Average Error: 16.1 → 6.1
Time: 20.8s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 1190096873705876480:\\ \;\;\;\;{e}^{\left(\log \left(\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)}{2}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt[3]{\beta} \cdot \sqrt[3]{\beta}}{\sqrt[3]{\left(\alpha + \beta\right) + 2} \cdot \sqrt[3]{\left(\alpha + \beta\right) + 2}} \cdot \frac{\sqrt[3]{\beta}}{\sqrt[3]{\left(\alpha + \beta\right) + 2}} - \left(\left(\frac{4}{{\alpha}^{2}} - \frac{2}{\alpha}\right) - \frac{8}{{\alpha}^{3}}\right)}{2}\\ \end{array}\]
\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \le 1190096873705876480:\\
\;\;\;\;{e}^{\left(\log \left(\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)}{2}\right)\right)}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{\sqrt[3]{\beta} \cdot \sqrt[3]{\beta}}{\sqrt[3]{\left(\alpha + \beta\right) + 2} \cdot \sqrt[3]{\left(\alpha + \beta\right) + 2}} \cdot \frac{\sqrt[3]{\beta}}{\sqrt[3]{\left(\alpha + \beta\right) + 2}} - \left(\left(\frac{4}{{\alpha}^{2}} - \frac{2}{\alpha}\right) - \frac{8}{{\alpha}^{3}}\right)}{2}\\

\end{array}
double f(double alpha, double beta) {
        double r82831 = beta;
        double r82832 = alpha;
        double r82833 = r82831 - r82832;
        double r82834 = r82832 + r82831;
        double r82835 = 2.0;
        double r82836 = r82834 + r82835;
        double r82837 = r82833 / r82836;
        double r82838 = 1.0;
        double r82839 = r82837 + r82838;
        double r82840 = r82839 / r82835;
        return r82840;
}
↓
double f(double alpha, double beta) {
        double r82841 = alpha;
        double r82842 = 1.1900968737058765e+18;
        bool r82843 = r82841 <= r82842;
        double r82844 = exp(1.0);
        double r82845 = beta;
        double r82846 = r82841 + r82845;
        double r82847 = 2.0;
        double r82848 = r82846 + r82847;
        double r82849 = r82845 / r82848;
        double r82850 = r82841 / r82848;
        double r82851 = 1.0;
        double r82852 = r82850 - r82851;
        double r82853 = r82849 - r82852;
        double r82854 = r82853 / r82847;
        double r82855 = log(r82854);
        double r82856 = pow(r82844, r82855);
        double r82857 = cbrt(r82845);
        double r82858 = r82857 * r82857;
        double r82859 = cbrt(r82848);
        double r82860 = r82859 * r82859;
        double r82861 = r82858 / r82860;
        double r82862 = r82857 / r82859;
        double r82863 = r82861 * r82862;
        double r82864 = 4.0;
        double r82865 = 2.0;
        double r82866 = pow(r82841, r82865);
        double r82867 = r82864 / r82866;
        double r82868 = r82847 / r82841;
        double r82869 = r82867 - r82868;
        double r82870 = 8.0;
        double r82871 = 3.0;
        double r82872 = pow(r82841, r82871);
        double r82873 = r82870 / r82872;
        double r82874 = r82869 - r82873;
        double r82875 = r82863 - r82874;
        double r82876 = r82875 / r82847;
        double r82877 = r82843 ? r82856 : r82876;
        return r82877;
}

Error

Bits error versus alpha

Bits error versus beta

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if alpha < 1.1900968737058765e+18

    1. Initial program 0.5

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
    2. Using strategy rm

    3. Applied div-sub0.5

      \[\leadsto \frac{\color{blue}{\left(\frac{\beta}{\left(\alpha + \beta\right) + 2} - \frac{\alpha}{\left(\alpha + \beta\right) + 2}\right)} + 1}{2}\]

