Average Error: 29.8 → 1.2
Time: 7.3s
Precision: 64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 2.2371703346309433:\\ \;\;\;\;\frac{\left(\left(1 - 0.5 \cdot {x}^{2}\right) + 0.33333333333333337 \cdot {x}^{3}\right) \cdot \left(\left(1 + 0.5 \cdot {x}^{2}\right) + 0.33333333333333337 \cdot {x}^{3}\right)}{\left(1 + 0.5 \cdot {x}^{2}\right) + 0.33333333333333337 \cdot {x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{1 + \frac{1}{\varepsilon}}{e^{\left(1 - \varepsilon\right) \cdot x}}}{2} - \frac{\frac{\frac{1}{\varepsilon}}{e^{\left(1 + \varepsilon\right) \cdot x}}}{2}\right) + \frac{\frac{1}{e^{\left(1 + \varepsilon\right) \cdot x}}}{2}\\ \end{array}\]
\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 2.2371703346309433:\\
\;\;\;\;\frac{\left(\left(1 - 0.5 \cdot {x}^{2}\right) + 0.33333333333333337 \cdot {x}^{3}\right) \cdot \left(\left(1 + 0.5 \cdot {x}^{2}\right) + 0.33333333333333337 \cdot {x}^{3}\right)}{\left(1 + 0.5 \cdot {x}^{2}\right) + 0.33333333333333337 \cdot {x}^{3}}\\

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

\end{array}
double f(double x, double eps) {
        double r43960 = 1.0;
        double r43961 = eps;
        double r43962 = r43960 / r43961;
        double r43963 = r43960 + r43962;
        double r43964 = r43960 - r43961;
        double r43965 = x;
        double r43966 = r43964 * r43965;
        double r43967 = -r43966;
        double r43968 = exp(r43967);
        double r43969 = r43963 * r43968;
        double r43970 = r43962 - r43960;
        double r43971 = r43960 + r43961;
        double r43972 = r43971 * r43965;
        double r43973 = -r43972;
        double r43974 = exp(r43973);
        double r43975 = r43970 * r43974;
        double r43976 = r43969 - r43975;
        double r43977 = 2.0;
        double r43978 = r43976 / r43977;
        return r43978;
}
↓
double f(double x, double eps) {
        double r43979 = x;
        double r43980 = 2.2371703346309433;
        bool r43981 = r43979 <= r43980;
        double r43982 = 1.0;
        double r43983 = 0.5;
        double r43984 = 2.0;
        double r43985 = pow(r43979, r43984);
        double r43986 = r43983 * r43985;
        double r43987 = r43982 - r43986;
        double r43988 = 0.33333333333333337;
        double r43989 = 3.0;
        double r43990 = pow(r43979, r43989);
        double r43991 = r43988 * r43990;
        double r43992 = r43987 + r43991;
        double r43993 = r43982 + r43986;
        double r43994 = r43993 + r43991;
        double r43995 = r43992 * r43994;
        double r43996 = r43995 / r43994;
        double r43997 = eps;
        double r43998 = r43982 / r43997;
        double r43999 = r43982 + r43998;
        double r44000 = r43982 - r43997;
        double r44001 = r44000 * r43979;
        double r44002 = exp(r44001);
        double r44003 = r43999 / r44002;
        double r44004 = 2.0;
        double r44005 = r44003 / r44004;
        double r44006 = r43982 + r43997;
        double r44007 = r44006 * r43979;
        double r44008 = exp(r44007);
        double r44009 = r43998 / r44008;
        double r44010 = r44009 / r44004;
        double r44011 = r44005 - r44010;
        double r44012 = r43982 / r44008;
        double r44013 = r44012 / r44004;
        double r44014 = r44011 + r44013;
        double r44015 = r43981 ? r43996 : r44014;
        return r44015;
}

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 < 2.2371703346309433

    1. Initial program 39.4

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

    2. Simplified39.4

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

    3. Taylor expanded around 0 1.4

      \[\leadsto \color{blue}{\left(0.33333333333333337 \cdot {x}^{3} + 1\right) - 0.5 \cdot {x}^{2}}\]
    4. Using strategy rm

