Average Error: 29.2 → 1.0
Time: 19.0s
Precision: 64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 162.5604579277316474872350227087736129761:\\ \;\;\;\;\frac{\left(x \cdot 0.6666666666666667406815349750104360282421\right) \cdot {\left(\sqrt[3]{x} \cdot \left(\sqrt[3]{\sqrt[3]{x}} \cdot \left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right)\right)\right)}^{3} + \left(2 - \left(x \cdot x\right) \cdot 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\left(\varepsilon - 1\right) \cdot x} \cdot \left(\frac{1}{\varepsilon} + 1\right) - e^{\left(-x\right) \cdot \left(1 + \varepsilon\right)} \cdot \left(\frac{1}{\varepsilon} - 1\right)}{2}\\ \end{array}\]
\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 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^{\left(\varepsilon - 1\right) \cdot x} \cdot \left(\frac{1}{\varepsilon} + 1\right) - e^{\left(-x\right) \cdot \left(1 + \varepsilon\right)} \cdot \left(\frac{1}{\varepsilon} - 1\right)}{2}\\

\end{array}
double f(double x, double eps) {
        double r45518 = 1.0;
        double r45519 = eps;
        double r45520 = r45518 / r45519;
        double r45521 = r45518 + r45520;
        double r45522 = r45518 - r45519;
        double r45523 = x;
        double r45524 = r45522 * r45523;
        double r45525 = -r45524;
        double r45526 = exp(r45525);
        double r45527 = r45521 * r45526;
        double r45528 = r45520 - r45518;
        double r45529 = r45518 + r45519;
        double r45530 = r45529 * r45523;
        double r45531 = -r45530;
        double r45532 = exp(r45531);
        double r45533 = r45528 * r45532;
        double r45534 = r45527 - r45533;
        double r45535 = 2.0;
        double r45536 = r45534 / r45535;
        return r45536;
}
↓
double f(double x, double eps) {
        double r45537 = x;
        double r45538 = 162.56045792773165;
        bool r45539 = r45537 <= r45538;
        double r45540 = 0.6666666666666667;
        double r45541 = r45537 * r45540;
        double r45542 = cbrt(r45537);
        double r45543 = cbrt(r45542);
        double r45544 = r45543 * r45543;
        double r45545 = r45543 * r45544;
        double r45546 = r45542 * r45545;
        double r45547 = 3.0;
        double r45548 = pow(r45546, r45547);
        double r45549 = r45541 * r45548;
        double r45550 = 2.0;
        double r45551 = r45537 * r45537;
        double r45552 = 1.0;
        double r45553 = r45551 * r45552;
        double r45554 = r45550 - r45553;
        double r45555 = r45549 + r45554;
        double r45556 = r45555 / r45550;
        double r45557 = eps;
        double r45558 = r45557 - r45552;
        double r45559 = r45558 * r45537;
        double r45560 = exp(r45559);
        double r45561 = r45552 / r45557;
        double r45562 = r45561 + r45552;
        double r45563 = r45560 * r45562;
        double r45564 = -r45537;
        double r45565 = r45552 + r45557;
        double r45566 = r45564 * r45565;
        double r45567 = exp(r45566);
        double r45568 = r45561 - r45552;
        double r45569 = r45567 * r45568;
        double r45570 = r45563 - r45569;
        double r45571 = r45570 / r45550;
        double r45572 = r45539 ? r45556 : r45571;
        return r45572;
}

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. Taylor expanded around inf 0.1

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

    3. Simplified0.1

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

  4. Final simplification1.0

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

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))