Average Error: 29.3 → 1.2
Time: 58.6s
Precision: 64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 1.9449410912425849:\\ \;\;\;\;\frac{\left(2 - x \cdot x\right) - \frac{-2}{3} \cdot \left(\left(x \cdot x\right) \cdot x\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(-1 - \varepsilon\right)} + \left(\left(e^{\left(-1 + \varepsilon\right) \cdot x} + \frac{e^{\left(-1 + \varepsilon\right) \cdot x}}{\varepsilon}\right) - \frac{e^{x \cdot \left(-1 - \varepsilon\right)}}{\varepsilon}\right)}{2}\\ \end{array}\]
\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 1.9449410912425849:\\
\;\;\;\;\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 r4816367 = 1.0;
        double r4816368 = eps;
        double r4816369 = r4816367 / r4816368;
        double r4816370 = r4816367 + r4816369;
        double r4816371 = r4816367 - r4816368;
        double r4816372 = x;
        double r4816373 = r4816371 * r4816372;
        double r4816374 = -r4816373;
        double r4816375 = exp(r4816374);
        double r4816376 = r4816370 * r4816375;
        double r4816377 = r4816369 - r4816367;
        double r4816378 = r4816367 + r4816368;
        double r4816379 = r4816378 * r4816372;
        double r4816380 = -r4816379;
        double r4816381 = exp(r4816380);
        double r4816382 = r4816377 * r4816381;
        double r4816383 = r4816376 - r4816382;
        double r4816384 = 2.0;
        double r4816385 = r4816383 / r4816384;
        return r4816385;
}
↓
double f(double x, double eps) {
        double r4816386 = x;
        double r4816387 = 1.9449410912425849;
        bool r4816388 = r4816386 <= r4816387;
        double r4816389 = 2.0;
        double r4816390 = r4816386 * r4816386;
        double r4816391 = r4816389 - r4816390;
        double r4816392 = -0.6666666666666666;
        double r4816393 = r4816390 * r4816386;
        double r4816394 = r4816392 * r4816393;
        double r4816395 = r4816391 - r4816394;
        double r4816396 = r4816395 / r4816389;
        double r4816397 = -1.0;
        double r4816398 = eps;
        double r4816399 = r4816397 - r4816398;
        double r4816400 = r4816386 * r4816399;
        double r4816401 = exp(r4816400);
        double r4816402 = r4816397 + r4816398;
        double r4816403 = r4816402 * r4816386;
        double r4816404 = exp(r4816403);
        double r4816405 = r4816404 / r4816398;
        double r4816406 = r4816404 + r4816405;
        double r4816407 = r4816401 / r4816398;
        double r4816408 = r4816406 - r4816407;
        double r4816409 = r4816401 + r4816408;
        double r4816410 = r4816409 / r4816389;
        double r4816411 = r4816388 ? r4816396 : r4816410;
        return r4816411;
}

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 < 1.9449410912425849

    1. Initial program 38.7

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

    2. Simplified38.7

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

    3. Taylor expanded around 0 1.4

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

    4. Simplified1.4

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

    if 1.9449410912425849 < 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 simplification1.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 1.9449410912425849:\\ \;\;\;\;\frac{\left(2 - x \cdot x\right) - \frac{-2}{3} \cdot \left(\left(x \cdot x\right) \cdot x\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(-1 - \varepsilon\right)} + \left(\left(e^{\left(-1 + \varepsilon\right) \cdot x} + \frac{e^{\left(-1 + \varepsilon\right) \cdot x}}{\varepsilon}\right) - \frac{e^{x \cdot \left(-1 - \varepsilon\right)}}{\varepsilon}\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019107 
(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))