Average Error: 29.5 → 0.9
Time: 19.7s
Precision: 64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 41.6575809018513:\\ \;\;\;\;\frac{\left(2 - x \cdot x\right) + \left(x \cdot x\right) \cdot \left(\frac{2}{3} \cdot x\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(e^{\left(\varepsilon + -1\right) \cdot x} + \frac{e^{\left(\varepsilon + -1\right) \cdot x}}{\varepsilon}\right) - \left(\frac{e^{x \cdot \left(-1 - \varepsilon\right)}}{\varepsilon} - \left(\sqrt[3]{e^{x \cdot \left(-1 - \varepsilon\right)}} \cdot \sqrt[3]{e^{x \cdot \left(-1 - \varepsilon\right)}}\right) \cdot \sqrt[3]{e^{x \cdot \left(-1 - \varepsilon\right)}}\right)}{2}\\ \end{array}\]
\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 41.6575809018513:\\
\;\;\;\;\frac{\left(2 - x \cdot x\right) + \left(x \cdot x\right) \cdot \left(\frac{2}{3} \cdot x\right)}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\left(e^{\left(\varepsilon + -1\right) \cdot x} + \frac{e^{\left(\varepsilon + -1\right) \cdot x}}{\varepsilon}\right) - \left(\frac{e^{x \cdot \left(-1 - \varepsilon\right)}}{\varepsilon} - \left(\sqrt[3]{e^{x \cdot \left(-1 - \varepsilon\right)}} \cdot \sqrt[3]{e^{x \cdot \left(-1 - \varepsilon\right)}}\right) \cdot \sqrt[3]{e^{x \cdot \left(-1 - \varepsilon\right)}}\right)}{2}\\

\end{array}
double f(double x, double eps) {
        double r1075474 = 1.0;
        double r1075475 = eps;
        double r1075476 = r1075474 / r1075475;
        double r1075477 = r1075474 + r1075476;
        double r1075478 = r1075474 - r1075475;
        double r1075479 = x;
        double r1075480 = r1075478 * r1075479;
        double r1075481 = -r1075480;
        double r1075482 = exp(r1075481);
        double r1075483 = r1075477 * r1075482;
        double r1075484 = r1075476 - r1075474;
        double r1075485 = r1075474 + r1075475;
        double r1075486 = r1075485 * r1075479;
        double r1075487 = -r1075486;
        double r1075488 = exp(r1075487);
        double r1075489 = r1075484 * r1075488;
        double r1075490 = r1075483 - r1075489;
        double r1075491 = 2.0;
        double r1075492 = r1075490 / r1075491;
        return r1075492;
}
↓
double f(double x, double eps) {
        double r1075493 = x;
        double r1075494 = 41.6575809018513;
        bool r1075495 = r1075493 <= r1075494;
        double r1075496 = 2.0;
        double r1075497 = r1075493 * r1075493;
        double r1075498 = r1075496 - r1075497;
        double r1075499 = 0.6666666666666666;
        double r1075500 = r1075499 * r1075493;
        double r1075501 = r1075497 * r1075500;
        double r1075502 = r1075498 + r1075501;
        double r1075503 = r1075502 / r1075496;
        double r1075504 = eps;
        double r1075505 = -1.0;
        double r1075506 = r1075504 + r1075505;
        double r1075507 = r1075506 * r1075493;
        double r1075508 = exp(r1075507);
        double r1075509 = r1075508 / r1075504;
        double r1075510 = r1075508 + r1075509;
        double r1075511 = r1075505 - r1075504;
        double r1075512 = r1075493 * r1075511;
        double r1075513 = exp(r1075512);
        double r1075514 = r1075513 / r1075504;
        double r1075515 = cbrt(r1075513);
        double r1075516 = r1075515 * r1075515;
        double r1075517 = r1075516 * r1075515;
        double r1075518 = r1075514 - r1075517;
        double r1075519 = r1075510 - r1075518;
        double r1075520 = r1075519 / r1075496;
        double r1075521 = r1075495 ? r1075503 : r1075520;
        return r1075521;
}

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 < 41.6575809018513

    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. Simplified39.1

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

    3. Taylor expanded around 0 1.1

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

    4. Simplified1.1

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

    6. Applied *-commutative1.1

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

    if 41.6575809018513 < 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. Simplified0.1

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

    4. Applied add-cube-cbrt0.1

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

  4. Final simplification0.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 41.6575809018513:\\ \;\;\;\;\frac{\left(2 - x \cdot x\right) + \left(x \cdot x\right) \cdot \left(\frac{2}{3} \cdot x\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(e^{\left(\varepsilon + -1\right) \cdot x} + \frac{e^{\left(\varepsilon + -1\right) \cdot x}}{\varepsilon}\right) - \left(\frac{e^{x \cdot \left(-1 - \varepsilon\right)}}{\varepsilon} - \left(\sqrt[3]{e^{x \cdot \left(-1 - \varepsilon\right)}} \cdot \sqrt[3]{e^{x \cdot \left(-1 - \varepsilon\right)}}\right) \cdot \sqrt[3]{e^{x \cdot \left(-1 - \varepsilon\right)}}\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019156 
(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))