Average Error: 29.3 → 0.9
Time: 28.4s
Precision: 64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 307.0243008404694:\\ \;\;\;\;\frac{\left(2 - x \cdot x\right) + \frac{2}{3} \cdot \left(\left(x \cdot x\right) \cdot x\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(e^{\left(-x\right) \cdot \varepsilon + \left(-x\right)} + e^{\varepsilon \cdot x - x}\right) + \frac{e^{\varepsilon \cdot x - x}}{\varepsilon}\right) - \frac{e^{\left(-x\right) \cdot \varepsilon + \left(-x\right)}}{\varepsilon}}{2}\\ \end{array}\]
\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 307.0243008404694:\\
\;\;\;\;\frac{\left(2 - x \cdot x\right) + \frac{2}{3} \cdot \left(\left(x \cdot x\right) \cdot x\right)}{2}\\

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

\end{array}
double f(double x, double eps) {
        double r1521074 = 1.0;
        double r1521075 = eps;
        double r1521076 = r1521074 / r1521075;
        double r1521077 = r1521074 + r1521076;
        double r1521078 = r1521074 - r1521075;
        double r1521079 = x;
        double r1521080 = r1521078 * r1521079;
        double r1521081 = -r1521080;
        double r1521082 = exp(r1521081);
        double r1521083 = r1521077 * r1521082;
        double r1521084 = r1521076 - r1521074;
        double r1521085 = r1521074 + r1521075;
        double r1521086 = r1521085 * r1521079;
        double r1521087 = -r1521086;
        double r1521088 = exp(r1521087);
        double r1521089 = r1521084 * r1521088;
        double r1521090 = r1521083 - r1521089;
        double r1521091 = 2.0;
        double r1521092 = r1521090 / r1521091;
        return r1521092;
}
↓
double f(double x, double eps) {
        double r1521093 = x;
        double r1521094 = 307.0243008404694;
        bool r1521095 = r1521093 <= r1521094;
        double r1521096 = 2.0;
        double r1521097 = r1521093 * r1521093;
        double r1521098 = r1521096 - r1521097;
        double r1521099 = 0.6666666666666666;
        double r1521100 = r1521097 * r1521093;
        double r1521101 = r1521099 * r1521100;
        double r1521102 = r1521098 + r1521101;
        double r1521103 = r1521102 / r1521096;
        double r1521104 = -r1521093;
        double r1521105 = eps;
        double r1521106 = r1521104 * r1521105;
        double r1521107 = r1521106 + r1521104;
        double r1521108 = exp(r1521107);
        double r1521109 = r1521105 * r1521093;
        double r1521110 = r1521109 - r1521093;
        double r1521111 = exp(r1521110);
        double r1521112 = r1521108 + r1521111;
        double r1521113 = r1521111 / r1521105;
        double r1521114 = r1521112 + r1521113;
        double r1521115 = r1521108 / r1521105;
        double r1521116 = r1521114 - r1521115;
        double r1521117 = r1521116 / r1521096;
        double r1521118 = r1521095 ? r1521103 : r1521117;
        return r1521118;
}

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 < 307.0243008404694

    1. Initial program 38.9

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

    2. Taylor expanded around 0 1.2

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

    3. Simplified1.2

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

    4. Taylor expanded around 0 1.2

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

    5. Simplified1.2

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

    if 307.0243008404694 < x

    1. Initial program 0.0

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

    2. Taylor expanded around -inf 0.0

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

  4. Final simplification0.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 307.0243008404694:\\ \;\;\;\;\frac{\left(2 - x \cdot x\right) + \frac{2}{3} \cdot \left(\left(x \cdot x\right) \cdot x\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(e^{\left(-x\right) \cdot \varepsilon + \left(-x\right)} + e^{\varepsilon \cdot x - x}\right) + \frac{e^{\varepsilon \cdot x - x}}{\varepsilon}\right) - \frac{e^{\left(-x\right) \cdot \varepsilon + \left(-x\right)}}{\varepsilon}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019141 
(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))