Average Error: 28.8 → 0.9
Time: 27.2s
Precision: 64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 30.43326508471591:\\ \;\;\;\;\frac{\left(2 - x \cdot x\right) + \left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{2}{3}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{1}{\varepsilon} + 1\right) \cdot e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)} - {e}^{\left(\left(\left(-\varepsilon\right) + -1\right) \cdot x\right)} \cdot \left(\frac{1}{\varepsilon} - 1\right)}{2}\\ \end{array}\]
\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 30.43326508471591:\\
\;\;\;\;\frac{\left(2 - x \cdot x\right) + \left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{2}{3}}{2}\\

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

\end{array}
double f(double x, double eps) {
        double r2279203 = 1.0;
        double r2279204 = eps;
        double r2279205 = r2279203 / r2279204;
        double r2279206 = r2279203 + r2279205;
        double r2279207 = r2279203 - r2279204;
        double r2279208 = x;
        double r2279209 = r2279207 * r2279208;
        double r2279210 = -r2279209;
        double r2279211 = exp(r2279210);
        double r2279212 = r2279206 * r2279211;
        double r2279213 = r2279205 - r2279203;
        double r2279214 = r2279203 + r2279204;
        double r2279215 = r2279214 * r2279208;
        double r2279216 = -r2279215;
        double r2279217 = exp(r2279216);
        double r2279218 = r2279213 * r2279217;
        double r2279219 = r2279212 - r2279218;
        double r2279220 = 2.0;
        double r2279221 = r2279219 / r2279220;
        return r2279221;
}
↓
double f(double x, double eps) {
        double r2279222 = x;
        double r2279223 = 30.43326508471591;
        bool r2279224 = r2279222 <= r2279223;
        double r2279225 = 2.0;
        double r2279226 = r2279222 * r2279222;
        double r2279227 = r2279225 - r2279226;
        double r2279228 = r2279226 * r2279222;
        double r2279229 = 0.6666666666666666;
        double r2279230 = r2279228 * r2279229;
        double r2279231 = r2279227 + r2279230;
        double r2279232 = r2279231 / r2279225;
        double r2279233 = 1.0;
        double r2279234 = eps;
        double r2279235 = r2279233 / r2279234;
        double r2279236 = r2279235 + r2279233;
        double r2279237 = -r2279222;
        double r2279238 = r2279233 - r2279234;
        double r2279239 = r2279237 * r2279238;
        double r2279240 = exp(r2279239);
        double r2279241 = r2279236 * r2279240;
        double r2279242 = exp(1.0);
        double r2279243 = -r2279234;
        double r2279244 = -1.0;
        double r2279245 = r2279243 + r2279244;
        double r2279246 = r2279245 * r2279222;
        double r2279247 = pow(r2279242, r2279246);
        double r2279248 = r2279235 - r2279233;
        double r2279249 = r2279247 * r2279248;
        double r2279250 = r2279241 - r2279249;
        double r2279251 = r2279250 / r2279225;
        double r2279252 = r2279224 ? r2279232 : r2279251;
        return r2279252;
}

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 < 30.43326508471591

    1. Initial program 37.8

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

    2. Taylor expanded around 0 1.1

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

    3. Simplified1.1

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

    if 30.43326508471591 < x

    1. Initial program 0.3

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

    3. Applied *-un-lft-identity0.3

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

    4. Applied exp-prod0.3

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

    5. Simplified0.3

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

  4. Final simplification0.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 30.43326508471591:\\ \;\;\;\;\frac{\left(2 - x \cdot x\right) + \left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{2}{3}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{1}{\varepsilon} + 1\right) \cdot e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)} - {e}^{\left(\left(\left(-\varepsilon\right) + -1\right) \cdot x\right)} \cdot \left(\frac{1}{\varepsilon} - 1\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019165 
(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))