Average Error: 29.6 → 1.0
Time: 30.6s
Precision: 64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 1.540523375218099:\\ \;\;\;\;\frac{\left(2 - x \cdot x\right) + \left(\frac{2}{3} \cdot \left(x \cdot x\right)\right) \cdot x}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{e^{\varepsilon \cdot x + \left(-x\right)}}{\varepsilon} - \left(\frac{e^{\left(-x\right) - \varepsilon \cdot x}}{\varepsilon} - e^{\left(-x\right) - \varepsilon \cdot x}\right)\right) + e^{\varepsilon \cdot x + \left(-x\right)}}{2}\\ \end{array}\]
\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 1.540523375218099:\\
\;\;\;\;\frac{\left(2 - x \cdot x\right) + \left(\frac{2}{3} \cdot \left(x \cdot x\right)\right) \cdot x}{2}\\

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

\end{array}
double f(double x, double eps) {
        double r1396213 = 1.0;
        double r1396214 = eps;
        double r1396215 = r1396213 / r1396214;
        double r1396216 = r1396213 + r1396215;
        double r1396217 = r1396213 - r1396214;
        double r1396218 = x;
        double r1396219 = r1396217 * r1396218;
        double r1396220 = -r1396219;
        double r1396221 = exp(r1396220);
        double r1396222 = r1396216 * r1396221;
        double r1396223 = r1396215 - r1396213;
        double r1396224 = r1396213 + r1396214;
        double r1396225 = r1396224 * r1396218;
        double r1396226 = -r1396225;
        double r1396227 = exp(r1396226);
        double r1396228 = r1396223 * r1396227;
        double r1396229 = r1396222 - r1396228;
        double r1396230 = 2.0;
        double r1396231 = r1396229 / r1396230;
        return r1396231;
}
↓
double f(double x, double eps) {
        double r1396232 = x;
        double r1396233 = 1.540523375218099;
        bool r1396234 = r1396232 <= r1396233;
        double r1396235 = 2.0;
        double r1396236 = r1396232 * r1396232;
        double r1396237 = r1396235 - r1396236;
        double r1396238 = 0.6666666666666666;
        double r1396239 = r1396238 * r1396236;
        double r1396240 = r1396239 * r1396232;
        double r1396241 = r1396237 + r1396240;
        double r1396242 = r1396241 / r1396235;
        double r1396243 = eps;
        double r1396244 = r1396243 * r1396232;
        double r1396245 = -r1396232;
        double r1396246 = r1396244 + r1396245;
        double r1396247 = exp(r1396246);
        double r1396248 = r1396247 / r1396243;
        double r1396249 = r1396245 - r1396244;
        double r1396250 = exp(r1396249);
        double r1396251 = r1396250 / r1396243;
        double r1396252 = r1396251 - r1396250;
        double r1396253 = r1396248 - r1396252;
        double r1396254 = r1396253 + r1396247;
        double r1396255 = r1396254 / r1396235;
        double r1396256 = r1396234 ? r1396242 : r1396255;
        return r1396256;
}

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 < 1.540523375218099

    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. Simplified39.5

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

    3. Taylor expanded around 0 1.1

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

    4. Simplified1.1

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

    if 1.540523375218099 < x

    1. Initial program 0.5

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

    2. Simplified0.5

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

    4. Applied associate--l+0.4

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

  4. Final simplification1.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 1.540523375218099:\\ \;\;\;\;\frac{\left(2 - x \cdot x\right) + \left(\frac{2}{3} \cdot \left(x \cdot x\right)\right) \cdot x}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{e^{\varepsilon \cdot x + \left(-x\right)}}{\varepsilon} - \left(\frac{e^{\left(-x\right) - \varepsilon \cdot x}}{\varepsilon} - e^{\left(-x\right) - \varepsilon \cdot x}\right)\right) + e^{\varepsilon \cdot x + \left(-x\right)}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019142 
(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))