Average Error: 29.3 → 0.9
Time: 16.1s
Precision: 64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 33.941483957658718:\\ \;\;\;\;\frac{\left(\left(\left(0.66666666666666674 \cdot x\right) \cdot x\right) \cdot x + 2\right) - 1 \cdot {x}^{2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot {\left(e^{1 + \varepsilon}\right)}^{\left(-x\right)}}{2}\\ \end{array}\]
\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 33.941483957658718:\\
\;\;\;\;\frac{\left(\left(\left(0.66666666666666674 \cdot x\right) \cdot x\right) \cdot x + 2\right) - 1 \cdot {x}^{2}}{2}\\

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

\end{array}
double f(double x, double eps) {
        double r41252 = 1.0;
        double r41253 = eps;
        double r41254 = r41252 / r41253;
        double r41255 = r41252 + r41254;
        double r41256 = r41252 - r41253;
        double r41257 = x;
        double r41258 = r41256 * r41257;
        double r41259 = -r41258;
        double r41260 = exp(r41259);
        double r41261 = r41255 * r41260;
        double r41262 = r41254 - r41252;
        double r41263 = r41252 + r41253;
        double r41264 = r41263 * r41257;
        double r41265 = -r41264;
        double r41266 = exp(r41265);
        double r41267 = r41262 * r41266;
        double r41268 = r41261 - r41267;
        double r41269 = 2.0;
        double r41270 = r41268 / r41269;
        return r41270;
}
↓
double f(double x, double eps) {
        double r41271 = x;
        double r41272 = 33.94148395765872;
        bool r41273 = r41271 <= r41272;
        double r41274 = 0.6666666666666667;
        double r41275 = r41274 * r41271;
        double r41276 = r41275 * r41271;
        double r41277 = r41276 * r41271;
        double r41278 = 2.0;
        double r41279 = r41277 + r41278;
        double r41280 = 1.0;
        double r41281 = 2.0;
        double r41282 = pow(r41271, r41281);
        double r41283 = r41280 * r41282;
        double r41284 = r41279 - r41283;
        double r41285 = r41284 / r41278;
        double r41286 = eps;
        double r41287 = r41280 / r41286;
        double r41288 = r41280 + r41287;
        double r41289 = r41280 - r41286;
        double r41290 = r41289 * r41271;
        double r41291 = -r41290;
        double r41292 = exp(r41291);
        double r41293 = r41288 * r41292;
        double r41294 = r41287 - r41280;
        double r41295 = r41280 + r41286;
        double r41296 = exp(r41295);
        double r41297 = -r41271;
        double r41298 = pow(r41296, r41297);
        double r41299 = r41294 * r41298;
        double r41300 = r41293 - r41299;
        double r41301 = r41300 / r41278;
        double r41302 = r41273 ? r41285 : r41301;
        return r41302;
}

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 < 33.94148395765872

    1. Initial program 39.3

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

    2. Taylor expanded around 0 1.1

      \[\leadsto \frac{\color{blue}{\left(0.66666666666666674 \cdot {x}^{3} + 2\right) - 1 \cdot {x}^{2}}}{2}\]
    3. Using strategy rm

    4. Applied unpow31.1

      \[\leadsto \frac{\left(0.66666666666666674 \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot x\right)} + 2\right) - 1 \cdot {x}^{2}}{2}\]

    5. Applied associate-*r*1.1

      \[\leadsto \frac{\left(\color{blue}{\left(0.66666666666666674 \cdot \left(x \cdot x\right)\right) \cdot x} + 2\right) - 1 \cdot {x}^{2}}{2}\]

    6. Simplified1.1

      \[\leadsto \frac{\left(\color{blue}{\left(\left(0.66666666666666674 \cdot x\right) \cdot x\right)} \cdot x + 2\right) - 1 \cdot {x}^{2}}{2}\]

    if 33.94148395765872 < x

    1. Initial program 0.2

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

    3. Applied distribute-rgt-neg-in0.2

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

    4. Applied exp-prod0.2

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

  4. Final simplification0.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 33.941483957658718:\\ \;\;\;\;\frac{\left(\left(\left(0.66666666666666674 \cdot x\right) \cdot x\right) \cdot x + 2\right) - 1 \cdot {x}^{2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot {\left(e^{1 + \varepsilon}\right)}^{\left(-x\right)}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020042 
(FPCore (x eps)
  :name "NMSE Section 6.1 mentioned, A"
  :precision binary64
  (/ (- (* (+ 1 (/ 1 eps)) (exp (- (* (- 1 eps) x)))) (* (- (/ 1 eps) 1) (exp (- (* (+ 1 eps) x))))) 2))