Average Error: 29.5 → 1.1
Time: 26.7s
Precision: 64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 1.9334006562055708:\\ \;\;\;\;\frac{\left(2 + \left(\frac{2}{3} \cdot x\right) \cdot \left(x \cdot x\right)\right) - x \cdot x}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(e^{\left(-1 + \varepsilon\right) \cdot x} - \frac{e^{x \cdot \left(-1 - \varepsilon\right)}}{\varepsilon}\right) + \frac{e^{\left(-1 + \varepsilon\right) \cdot x}}{\varepsilon}\right) + e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\\ \end{array}\]
\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 1.9334006562055708:\\
\;\;\;\;\frac{\left(2 + \left(\frac{2}{3} \cdot x\right) \cdot \left(x \cdot x\right)\right) - x \cdot x}{2}\\

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

\end{array}
double f(double x, double eps) {
        double r1909968 = 1.0;
        double r1909969 = eps;
        double r1909970 = r1909968 / r1909969;
        double r1909971 = r1909968 + r1909970;
        double r1909972 = r1909968 - r1909969;
        double r1909973 = x;
        double r1909974 = r1909972 * r1909973;
        double r1909975 = -r1909974;
        double r1909976 = exp(r1909975);
        double r1909977 = r1909971 * r1909976;
        double r1909978 = r1909970 - r1909968;
        double r1909979 = r1909968 + r1909969;
        double r1909980 = r1909979 * r1909973;
        double r1909981 = -r1909980;
        double r1909982 = exp(r1909981);
        double r1909983 = r1909978 * r1909982;
        double r1909984 = r1909977 - r1909983;
        double r1909985 = 2.0;
        double r1909986 = r1909984 / r1909985;
        return r1909986;
}
↓
double f(double x, double eps) {
        double r1909987 = x;
        double r1909988 = 1.9334006562055708;
        bool r1909989 = r1909987 <= r1909988;
        double r1909990 = 2.0;
        double r1909991 = 0.6666666666666666;
        double r1909992 = r1909991 * r1909987;
        double r1909993 = r1909987 * r1909987;
        double r1909994 = r1909992 * r1909993;
        double r1909995 = r1909990 + r1909994;
        double r1909996 = r1909995 - r1909993;
        double r1909997 = r1909996 / r1909990;
        double r1909998 = -1.0;
        double r1909999 = eps;
        double r1910000 = r1909998 + r1909999;
        double r1910001 = r1910000 * r1909987;
        double r1910002 = exp(r1910001);
        double r1910003 = r1909998 - r1909999;
        double r1910004 = r1909987 * r1910003;
        double r1910005 = exp(r1910004);
        double r1910006 = r1910005 / r1909999;
        double r1910007 = r1910002 - r1910006;
        double r1910008 = r1910002 / r1909999;
        double r1910009 = r1910007 + r1910008;
        double r1910010 = r1910009 + r1910005;
        double r1910011 = r1910010 / r1909990;
        double r1910012 = r1909989 ? r1909997 : r1910011;
        return r1910012;
}

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 < 1.9334006562055708

    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. Simplified39.0

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

    3. Taylor expanded around 0 1.2

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

    4. Simplified1.2

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

    if 1.9334006562055708 < x

    1. Initial program 0.6

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

    2. Simplified0.6

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

    4. Applied associate-+r+0.5

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

  4. Final simplification1.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 1.9334006562055708:\\ \;\;\;\;\frac{\left(2 + \left(\frac{2}{3} \cdot x\right) \cdot \left(x \cdot x\right)\right) - x \cdot x}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(e^{\left(-1 + \varepsilon\right) \cdot x} - \frac{e^{x \cdot \left(-1 - \varepsilon\right)}}{\varepsilon}\right) + \frac{e^{\left(-1 + \varepsilon\right) \cdot x}}{\varepsilon}\right) + e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019163 
(FPCore (x eps)
  :name "NMSE Section 6.1 mentioned, A"
  (/ (- (* (+ 1 (/ 1 eps)) (exp (- (* (- 1 eps) x)))) (* (- (/ 1 eps) 1) (exp (- (* (+ 1 eps) x))))) 2))