {x}^{2}}}{2}\]

    7. Simplified1.3

      \[\leadsto \frac{\color{blue}{{x}^{2} \cdot \left(x \cdot \frac{3002399751580331}{4503599627370496} - 1\right) + 2}}{2}\]

    if 84.47417936124704 < 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. Taylor expanded around inf 0.3

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

    3. Simplified0.3

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

  4. Final simplification1.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le \frac{5944341917089697}{70368744177664}:\\ \;\;\;\;\frac{{x}^{2} \cdot \left(x \cdot \frac{3002399751580331}{4503599627370496} - 1\right) + 2}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - 1 \cdot \left(\frac{e^{-\left(x \cdot \varepsilon + 1 \cdot x\right)}}{\varepsilon} - e^{-\left(x \cdot \varepsilon + 1 \cdot x\right)}\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019303 
(FPCore (x eps)
  :name "NMSE Section 6.1 mentioned, A"
  :precision binary64
  (/ (- (* (+ 1 (/ 1 eps)) (exp (- (* (- 1 eps) x)))) (* (- (/ 1 eps) 1) (exp (- (* (+ 1 eps) x))))) 2))