Average Error: 29.5 → 1.1
Time: 30.5s
Precision: 64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 1.9334006562055708:\\ \;\;\;\;\frac{\left(2 + \left(\frac{2}{3} \cdot x\right) \cdot \left(x \cdot x\right)\right) - x \cdot x}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(e^{\left(-1 + \varepsilon\right) \cdot x} - \frac{e^{x \cdot \left(-1 - \varepsilon\right)}}{\varepsilon}\right) + \frac{e^{\left(-1 + \varepsilon\right) \cdot x}}{\varepsilon}\right) + e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\\ \end{array}\]
\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 1.9334006562055708:\\
\;\;\;\;\frac{\left(2 + \left(\frac{2}{3} \cdot x\right) \cdot \left(x \cdot x\right)\right) - x \cdot x}{2}\\

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

\end{array}
double f(double x, double eps) {
        double r1826441 = 1.0;
        double r1826442 = eps;
        double r1826443 = r1826441 / r1826442;
        double r1826444 = r1826441 + r1826443;
        double r1826445 = r1826441 - r1826442;
        double r1826446 = x;
        double r1826447 = r1826445 * r1826446;
        double r1826448 = -r1826447;
        double r1826449 = exp(r1826448);
        double r1826450 = r1826444 * r1826449;
        double r1826451 = r1826443 - r1826441;
        double r1826452 = r1826441 + r1826442;
        double r1826453 = r1826452 * r1826446;
        double r1826454 = -r1826453;
        double r1826455 = exp(r1826454);
        double r1826456 = r1826451 * r1826455;
        double r1826457 = r1826450 - r1826456;
        double r1826458 = 2.0;
        double r1826459 = r1826457 / r1826458;
        return r1826459;
}
↓
double f(double x, double eps) {
        double r1826460 = x;
        double r1826461 = 1.9334006562055708;
        bool r1826462 = r1826460 <= r1826461;
        double r1826463 = 2.0;
        double r1826464 = 0.6666666666666666;
        double r1826465 = r1826464 * r1826460;
        double r1826466 = r1826460 * r1826460;
        double r1826467 = r1826465 * r1826466;
        double r1826468 = r1826463 + r1826467;
        double r1826469 = r1826468 - r1826466;
        double r1826470 = r1826469 / r1826463;
        double r1826471 = -1.0;
        double r1826472 = eps;
        double r1826473 = r1826471 + r1826472;
        double r1826474 = r1826473 * r1826460;
        double r1826475 = exp(r1826474);
        double r1826476 = r1826471 - r1826472;
        double r1826477 = r1826460 * r1826476;
        double r1826478 = exp(r1826477);
        double r1826479 = r1826478 / r1826472;
        double r1826480 = r1826475 - r1826479;
        double r1826481 = r1826475 / r1826472;
        double r1826482 = r1826480 + r1826481;
        double r1826483 = r1826482 + r1826478;
        double r1826484 = r1826483 / r1826463;
        double r1826485 = r1826462 ? r1826470 : r1826484;
        return r1826485;
}

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 < 1.9334006562055708

    1. Initial program 39.0

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

    2. Simplified39.0

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

    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

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

    if 1.9334006562055708 < x

    1. Initial program 0.6

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

    2. Simplified0.6

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

    4. Applied associate-+r+0.5

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

  4. Final simplification1.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 1.9334006562055708:\\ \;\;\;\;\frac{\left(2 + \left(\frac{2}{3} \cdot x\right) \cdot \left(x \cdot x\right)\right) - x \cdot x}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(e^{\left(-1 + \varepsilon\right) \cdot x} - \frac{e^{x \cdot \left(-1 - \varepsilon\right)}}{\varepsilon}\right) + \frac{e^{\left(-1 + \varepsilon\right) \cdot x}}{\varepsilon}\right) + e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019163 
(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))