Average Error: 28.9 → 1.8
Time: 26.1s
Precision: 64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 3.375282683015706291841696417845614769891 \cdot 10^{-23}:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt[3]{{\left(\sqrt[3]{x \cdot x}\right)}^{2}} \cdot \sqrt[3]{{\left(\sqrt[3]{x \cdot x}\right)}^{2}}\right) \cdot \sqrt[3]{{\left(\sqrt[3]{x \cdot x}\right)}^{2}}\right) \cdot \left(\left(\sqrt[3]{\sqrt[3]{{x}^{2}} \cdot \sqrt[3]{{x}^{2}}} \cdot \sqrt[3]{\sqrt[3]{x \cdot x}}\right) \cdot \left(x \cdot 0.6666666666666667406815349750104360282421 - 1\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{\frac{1}{\varepsilon} + 1}{e^{\left(1 - \varepsilon\right) \cdot x}} - \frac{\frac{1}{\varepsilon}}{e^{\left(1 + \varepsilon\right) \cdot x}}\right) + \frac{1}{e^{\left(1 + \varepsilon\right) \cdot x}}}{2}\\ \end{array}\]
\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 3.375282683015706291841696417845614769891 \cdot 10^{-23}:\\
\;\;\;\;\frac{2 + \left(\left(\sqrt[3]{{\left(\sqrt[3]{x \cdot x}\right)}^{2}} \cdot \sqrt[3]{{\left(\sqrt[3]{x \cdot x}\right)}^{2}}\right) \cdot \sqrt[3]{{\left(\sqrt[3]{x \cdot x}\right)}^{2}}\right) \cdot \left(\left(\sqrt[3]{\sqrt[3]{{x}^{2}} \cdot \sqrt[3]{{x}^{2}}} \cdot \sqrt[3]{\sqrt[3]{x \cdot x}}\right) \cdot \left(x \cdot 0.6666666666666667406815349750104360282421 - 1\right)\right)}{2}\\

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

\end{array}
double f(double x, double eps) {
        double r39003 = 1.0;
        double r39004 = eps;
        double r39005 = r39003 / r39004;
        double r39006 = r39003 + r39005;
        double r39007 = r39003 - r39004;
        double r39008 = x;
        double r39009 = r39007 * r39008;
        double r39010 = -r39009;
        double r39011 = exp(r39010);
        double r39012 = r39006 * r39011;
        double r39013 = r39005 - r39003;
        double r39014 = r39003 + r39004;
        double r39015 = r39014 * r39008;
        double r39016 = -r39015;
        double r39017 = exp(r39016);
        double r39018 = r39013 * r39017;
        double r39019 = r39012 - r39018;
        double r39020 = 2.0;
        double r39021 = r39019 / r39020;
        return r39021;
}
↓
double f(double x, double eps) {
        double r39022 = x;
        double r39023 = 3.375282683015706e-23;
        bool r39024 = r39022 <= r39023;
        double r39025 = 2.0;
        double r39026 = r39022 * r39022;
        double r39027 = cbrt(r39026);
        double r39028 = 2.0;
        double r39029 = pow(r39027, r39028);
        double r39030 = cbrt(r39029);
        double r39031 = r39030 * r39030;
        double r39032 = r39031 * r39030;
        double r39033 = pow(r39022, r39028);
        double r39034 = cbrt(r39033);
        double r39035 = r39034 * r39034;
        double r39036 = cbrt(r39035);
        double r39037 = cbrt(r39027);
        double r39038 = r39036 * r39037;
        double r39039 = 0.6666666666666667;
        double r39040 = r39022 * r39039;
        double r39041 = 1.0;
        double r39042 = r39040 - r39041;
        double r39043 = r39038 * r39042;
        double r39044 = r39032 * r39043;
        double r39045 = r39025 + r39044;
        double r39046 = r39045 / r39025;
        double r39047 = eps;
        double r39048 = r39041 / r39047;
        double r39049 = r39048 + r39041;
        double r39050 = r39041 - r39047;
        double r39051 = r39050 * r39022;
        double r39052 = exp(r39051);
        double r39053 = r39049 / r39052;
        double r39054 = r39041 + r39047;
        double r39055 = r39054 * r39022;
        double r39056 = exp(r39055);
        double r39057 = r39048 / r39056;
        double r39058 = r39053 - r39057;
        double r39059 = r39041 / r39056;
        double r39060 = r39058 + r39059;
        double r39061 = r39060 / r39025;
        double r39062 = r39024 ? r39046 : r39061;
        return r39062;
}

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 < 3.375282683015706e-23

    1. Initial program 38.2

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

    2. Simplified38.2

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

    3. Taylor expanded around 0 1.2

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

    4. Simplified1.2

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

    6. Applied add-cube-cbrt1.2

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

    7. Applied associate-*l*1.2

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

    9. Applied add-cube-cbrt1.2

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

    10. Simplified1.2

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

    11. Simplified1.2

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

    13. Applied add-cube-cbrt1.2

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

    14. Applied cbrt-prod1.2

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

    15. Simplified1.2

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

    if 3.375282683015706e-23 < x

    1. Initial program 4.1

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

    2. Simplified4.1

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

    4. Applied div-sub4.1

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

    5. Applied associate--r-3.5

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

  4. Final simplification1.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 3.375282683015706291841696417845614769891 \cdot 10^{-23}:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt[3]{{\left(\sqrt[3]{x \cdot x}\right)}^{2}} \cdot \sqrt[3]{{\left(\sqrt[3]{x \cdot x}\right)}^{2}}\right) \cdot \sqrt[3]{{\left(\sqrt[3]{x \cdot x}\right)}^{2}}\right) \cdot \left(\left(\sqrt[3]{\sqrt[3]{{x}^{2}} \cdot \sqrt[3]{{x}^{2}}} \cdot \sqrt[3]{\sqrt[3]{x \cdot x}}\right) \cdot \left(x \cdot 0.6666666666666667406815349750104360282421 - 1\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{\frac{1}{\varepsilon} + 1}{e^{\left(1 - \varepsilon\right) \cdot x}} - \frac{\frac{1}{\varepsilon}}{e^{\left(1 + \varepsilon\right) \cdot x}}\right) + \frac{1}{e^{\left(1 + \varepsilon\right) \cdot x}}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019322 
(FPCore (x eps)
  :name "NMSE Section 6.1 mentioned, A"
  :precision binary64
  (/ (- (* (+ 1 (/ 1 eps)) (exp (- (* (- 1 eps) x)))) (* (- (/ 1 eps) 1) (exp (- (* (+ 1 eps) x))))) 2))