Average Error: 29.6 → 1.0
Time: 56.5s
Precision: 64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 1.7923582177702537:\\ \;\;\;\;\frac{x \cdot \left(\left(\frac{2}{3} \cdot x\right) \cdot x - x\right) + 2}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\left(-x\right) \cdot \varepsilon - x} - \left(\frac{e^{\left(-x\right) \cdot \varepsilon - x}}{\varepsilon} - \left(e^{x \cdot \left(-1 + \varepsilon\right)} + \frac{e^{x \cdot \left(-1 + \varepsilon\right)}}{\varepsilon}\right)\right)}{2}\\ \end{array}\]
\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 1.7923582177702537:\\
\;\;\;\;\frac{x \cdot \left(\left(\frac{2}{3} \cdot x\right) \cdot x - x\right) + 2}{2}\\

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

\end{array}
double f(double x, double eps) {
        double r4786474 = 1.0;
        double r4786475 = eps;
        double r4786476 = r4786474 / r4786475;
        double r4786477 = r4786474 + r4786476;
        double r4786478 = r4786474 - r4786475;
        double r4786479 = x;
        double r4786480 = r4786478 * r4786479;
        double r4786481 = -r4786480;
        double r4786482 = exp(r4786481);
        double r4786483 = r4786477 * r4786482;
        double r4786484 = r4786476 - r4786474;
        double r4786485 = r4786474 + r4786475;
        double r4786486 = r4786485 * r4786479;
        double r4786487 = -r4786486;
        double r4786488 = exp(r4786487);
        double r4786489 = r4786484 * r4786488;
        double r4786490 = r4786483 - r4786489;
        double r4786491 = 2.0;
        double r4786492 = r4786490 / r4786491;
        return r4786492;
}
↓
double f(double x, double eps) {
        double r4786493 = x;
        double r4786494 = 1.7923582177702537;
        bool r4786495 = r4786493 <= r4786494;
        double r4786496 = 0.6666666666666666;
        double r4786497 = r4786496 * r4786493;
        double r4786498 = r4786497 * r4786493;
        double r4786499 = r4786498 - r4786493;
        double r4786500 = r4786493 * r4786499;
        double r4786501 = 2.0;
        double r4786502 = r4786500 + r4786501;
        double r4786503 = r4786502 / r4786501;
        double r4786504 = -r4786493;
        double r4786505 = eps;
        double r4786506 = r4786504 * r4786505;
        double r4786507 = r4786506 - r4786493;
        double r4786508 = exp(r4786507);
        double r4786509 = r4786508 / r4786505;
        double r4786510 = -1.0;
        double r4786511 = r4786510 + r4786505;
        double r4786512 = r4786493 * r4786511;
        double r4786513 = exp(r4786512);
        double r4786514 = r4786513 / r4786505;
        double r4786515 = r4786513 + r4786514;
        double r4786516 = r4786509 - r4786515;
        double r4786517 = r4786508 - r4786516;
        double r4786518 = r4786517 / r4786501;
        double r4786519 = r4786495 ? r4786503 : r4786518;
        return r4786519;
}

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 < 1.7923582177702537

    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. Simplified39.3

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

    3. Taylor expanded around 0 1.2

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

    4. Simplified1.2

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

    if 1.7923582177702537 < 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. Simplified0.4

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

    3. Taylor expanded around -inf 0.4

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

    4. Simplified0.3

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

  4. Final simplification1.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 1.7923582177702537:\\ \;\;\;\;\frac{x \cdot \left(\left(\frac{2}{3} \cdot x\right) \cdot x - x\right) + 2}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\left(-x\right) \cdot \varepsilon - x} - \left(\frac{e^{\left(-x\right) \cdot \varepsilon - x}}{\varepsilon} - \left(e^{x \cdot \left(-1 + \varepsilon\right)} + \frac{e^{x \cdot \left(-1 + \varepsilon\right)}}{\varepsilon}\right)\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019124 
(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))