Average Error: 29.2 → 1.0
Time: 19.8s
Precision: 64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 162.5604579277316474872350227087736129761:\\ \;\;\;\;\frac{\left(x \cdot 0.6666666666666667406815349750104360282421\right) \cdot {\left(\sqrt[3]{x} \cdot \left(\sqrt[3]{\sqrt[3]{x}} \cdot \left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right)\right)\right)}^{3} + \left(2 - \left(x \cdot x\right) \cdot 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(-\left(1 - \varepsilon\right)\right)} \cdot \left(\frac{1}{\varepsilon} + 1\right) - \left(\sqrt[3]{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{x \cdot \left(-\left(1 + \varepsilon\right)\right)}} \cdot \sqrt[3]{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{x \cdot \left(-\left(1 + \varepsilon\right)\right)}}\right) \cdot \sqrt[3]{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{x \cdot \left(-\left(1 + \varepsilon\right)\right)}}}{2}\\ \end{array}\]
\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 162.5604579277316474872350227087736129761:\\
\;\;\;\;\frac{\left(x \cdot 0.6666666666666667406815349750104360282421\right) \cdot {\left(\sqrt[3]{x} \cdot \left(\sqrt[3]{\sqrt[3]{x}} \cdot \left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right)\right)\right)}^{3} + \left(2 - \left(x \cdot x\right) \cdot 1\right)}{2}\\

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

\end{array}
double f(double x, double eps) {
        double r37513 = 1.0;
        double r37514 = eps;
        double r37515 = r37513 / r37514;
        double r37516 = r37513 + r37515;
        double r37517 = r37513 - r37514;
        double r37518 = x;
        double r37519 = r37517 * r37518;
        double r37520 = -r37519;
        double r37521 = exp(r37520);
        double r37522 = r37516 * r37521;
        double r37523 = r37515 - r37513;
        double r37524 = r37513 + r37514;
        double r37525 = r37524 * r37518;
        double r37526 = -r37525;
        double r37527 = exp(r37526);
        double r37528 = r37523 * r37527;
        double r37529 = r37522 - r37528;
        double r37530 = 2.0;
        double r37531 = r37529 / r37530;
        return r37531;
}
↓
double f(double x, double eps) {
        double r37532 = x;
        double r37533 = 162.56045792773165;
        bool r37534 = r37532 <= r37533;
        double r37535 = 0.6666666666666667;
        double r37536 = r37532 * r37535;
        double r37537 = cbrt(r37532);
        double r37538 = cbrt(r37537);
        double r37539 = r37538 * r37538;
        double r37540 = r37538 * r37539;
        double r37541 = r37537 * r37540;
        double r37542 = 3.0;
        double r37543 = pow(r37541, r37542);
        double r37544 = r37536 * r37543;
        double r37545 = 2.0;
        double r37546 = r37532 * r37532;
        double r37547 = 1.0;
        double r37548 = r37546 * r37547;
        double r37549 = r37545 - r37548;
        double r37550 = r37544 + r37549;
        double r37551 = r37550 / r37545;
        double r37552 = eps;
        double r37553 = r37547 - r37552;
        double r37554 = -r37553;
        double r37555 = r37532 * r37554;
        double r37556 = exp(r37555);
        double r37557 = r37547 / r37552;
        double r37558 = r37557 + r37547;
        double r37559 = r37556 * r37558;
        double r37560 = r37557 - r37547;
        double r37561 = r37547 + r37552;
        double r37562 = -r37561;
        double r37563 = r37532 * r37562;
        double r37564 = exp(r37563);
        double r37565 = r37560 * r37564;
        double r37566 = cbrt(r37565);
        double r37567 = r37566 * r37566;
        double r37568 = r37567 * r37566;
        double r37569 = r37559 - r37568;
        double r37570 = r37569 / r37545;
        double r37571 = r37534 ? r37551 : r37570;
        return r37571;
}

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 < 162.56045792773165

    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 1.4

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

    3. Simplified1.4

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

    5. Applied add-cube-cbrt1.4

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

    6. Applied unpow-prod-down1.4

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

    7. Applied associate-*l*1.4

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

    8. Simplified1.4

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

    10. Applied add-cube-cbrt1.4

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

    if 162.56045792773165 < 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-cube-cbrt0.1

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

    4. Simplified0.1

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

    5. Simplified0.1

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

  4. Final simplification1.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 162.5604579277316474872350227087736129761:\\ \;\;\;\;\frac{\left(x \cdot 0.6666666666666667406815349750104360282421\right) \cdot {\left(\sqrt[3]{x} \cdot \left(\sqrt[3]{\sqrt[3]{x}} \cdot \left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right)\right)\right)}^{3} + \left(2 - \left(x \cdot x\right) \cdot 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(-\left(1 - \varepsilon\right)\right)} \cdot \left(\frac{1}{\varepsilon} + 1\right) - \left(\sqrt[3]{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{x \cdot \left(-\left(1 + \varepsilon\right)\right)}} \cdot \sqrt[3]{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{x \cdot \left(-\left(1 + \varepsilon\right)\right)}}\right) \cdot \sqrt[3]{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{x \cdot \left(-\left(1 + \varepsilon\right)\right)}}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019194 
(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))