Average Error: 30.1 → 1.1
Time: 30.7s
Precision: 64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 381.9625649006933940654562320560216903687:\\ \;\;\;\;\left(\left(\sqrt[3]{\sqrt[3]{\frac{\left(2 - \left(x \cdot x\right) \cdot 1\right) + \left(\left(x \cdot x\right) \cdot x\right) \cdot 0.6666666666666667406815349750104360282421}{2}} \cdot \sqrt[3]{\frac{\left(2 - \left(x \cdot x\right) \cdot 1\right) + \left(\left(x \cdot x\right) \cdot x\right) \cdot 0.6666666666666667406815349750104360282421}{2}}} \cdot \sqrt[3]{\sqrt[3]{\frac{\left(2 - \left(x \cdot x\right) \cdot 1\right) + \left(\left(x \cdot x\right) \cdot x\right) \cdot 0.6666666666666667406815349750104360282421}{2}}}\right) \cdot \sqrt[3]{\frac{\left(2 - \left(x \cdot x\right) \cdot 1\right) + \left(\left(x \cdot x\right) \cdot x\right) \cdot 0.6666666666666667406815349750104360282421}{2}}\right) \cdot \sqrt[3]{\frac{\left(2 - \left(x \cdot x\right) \cdot 1\right) + \left(\left(x \cdot x\right) \cdot x\right) \cdot 0.6666666666666667406815349750104360282421}{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(\varepsilon - 1\right) \cdot x} + \frac{1 - \frac{1}{\varepsilon}}{e^{\left(\varepsilon + 1\right) \cdot x}}}{2}\\ \end{array}\]
\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 381.9625649006933940654562320560216903687:\\
\;\;\;\;\left(\left(\sqrt[3]{\sqrt[3]{\frac{\left(2 - \left(x \cdot x\right) \cdot 1\right) + \left(\left(x \cdot x\right) \cdot x\right) \cdot 0.6666666666666667406815349750104360282421}{2}} \cdot \sqrt[3]{\frac{\left(2 - \left(x \cdot x\right) \cdot 1\right) + \left(\left(x \cdot x\right) \cdot x\right) \cdot 0.6666666666666667406815349750104360282421}{2}}} \cdot \sqrt[3]{\sqrt[3]{\frac{\left(2 - \left(x \cdot x\right) \cdot 1\right) + \left(\left(x \cdot x\right) \cdot x\right) \cdot 0.6666666666666667406815349750104360282421}{2}}}\right) \cdot \sqrt[3]{\frac{\left(2 - \left(x \cdot x\right) \cdot 1\right) + \left(\left(x \cdot x\right) \cdot x\right) \cdot 0.6666666666666667406815349750104360282421}{2}}\right) \cdot \sqrt[3]{\frac{\left(2 - \left(x \cdot x\right) \cdot 1\right) + \left(\left(x \cdot x\right) \cdot x\right) \cdot 0.6666666666666667406815349750104360282421}{2}}\\

