Average Error: 29.5 → 1.0
Time: 28.6s
Precision: 64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 226.33586080847886:\\ \;\;\;\;\frac{\left(2 - x \cdot x\right) + \frac{2}{3} \cdot \left(\left(x \cdot x\right) \cdot x\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{1}{\varepsilon} + 1\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\sqrt{e^{\left(-x\right) \cdot \left(\varepsilon + 1\right)}} \cdot \sqrt{e^{\left(-x\right) \cdot \left(\varepsilon + 1\right)}}\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)}{2}\\ \end{array}\]
\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 226.33586080847886:\\
\;\;\;\;\frac{\left(2 - x \cdot x\right) + \frac{2}{3} \cdot \left(\left(x \cdot x\right) \cdot x\right)}{2}\\

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

\end{array}
double f(double x, double eps) {
        double r1497475 = 1.0;
        double r1497476 = eps;
        double r1497477 = r1497475 / r1497476;
        double r1497478 = r1497475 + r1497477;
        double r1497479 = r1497475 - r1497476;
        double r1497480 = x;
        double r1497481 = r1497479 * r1497480;
        double r1497482 = -r1497481;
        double r1497483 = exp(r1497482);
        double r1497484 = r1497478 * r1497483;
        double r1497485 = r1497477 - r1497475;
        double r1497486 = r1497475 + r1497476;
        double r1497487 = r1497486 * r1497480;
        double r1497488 = -r1497487;
        double r1497489 = exp(r1497488);
        double r1497490 = r1497485 * r1497489;
        double r1497491 = r1497484 - r1497490;
        double r1497492 = 2.0;
        double r1497493 = r1497491 / r1497492;
        return r1497493;
}
↓
double f(double x, double eps) {
        double r1497494 = x;
        double r1497495 = 226.33586080847886;
        bool r1497496 = r1497494 <= r1497495;
        double r1497497 = 2.0;
        double r1497498 = r1497494 * r1497494;
        double r1497499 = r1497497 - r1497498;
        double r1497500 = 0.6666666666666666;
        double r1497501 = r1497498 * r1497494;
        double r1497502 = r1497500 * r1497501;
        double r1497503 = r1497499 + r1497502;
        double r1497504 = r1497503 / r1497497;
        double r1497505 = 1.0;
        double r1497506 = eps;
        double r1497507 = r1497505 / r1497506;
        double r1497508 = r1497507 + r1497505;
        double r1497509 = r1497505 - r1497506;
        double r1497510 = -r1497494;
        double r1497511 = r1497509 * r1497510;
        double r1497512 = exp(r1497511);
        double r1497513 = r1497508 * r1497512;
        double r1497514 = r1497506 + r1497505;
        double r1497515 = r1497510 * r1497514;
        double r1497516 = exp(r1497515);
        double r1497517 = sqrt(r1497516);
        double r1497518 = r1497517 * r1497517;
        double r1497519 = r1497507 - r1497505;
        double r1497520 = r1497518 * r1497519;
        double r1497521 = r1497513 - r1497520;
        double r1497522 = r1497521 / r1497497;
        double r1497523 = r1497496 ? r1497504 : r1497522;
        return r1497523;
}

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 < 226.33586080847886

    1. Initial program 39.1

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

    2. Taylor expanded around 0 1.3

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

    3. Simplified1.3

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

    if 226.33586080847886 < x

    1. Initial program 0.1

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

    3. Applied add-sqr-sqrt0.1

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

  4. Final simplification1.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 226.33586080847886:\\ \;\;\;\;\frac{\left(2 - x \cdot x\right) + \frac{2}{3} \cdot \left(\left(x \cdot x\right) \cdot x\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{1}{\varepsilon} + 1\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\sqrt{e^{\left(-x\right) \cdot \left(\varepsilon + 1\right)}} \cdot \sqrt{e^{\left(-x\right) \cdot \left(\varepsilon + 1\right)}}\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019143 
(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))