Average Error: 29.7 → 1.1
Time: 29.8s
Precision: 64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 331.7599664969623631805006880313158035278:\\ \;\;\;\;\frac{\left(0.6666666666666667406815349750104360282421 \cdot {x}^{3} + 2\right) - 1 \cdot {x}^{2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\left(\sqrt[3]{\frac{1}{\varepsilon} - 1} \cdot \sqrt[3]{\frac{1}{\varepsilon} - 1}\right) \cdot \sqrt[3]{\frac{1}{\varepsilon} - 1}\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\\ \end{array}\]
\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 331.7599664969623631805006880313158035278:\\
\;\;\;\;\frac{\left(0.6666666666666667406815349750104360282421 \cdot {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(\left(\sqrt[3]{\frac{1}{\varepsilon} - 1} \cdot \sqrt[3]{\frac{1}{\varepsilon} - 1}\right) \cdot \sqrt[3]{\frac{1}{\varepsilon} - 1}\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\\

\end{array}
double f(double x, double eps) {
        double r40970 = 1.0;
        double r40971 = eps;
        double r40972 = r40970 / r40971;
        double r40973 = r40970 + r40972;
        double r40974 = r40970 - r40971;
        double r40975 = x;
        double r40976 = r40974 * r40975;
        double r40977 = -r40976;
        double r40978 = exp(r40977);
        double r40979 = r40973 * r40978;
        double r40980 = r40972 - r40970;
        double r40981 = r40970 + r40971;
        double r40982 = r40981 * r40975;
        double r40983 = -r40982;
        double r40984 = exp(r40983);
        double r40985 = r40980 * r40984;
        double r40986 = r40979 - r40985;
        double r40987 = 2.0;
        double r40988 = r40986 / r40987;
        return r40988;
}
↓
double f(double x, double eps) {
        double r40989 = x;
        double r40990 = 331.75996649696236;
        bool r40991 = r40989 <= r40990;
        double r40992 = 0.6666666666666667;
        double r40993 = 3.0;
        double r40994 = pow(r40989, r40993);
        double r40995 = r40992 * r40994;
        double r40996 = 2.0;
        double r40997 = r40995 + r40996;
        double r40998 = 1.0;
        double r40999 = 2.0;
        double r41000 = pow(r40989, r40999);
        double r41001 = r40998 * r41000;
        double r41002 = r40997 - r41001;
        double r41003 = r41002 / r40996;
        double r41004 = eps;
        double r41005 = r40998 / r41004;
        double r41006 = r40998 + r41005;
        double r41007 = r40998 - r41004;
        double r41008 = r41007 * r40989;
        double r41009 = -r41008;
        double r41010 = exp(r41009);
        double r41011 = r41006 * r41010;
        double r41012 = r41005 - r40998;
        double r41013 = cbrt(r41012);
        double r41014 = r41013 * r41013;
        double r41015 = r41014 * r41013;
        double r41016 = r40998 + r41004;
        double r41017 = r41016 * r40989;
        double r41018 = -r41017;
        double r41019 = exp(r41018);
        double r41020 = r41015 * r41019;
        double r41021 = r41011 - r41020;
        double r41022 = r41021 / r40996;
        double r41023 = r40991 ? r41003 : r41022;
        return r41023;
}

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 < 331.75996649696236

    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. Taylor expanded around 0 1.4

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

    if 331.75996649696236 < 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 add-cube-cbrt0.2

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

  4. Final simplification1.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 331.7599664969623631805006880313158035278:\\ \;\;\;\;\frac{\left(0.6666666666666667406815349750104360282421 \cdot {x}^{3} + 2\right) - 1 \cdot {x}^{2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\left(\sqrt[3]{\frac{1}{\varepsilon} - 1} \cdot \sqrt[3]{\frac{1}{\varepsilon} - 1}\right) \cdot \sqrt[3]{\frac{1}{\varepsilon} - 1}\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019325 
(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))