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

\end{array}
double f(double x, double eps) {
        double r3060431 = 1.0;
        double r3060432 = eps;
        double r3060433 = r3060431 / r3060432;
        double r3060434 = r3060431 + r3060433;
        double r3060435 = r3060431 - r3060432;
        double r3060436 = x;
        double r3060437 = r3060435 * r3060436;
        double r3060438 = -r3060437;
        double r3060439 = exp(r3060438);
        double r3060440 = r3060434 * r3060439;
        double r3060441 = r3060433 - r3060431;
        double r3060442 = r3060431 + r3060432;
        double r3060443 = r3060442 * r3060436;
        double r3060444 = -r3060443;
        double r3060445 = exp(r3060444);
        double r3060446 = r3060441 * r3060445;
        double r3060447 = r3060440 - r3060446;
        double r3060448 = 2.0;
        double r3060449 = r3060447 / r3060448;
        return r3060449;
}
↓
double f(double x, double eps) {
        double r3060450 = x;
        double r3060451 = 381.9625649006934;
        bool r3060452 = r3060450 <= r3060451;
        double r3060453 = 2.0;
        double r3060454 = r3060450 * r3060450;
        double r3060455 = 1.0;
        double r3060456 = r3060454 * r3060455;
        double r3060457 = r3060453 - r3060456;
        double r3060458 = r3060454 * r3060450;
        double r3060459 = 0.6666666666666667;
        double r3060460 = r3060458 * r3060459;
        double r3060461 = r3060457 + r3060460;
        double r3060462 = r3060461 / r3060453;
        double r3060463 = cbrt(r3060462);
        double r3060464 = r3060463 * r3060463;
        double r3060465 = cbrt(r3060464);
        double r3060466 = cbrt(r3060463);
        double r3060467 = r3060465 * r3060466;
        double r3060468 = r3060467 * r3060463;
        double r3060469 = r3060468 * r3060463;
        double r3060470 = eps;
        double r3060471 = r3060455 / r3060470;
        double r3060472 = r3060455 + r3060471;
        double r3060473 = r3060470 - r3060455;
        double r3060474 = r3060473 * r3060450;
        double r3060475 = exp(r3060474);
        double r3060476 = r3060472 * r3060475;
        double r3060477 = r3060455 - r3060471;
        double r3060478 = r3060470 + r3060455;
        double r3060479 = r3060478 * r3060450;
        double r3060480 = exp(r3060479);
        double r3060481 = r3060477 / r3060480;
        double r3060482 = r3060476 + r3060481;
        double r3060483 = r3060482 / r3060453;
        double r3060484 = r3060452 ? r3060469 : r3060483;
        return r3060484;
}

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 < 381.9625649006934

    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. Taylor expanded around 0 1.3

      \[\leadsto \frac{\color{blue}{\left(0.6666666666666667406815349750104360282421 \cdot {x}^{3} + 2\right) - 1 \cdot {x}^{2}}}{2}\]

    3. Simplified1.3

      \[\leadsto \frac{\color{blue}{\left(2 - \left(x \cdot x\right) \cdot 1\right) + 0.6666666666666667406815349750104360282421 \cdot \left(\left(x \cdot x\right) \cdot x\right)}}{2}\]
    4. Using strategy rm

    5. Applied add-cube-cbrt1.4

      \[\leadsto \color{blue}{\left(\sqrt[3]{\frac{\left(2 - \left(x \cdot x\right) \cdot 1\right) + 0.6666666666666667406815349750104360282421 \cdot \left(\left(x \cdot x\right) \cdot x\right)}{2}} \cdot \sqrt[3]{\frac{\left(2 - \left(x \cdot x\right) \cdot 1\right) + 0.6666666666666667406815349750104360282421 \cdot \left(\left(x \cdot x\right) \cdot x\right)}{2}}\right) \cdot \sqrt[3]{\frac{\left(2 - \left(x \cdot x\right) \cdot 1\right) + 0.6666666666666667406815349750104360282421 \cdot \left(\left(x \cdot x\right) \cdot x\right)}{2}}}\]
    6. Using strategy rm

    7. Applied add-cube-cbrt1.4

      \[\leadsto \left(\sqrt[3]{\color{blue}{\left(\sqrt[3]{\frac{\left(2 - \left(x \cdot x\right) \cdot 1\right) + 0.6666666666666667406815349750104360282421 \cdot \left(\left(x \cdot x\right) \cdot x\right)}{2}} \cdot \sqrt[3]{\frac{\left(2 - \left(x \cdot x\right) \cdot 1\right) + 0.6666666666666667406815349750104360282421 \cdot \left(\left(x \cdot x\right) \cdot x\right)}{2}}\right) \cdot \sqrt[3]{\frac{\left(2 - \left(x \cdot x\right) \cdot 1\right) + 0.6666666666666667406815349750104360282421 \cdot \left(\left(x \cdot x\right) \cdot x\right)}{2}}}} \cdot \sqrt[3]{\frac{\left(2 - \left(x \cdot x\right) \cdot 1\right) + 0.6666666666666667406815349750104360282421 \cdot \left(\left(x \cdot x\right) \cdot x\right)}{2}}\right) \cdot \sqrt[3]{\frac{\left(2 - \left(x \cdot x\right) \cdot 1\right) + 0.6666666666666667406815349750104360282421 \cdot \left(\left(x \cdot x\right) \cdot x\right)}{2}}\]