    4. Applied associate-+l-0.5

      \[\leadsto \frac{\color{blue}{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)}}{2}\]
    5. Using strategy rm

    6. Applied add-exp-log0.5

      \[\leadsto \frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)}{\color{blue}{e^{\log 2}}}\]

    7. Applied add-exp-log0.5

      \[\leadsto \frac{\color{blue}{e^{\log \left(\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)\right)}}}{e^{\log 2}}\]

    8. Applied div-exp0.5

      \[\leadsto \color{blue}{e^{\log \left(\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)\right) - \log 2}}\]

    9. Simplified0.5

      \[\leadsto e^{\color{blue}{\log \left(\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)}{2}\right)}}\]
    10. Using strategy rm

    11. Applied pow10.5

      \[\leadsto e^{\log \color{blue}{\left({\left(\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)}{2}\right)}^{1}\right)}}\]

    12. Applied log-pow0.5

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

    13. Applied exp-prod0.5

      \[\leadsto \color{blue}{{\left(e^{1}\right)}^{\left(\log \left(\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)}{2}\right)\right)}}\]

    14. Simplified0.5

      \[\leadsto {\color{blue}{e}}^{\left(\log \left(\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)}{2}\right)\right)}\]

    if 1.1900968737058765e+18 < alpha

    1. Initial program 50.4

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
    2. Using strategy rm

    3. Applied div-sub50.4

      \[\leadsto \frac{\color{blue}{\left(\frac{\beta}{\left(\alpha + \beta\right) + 2} - \frac{\alpha}{\left(\alpha + \beta\right) + 2}\right)} + 1}{2}\]

    4. Applied associate-+l-48.8

      \[\leadsto \frac{\color{blue}{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)}}{2}\]
    5. Using strategy rm

    6. Applied add-cube-cbrt48.9

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

    7. Applied add-cube-cbrt48.8

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

    8. Applied times-frac48.8

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

    9. Taylor expanded around inf 18.3

      \[\leadsto \frac{\frac{\sqrt[3]{\beta} \cdot \sqrt[3]{\beta}}{\sqrt[3]{\left(\alpha + \beta\right) + 2} \cdot \sqrt[3]{\left(\alpha + \beta\right) + 2}} \cdot \frac{\sqrt[3]{\beta}}{\sqrt[3]{\left(\alpha + \beta\right) + 2}} - \color{blue}{\left(4 \cdot \frac{1}{{\alpha}^{2}} - \left(2 \cdot \frac{1}{\alpha} + 8 \cdot \frac{1}{{\alpha}^{3}}\right)\right)}}{2}\]

    10. Simplified18.3

      \[\leadsto \frac{\frac{\sqrt[3]{\beta} \cdot \sqrt[3]{\beta}}{\sqrt[3]{\left(\alpha + \beta\right) + 2} \cdot \sqrt[3]{\left(\alpha + \beta\right) + 2}} \cdot \frac{\sqrt[3]{\beta}}{\sqrt[3]{\left(\alpha + \beta\right) + 2}} - \color{blue}{\left(\left(\frac{4}{{\alpha}^{2}} - \frac{2}{\alpha}\right) - \frac{8}{{\alpha}^{3}}\right)}}{2}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification6.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 1190096873705876480:\\ \;\;\;\;{e}^{\left(\log \left(\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)}{2}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt[3]{\beta} \cdot \sqrt[3]{\beta}}{\sqrt[3]{\left(\alpha + \beta\right) + 2} \cdot \sqrt[3]{\left(\alpha + \beta\right) + 2}} \cdot \frac{\sqrt[3]{\beta}}{\sqrt[3]{\left(\alpha + \beta\right) + 2}} - \left(\left(\frac{4}{{\alpha}^{2}} - \frac{2}{\alpha}\right) - \frac{8}{{\alpha}^{3}}\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019209 +o rules:numerics
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/1"
  :precision binary64
  :pre (and (> alpha -1) (> beta -1))
  (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2)) 1) 2))