Average Error: 29.1 → 1.0
Time: 45.7s
Precision: 64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 0.8977144360588689:\\ \;\;\;\;\frac{\left(2 - x \cdot x\right) - \frac{-2}{3} \cdot \left(\left(x \cdot x\right) \cdot x\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(-1 - \varepsilon\right)} + \left(\left(e^{\left(-1 + \varepsilon\right) \cdot x} + \frac{e^{\left(-1 + \varepsilon\right) \cdot x}}{\varepsilon}\right) - \frac{e^{x \cdot \left(-1 - \varepsilon\right)}}{\varepsilon}\right)}{2}\\ \end{array}\]
\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 0.8977144360588689:\\
\;\;\;\;\frac{\left(2 - x \cdot x\right) - \frac{-2}{3} \cdot \left(\left(x \cdot x\right) \cdot x\right)}{2}\\

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

\end{array}
double f(double x, double eps) {
        double r3542019 = 1.0;
        double r3542020 = eps;
        double r3542021 = r3542019 / r3542020;
        double r3542022 = r3542019 + r3542021;
        double r3542023 = r3542019 - r3542020;
        double r3542024 = x;
        double r3542025 = r3542023 * r3542024;
        double r3542026 = -r3542025;
        double r3542027 = exp(r3542026);
        double r3542028 = r3542022 * r3542027;
        double r3542029 = r3542021 - r3542019;
        double r3542030 = r3542019 + r3542020;
        double r3542031 = r3542030 * r3542024;
        double r3542032 = -r3542031;
        double r3542033 = exp(r3542032);
        double r3542034 = r3542029 * r3542033;
        double r3542035 = r3542028 - r3542034;
        double r3542036 = 2.0;
        double r3542037 = r3542035 / r3542036;
        return r3542037;
}
↓
double f(double x, double eps) {
        double r3542038 = x;
        double r3542039 = 0.8977144360588689;
        bool r3542040 = r3542038 <= r3542039;
        double r3542041 = 2.0;
        double r3542042 = r3542038 * r3542038;
        double r3542043 = r3542041 - r3542042;
        double r3542044 = -0.6666666666666666;
        double r3542045 = r3542042 * r3542038;
        double r3542046 = r3542044 * r3542045;
        double r3542047 = r3542043 - r3542046;
        double r3542048 = r3542047 / r3542041;
        double r3542049 = -1.0;
        double r3542050 = eps;
        double r3542051 = r3542049 - r3542050;
        double r3542052 = r3542038 * r3542051;
        double r3542053 = exp(r3542052);
        double r3542054 = r3542049 + r3542050;
        double r3542055 = r3542054 * r3542038;
        double r3542056 = exp(r3542055);
        double r3542057 = r3542056 / r3542050;
        double r3542058 = r3542056 + r3542057;
        double r3542059 = r3542053 / r3542050;
        double r3542060 = r3542058 - r3542059;
        double r3542061 = r3542053 + r3542060;
        double r3542062 = r3542061 / r3542041;
        double r3542063 = r3542040 ? r3542048 : r3542062;
        return r3542063;
}

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 < 0.8977144360588689

    1. Initial program 38.8

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

    2. Simplified38.8

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

    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

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

    5. Taylor expanded around inf 1.2

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

    6. Simplified1.2

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

    if 0.8977144360588689 < x

    1. Initial program 0.7

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

    2. Simplified0.7

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

    4. Applied associate--r-0.6

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

  4. Final simplification1.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 0.8977144360588689:\\ \;\;\;\;\frac{\left(2 - x \cdot x\right) - \frac{-2}{3} \cdot \left(\left(x \cdot x\right) \cdot x\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(-1 - \varepsilon\right)} + \left(\left(e^{\left(-1 + \varepsilon\right) \cdot x} + \frac{e^{\left(-1 + \varepsilon\right) \cdot x}}{\varepsilon}\right) - \frac{e^{x \cdot \left(-1 - \varepsilon\right)}}{\varepsilon}\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019121 
(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))