    8. Applied cbrt-prod1.4

      \[\leadsto \left(\color{blue}{\left(\sqrt[3]{\sqrt[3]{\frac{\left(2 - \left(x \cdot x\right) \cdot 1\right) + 0.6666666666666667406815349750104360282421 \cdot \left(\left(x \cdot x\right) \cdot x\right)}{2}} \cdot \sqrt[3]{\frac{\left(2 - \left(x \cdot x\right) \cdot 1\right) + 0.6666666666666667406815349750104360282421 \cdot \left(\left(x \cdot x\right) \cdot x\right)}{2}}} \cdot \sqrt[3]{\sqrt[3]{\frac{\left(2 - \left(x \cdot x\right) \cdot 1\right) + 0.6666666666666667406815349750104360282421 \cdot \left(\left(x \cdot x\right) \cdot x\right)}{2}}}\right)} \cdot \sqrt[3]{\frac{\left(2 - \left(x \cdot x\right) \cdot 1\right) + 0.6666666666666667406815349750104360282421 \cdot \left(\left(x \cdot x\right) \cdot x\right)}{2}}\right) \cdot \sqrt[3]{\frac{\left(2 - \left(x \cdot x\right) \cdot 1\right) + 0.6666666666666667406815349750104360282421 \cdot \left(\left(x \cdot x\right) \cdot x\right)}{2}}\]

    if 381.9625649006934 < 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

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

  4. Final simplification1.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 381.9625649006933940654562320560216903687:\\ \;\;\;\;\left(\left(\sqrt[3]{\sqrt[3]{\frac{\left(2 - \left(x \cdot x\right) \cdot 1\right) + \left(\left(x \cdot x\right) \cdot x\right) \cdot 0.6666666666666667406815349750104360282421}{2}} \cdot \sqrt[3]{\frac{\left(2 - \left(x \cdot x\right) \cdot 1\right) + \left(\left(x \cdot x\right) \cdot x\right) \cdot 0.6666666666666667406815349750104360282421}{2}}} \cdot \sqrt[3]{\sqrt[3]{\frac{\left(2 - \left(x \cdot x\right) \cdot 1\right) + \left(\left(x \cdot x\right) \cdot x\right) \cdot 0.6666666666666667406815349750104360282421}{2}}}\right) \cdot \sqrt[3]{\frac{\left(2 - \left(x \cdot x\right) \cdot 1\right) + \left(\left(x \cdot x\right) \cdot x\right) \cdot 0.6666666666666667406815349750104360282421}{2}}\right) \cdot \sqrt[3]{\frac{\left(2 - \left(x \cdot x\right) \cdot 1\right) + \left(\left(x \cdot x\right) \cdot x\right) \cdot 0.6666666666666667406815349750104360282421}{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(\varepsilon - 1\right) \cdot x} + \frac{1 - \frac{1}{\varepsilon}}{e^{\left(\varepsilon + 1\right) \cdot x}}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 
(FPCore (x eps)
  :name "NMSE Section 6.1 mentioned, A"
  (/ (- (* (+ 1.0 (/ 1.0 eps)) (exp (- (* (- 1.0 eps) x)))) (* (- (/ 1.0 eps) 1.0) (exp (- (* (+ 1.0 eps) x))))) 2.0))