Average Error: 29.5 → 1.1
Time: 13.9s
Precision: 64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 178.4372176880553126920858630910515785217:\\ \;\;\;\;\frac{\sqrt[3]{{\left(\mathsf{fma}\left(0.6666666666666667406815349750104360282421, {x}^{3}, 2\right)\right)}^{3}} - 1 \cdot {x}^{2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 \cdot \left(\frac{e^{x \cdot \left(\varepsilon - 1\right)}}{\varepsilon} + e^{x \cdot \left(\varepsilon - 1\right)}\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\\ \end{array}\]
\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 178.4372176880553126920858630910515785217:\\
\;\;\;\;\frac{\sqrt[3]{{\left(\mathsf{fma}\left(0.6666666666666667406815349750104360282421, {x}^{3}, 2\right)\right)}^{3}} - 1 \cdot {x}^{2}}{2}\\

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

\end{array}
double f(double x, double eps) {
        double r38105 = 1.0;
        double r38106 = eps;
        double r38107 = r38105 / r38106;
        double r38108 = r38105 + r38107;
        double r38109 = r38105 - r38106;
        double r38110 = x;
        double r38111 = r38109 * r38110;
        double r38112 = -r38111;
        double r38113 = exp(r38112);
        double r38114 = r38108 * r38113;
        double r38115 = r38107 - r38105;
        double r38116 = r38105 + r38106;
        double r38117 = r38116 * r38110;
        double r38118 = -r38117;
        double r38119 = exp(r38118);
        double r38120 = r38115 * r38119;
        double r38121 = r38114 - r38120;
        double r38122 = 2.0;
        double r38123 = r38121 / r38122;
        return r38123;
}
↓
double f(double x, double eps) {
        double r38124 = x;
        double r38125 = 178.4372176880553;
        bool r38126 = r38124 <= r38125;
        double r38127 = 0.6666666666666667;
        double r38128 = 3.0;
        double r38129 = pow(r38124, r38128);
        double r38130 = 2.0;
        double r38131 = fma(r38127, r38129, r38130);
        double r38132 = pow(r38131, r38128);
        double r38133 = cbrt(r38132);
        double r38134 = 1.0;
        double r38135 = 2.0;
        double r38136 = pow(r38124, r38135);
        double r38137 = r38134 * r38136;
        double r38138 = r38133 - r38137;
        double r38139 = r38138 / r38130;
        double r38140 = eps;
        double r38141 = r38140 - r38134;
        double r38142 = r38124 * r38141;
        double r38143 = exp(r38142);
        double r38144 = r38143 / r38140;
        double r38145 = r38144 + r38143;
        double r38146 = r38134 * r38145;
        double r38147 = r38134 / r38140;
        double r38148 = r38147 - r38134;
        double r38149 = r38134 + r38140;
        double r38150 = r38149 * r38124;
        double r38151 = -r38150;
        double r38152 = exp(r38151);
        double r38153 = r38148 * r38152;
        double r38154 = r38146 - r38153;
        double r38155 = r38154 / r38130;
        double r38156 = r38126 ? r38139 : r38155;
        return r38156;
}

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 2 regimes

  2. if x < 178.4372176880553

    1. Initial program 39.0

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

    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

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

    5. Applied add-cbrt-cube1.3

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

    6. Simplified1.3

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

    if 178.4372176880553 < 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. Taylor expanded around inf 0.2

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

    3. Simplified0.2

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

  4. Final simplification1.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 178.4372176880553126920858630910515785217:\\ \;\;\;\;\frac{\sqrt[3]{{\left(\mathsf{fma}\left(0.6666666666666667406815349750104360282421, {x}^{3}, 2\right)\right)}^{3}} - 1 \cdot {x}^{2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 \cdot \left(\frac{e^{x \cdot \left(\varepsilon - 1\right)}}{\varepsilon} + e^{x \cdot \left(\varepsilon - 1\right)}\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019350 +o rules:numerics
(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))