    5. Applied add-cube-cbrt1.4

      \[\leadsto \left(0.33333333333333337 \cdot {\color{blue}{\left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}\right)}}^{3} + 1\right) - 0.5 \cdot {x}^{2}\]

    6. Applied unpow-prod-down1.4

      \[\leadsto \left(0.33333333333333337 \cdot \color{blue}{\left({\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{3} \cdot {\left(\sqrt[3]{x}\right)}^{3}\right)} + 1\right) - 0.5 \cdot {x}^{2}\]

    7. Applied associate-*r*1.4

      \[\leadsto \left(\color{blue}{\left(0.33333333333333337 \cdot {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{3}\right) \cdot {\left(\sqrt[3]{x}\right)}^{3}} + 1\right) - 0.5 \cdot {x}^{2}\]

    8. Simplified1.4

      \[\leadsto \left(\color{blue}{\left(\left(0.33333333333333337 \cdot x\right) \cdot x\right)} \cdot {\left(\sqrt[3]{x}\right)}^{3} + 1\right) - 0.5 \cdot {x}^{2}\]
    9. Using strategy rm

    10. Applied flip--1.4

      \[\leadsto \color{blue}{\frac{\left(\left(\left(0.33333333333333337 \cdot x\right) \cdot x\right) \cdot {\left(\sqrt[3]{x}\right)}^{3} + 1\right) \cdot \left(\left(\left(0.33333333333333337 \cdot x\right) \cdot x\right) \cdot {\left(\sqrt[3]{x}\right)}^{3} + 1\right) - \left(0.5 \cdot {x}^{2}\right) \cdot \left(0.5 \cdot {x}^{2}\right)}{\left(\left(\left(0.33333333333333337 \cdot x\right) \cdot x\right) \cdot {\left(\sqrt[3]{x}\right)}^{3} + 1\right) + 0.5 \cdot {x}^{2}}}\]

    11. Simplified1.4

      \[\leadsto \frac{\color{blue}{\left(\left(1 - 0.5 \cdot {x}^{2}\right) + 0.33333333333333337 \cdot {x}^{3}\right) \cdot \left(\left(1 + 0.5 \cdot {x}^{2}\right) + 0.33333333333333337 \cdot {x}^{3}\right)}}{\left(\left(\left(0.33333333333333337 \cdot x\right) \cdot x\right) \cdot {\left(\sqrt[3]{x}\right)}^{3} + 1\right) + 0.5 \cdot {x}^{2}}\]

    12. Simplified1.4

      \[\leadsto \frac{\left(\left(1 - 0.5 \cdot {x}^{2}\right) + 0.33333333333333337 \cdot {x}^{3}\right) \cdot \left(\left(1 + 0.5 \cdot {x}^{2}\right) + 0.33333333333333337 \cdot {x}^{3}\right)}{\color{blue}{\left(1 + 0.5 \cdot {x}^{2}\right) + 0.33333333333333337 \cdot {x}^{3}}}\]

    if 2.2371703346309433 < x

    1. Initial program 0.4

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

    2. Simplified0.4

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

    4. Applied div-sub0.4

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

    5. Applied div-sub0.4

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

    6. Applied associate--r-0.4

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

  4. Final simplification1.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 2.2371703346309433:\\ \;\;\;\;\frac{\left(\left(1 - 0.5 \cdot {x}^{2}\right) + 0.33333333333333337 \cdot {x}^{3}\right) \cdot \left(\left(1 + 0.5 \cdot {x}^{2}\right) + 0.33333333333333337 \cdot {x}^{3}\right)}{\left(1 + 0.5 \cdot {x}^{2}\right) + 0.33333333333333337 \cdot {x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{1 + \frac{1}{\varepsilon}}{e^{\left(1 - \varepsilon\right) \cdot x}}}{2} - \frac{\frac{\frac{1}{\varepsilon}}{e^{\left(1 + \varepsilon\right) \cdot x}}}{2}\right) + \frac{\frac{1}{e^{\left(1 + \varepsilon\right) \cdot x}}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020039 
(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))