Average Error: 29.2 → 0.9
Time: 23.7s
Precision: 64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 326.6122928500697639719874132424592971802:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.6666666666666667406815349750104360282421, {x}^{3}, 2\right) - 1 \cdot {x}^{2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\sqrt[3]{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}} \cdot \sqrt[3]{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}\right) \cdot \sqrt[3]{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}}{2}\\ \end{array}\]
\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 326.6122928500697639719874132424592971802:\\
\;\;\;\;\frac{\mathsf{fma}\left(0.6666666666666667406815349750104360282421, {x}^{3}, 2\right) - 1 \cdot {x}^{2}}{2}\\

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

\end{array}
double f(double x, double eps) {
        double r88088 = 1.0;
        double r88089 = eps;
        double r88090 = r88088 / r88089;
        double r88091 = r88088 + r88090;
        double r88092 = r88088 - r88089;
        double r88093 = x;
        double r88094 = r88092 * r88093;
        double r88095 = -r88094;
        double r88096 = exp(r88095);
        double r88097 = r88091 * r88096;
        double r88098 = r88090 - r88088;
        double r88099 = r88088 + r88089;
        double r88100 = r88099 * r88093;
        double r88101 = -r88100;
        double r88102 = exp(r88101);
        double r88103 = r88098 * r88102;
        double r88104 = r88097 - r88103;
        double r88105 = 2.0;
        double r88106 = r88104 / r88105;
        return r88106;
}
↓
double f(double x, double eps) {
        double r88107 = x;
        double r88108 = 326.61229285006976;
        bool r88109 = r88107 <= r88108;
        double r88110 = 0.6666666666666667;
        double r88111 = 3.0;
        double r88112 = pow(r88107, r88111);
        double r88113 = 2.0;
        double r88114 = fma(r88110, r88112, r88113);
        double r88115 = 1.0;
        double r88116 = 2.0;
        double r88117 = pow(r88107, r88116);
        double r88118 = r88115 * r88117;
        double r88119 = r88114 - r88118;
        double r88120 = r88119 / r88113;
        double r88121 = eps;
        double r88122 = r88115 / r88121;
        double r88123 = r88115 + r88122;
        double r88124 = r88115 - r88121;
        double r88125 = r88124 * r88107;
        double r88126 = -r88125;
        double r88127 = exp(r88126);
        double r88128 = r88123 * r88127;
        double r88129 = r88122 - r88115;
        double r88130 = r88115 + r88121;
        double r88131 = r88130 * r88107;
        double r88132 = -r88131;
        double r88133 = exp(r88132);
        double r88134 = r88129 * r88133;
        double r88135 = cbrt(r88134);
        double r88136 = r88135 * r88135;
        double r88137 = r88136 * r88135;
        double r88138 = r88128 - r88137;
        double r88139 = r88138 / r88113;
        double r88140 = r88109 ? r88120 : r88139;
        return r88140;
}

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 2 regimes

  2. if x < 326.61229285006976

    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 1.2

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

    3. Simplified1.2

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

    if 326.61229285006976 < 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. Using strategy rm

    3. Applied add-cube-cbrt0.1

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

  4. Final simplification0.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 326.6122928500697639719874132424592971802:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.6666666666666667406815349750104360282421, {x}^{3}, 2\right) - 1 \cdot {x}^{2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\sqrt[3]{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}} \cdot \sqrt[3]{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}\right) \cdot \sqrt[3]{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019212 +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))