Average Error: 30.1 → 1.1
Time: 23.4s
Precision: 64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 363.2798126170393402389890979975461959839:\\ \;\;\;\;\frac{\left(2 + \left({x}^{3} \cdot \sqrt[3]{0.6666666666666667406815349750104360282421}\right) \cdot \left(\sqrt[3]{0.6666666666666667406815349750104360282421} \cdot \sqrt[3]{0.6666666666666667406815349750104360282421}\right)\right) - \left(1 \cdot x\right) \cdot x}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{1}{\varepsilon} + 1\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \frac{\frac{1}{\varepsilon} - 1}{e^{x \cdot \left(1 + \varepsilon\right)}}}{2}\\ \end{array}\]
\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 363.2798126170393402389890979975461959839:\\
\;\;\;\;\frac{\left(2 + \left({x}^{3} \cdot \sqrt[3]{0.6666666666666667406815349750104360282421}\right) \cdot \left(\sqrt[3]{0.6666666666666667406815349750104360282421} \cdot \sqrt[3]{0.6666666666666667406815349750104360282421}\right)\right) - \left(1 \cdot x\right) \cdot x}{2}\\

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

\end{array}
double f(double x, double eps) {
        double r44104 = 1.0;
        double r44105 = eps;
        double r44106 = r44104 / r44105;
        double r44107 = r44104 + r44106;
        double r44108 = r44104 - r44105;
        double r44109 = x;
        double r44110 = r44108 * r44109;
        double r44111 = -r44110;
        double r44112 = exp(r44111);
        double r44113 = r44107 * r44112;
        double r44114 = r44106 - r44104;
        double r44115 = r44104 + r44105;
        double r44116 = r44115 * r44109;
        double r44117 = -r44116;
        double r44118 = exp(r44117);
        double r44119 = r44114 * r44118;
        double r44120 = r44113 - r44119;
        double r44121 = 2.0;
        double r44122 = r44120 / r44121;
        return r44122;
}
↓
double f(double x, double eps) {
        double r44123 = x;
        double r44124 = 363.27981261703934;
        bool r44125 = r44123 <= r44124;
        double r44126 = 2.0;
        double r44127 = 3.0;
        double r44128 = pow(r44123, r44127);
        double r44129 = 0.6666666666666667;
        double r44130 = cbrt(r44129);
        double r44131 = r44128 * r44130;
        double r44132 = r44130 * r44130;
        double r44133 = r44131 * r44132;
        double r44134 = r44126 + r44133;
        double r44135 = 1.0;
        double r44136 = r44135 * r44123;
        double r44137 = r44136 * r44123;
        double r44138 = r44134 - r44137;
        double r44139 = r44138 / r44126;
        double r44140 = eps;
        double r44141 = r44135 / r44140;
        double r44142 = r44141 + r44135;
        double r44143 = r44135 - r44140;
        double r44144 = -r44123;
        double r44145 = r44143 * r44144;
        double r44146 = exp(r44145);
        double r44147 = r44142 * r44146;
        double r44148 = r44141 - r44135;
        double r44149 = r44135 + r44140;
        double r44150 = r44123 * r44149;
        double r44151 = exp(r44150);
        double r44152 = r44148 / r44151;
        double r44153 = r44147 - r44152;
        double r44154 = r44153 / r44126;
        double r44155 = r44125 ? r44139 : r44154;
        return r44155;
}

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 < 363.27981261703934

    1. Initial program 39.5

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

    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

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

    5. Applied add-cube-cbrt1.3

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

    6. Applied associate-*l*1.3

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

    if 363.27981261703934 < x

    1. Initial program 0.2

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

    3. Applied exp-neg0.2

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

    4. Applied un-div-inv0.2

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

  4. Final simplification1.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 363.2798126170393402389890979975461959839:\\ \;\;\;\;\frac{\left(2 + \left({x}^{3} \cdot \sqrt[3]{0.6666666666666667406815349750104360282421}\right) \cdot \left(\sqrt[3]{0.6666666666666667406815349750104360282421} \cdot \sqrt[3]{0.6666666666666667406815349750104360282421}\right)\right) - \left(1 \cdot x\right) \cdot x}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{1}{\varepsilon} + 1\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \frac{\frac{1}{\varepsilon} - 1}{e^{x \cdot \left(1 + \varepsilon\right)}}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019195 
(FPCore (x eps)
  :name "NMSE Section 6.1 mentioned, A"
  (/ (- (* (+ 1.0 (/ 1.0 eps)) (exp (- (* (- 1.0 eps) x)))) (* (- (/ 1.0 eps) 1.0) (exp (- (* (+ 1.0 eps) x))))) 2.0))