Average Error: 16.1 → 6.1
Time: 19.7s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 10272132967178968:\\ \;\;\;\;\frac{\sqrt[3]{\frac{\beta}{2 + \left(\beta + \alpha\right)} \cdot \left(\frac{\beta}{2 + \left(\beta + \alpha\right)} \cdot \frac{\beta}{2 + \left(\beta + \alpha\right)}\right)} - \left(\frac{\alpha}{2 + \left(\beta + \alpha\right)} - 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{2 + \left(\beta + \alpha\right)}} \cdot \frac{\beta}{\sqrt{2 + \left(\beta + \alpha\right)}} - \left(\left(\frac{4}{\alpha \cdot \alpha} - \frac{\frac{8}{\alpha}}{\alpha \cdot \alpha}\right) - \frac{2}{\alpha}\right)}{2}\\ \end{array}\]
\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \le 10272132967178968:\\
\;\;\;\;\frac{\sqrt[3]{\frac{\beta}{2 + \left(\beta + \alpha\right)} \cdot \left(\frac{\beta}{2 + \left(\beta + \alpha\right)} \cdot \frac{\beta}{2 + \left(\beta + \alpha\right)}\right)} - \left(\frac{\alpha}{2 + \left(\beta + \alpha\right)} - 1\right)}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{1}{\sqrt{2 + \left(\beta + \alpha\right)}} \cdot \frac{\beta}{\sqrt{2 + \left(\beta + \alpha\right)}} - \left(\left(\frac{4}{\alpha \cdot \alpha} - \frac{\frac{8}{\alpha}}{\alpha \cdot \alpha}\right) - \frac{2}{\alpha}\right)}{2}\\

\end{array}
double f(double alpha, double beta) {
        double r3902867 = beta;
        double r3902868 = alpha;
        double r3902869 = r3902867 - r3902868;
        double r3902870 = r3902868 + r3902867;
        double r3902871 = 2.0;
        double r3902872 = r3902870 + r3902871;
        double r3902873 = r3902869 / r3902872;
        double r3902874 = 1.0;
        double r3902875 = r3902873 + r3902874;
        double r3902876 = r3902875 / r3902871;
        return r3902876;
}
↓
double f(double alpha, double beta) {
        double r3902877 = alpha;
        double r3902878 = 10272132967178968.0;
        bool r3902879 = r3902877 <= r3902878;
        double r3902880 = beta;
        double r3902881 = 2.0;
        double r3902882 = r3902880 + r3902877;
        double r3902883 = r3902881 + r3902882;
        double r3902884 = r3902880 / r3902883;
        double r3902885 = r3902884 * r3902884;
        double r3902886 = r3902884 * r3902885;
        double r3902887 = cbrt(r3902886);
        double r3902888 = r3902877 / r3902883;
        double r3902889 = 1.0;
        double r3902890 = r3902888 - r3902889;
        double r3902891 = r3902887 - r3902890;
        double r3902892 = r3902891 / r3902881;
        double r3902893 = 1.0;
        double r3902894 = sqrt(r3902883);
        double r3902895 = r3902893 / r3902894;
        double r3902896 = r3902880 / r3902894;
        double r3902897 = r3902895 * r3902896;
        double r3902898 = 4.0;
        double r3902899 = r3902877 * r3902877;
        double r3902900 = r3902898 / r3902899;
        double r3902901 = 8.0;
        double r3902902 = r3902901 / r3902877;
        double r3902903 = r3902902 / r3902899;
        double r3902904 = r3902900 - r3902903;
        double r3902905 = r3902881 / r3902877;
        double r3902906 = r3902904 - r3902905;
        double r3902907 = r3902897 - r3902906;
        double r3902908 = r3902907 / r3902881;
        double r3902909 = r3902879 ? r3902892 : r3902908;
        return r3902909;
}

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 < 10272132967178968.0

    1. Initial program 0.4

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

    3. Applied div-sub0.4

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

    4. Applied associate-+l-0.4

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

    6. Applied add-cbrt-cube0.5

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

    if 10272132967178968.0 < alpha

    1. Initial program 50.0

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

    3. Applied div-sub49.9

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

    4. Applied associate-+l-48.4

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

    6. Applied add-sqr-sqrt48.4

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

    7. Applied *-un-lft-identity48.4

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

    8. Applied times-frac48.4

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

    9. Taylor expanded around inf 18.4

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

    10. Simplified18.4

      \[\leadsto \frac{\frac{1}{\sqrt{\left(\alpha + \beta\right) + 2}} \cdot \frac{\beta}{\sqrt{\left(\alpha + \beta\right) + 2}} - \color{blue}{\left(\left(\frac{4}{\alpha \cdot \alpha} - \frac{\frac{8}{\alpha}}{\alpha \cdot \alpha}\right) - \frac{2}{\alpha}\right)}}{2}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification6.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 10272132967178968:\\ \;\;\;\;\frac{\sqrt[3]{\frac{\beta}{2 + \left(\beta + \alpha\right)} \cdot \left(\frac{\beta}{2 + \left(\beta + \alpha\right)} \cdot \frac{\beta}{2 + \left(\beta + \alpha\right)}\right)} - \left(\frac{\alpha}{2 + \left(\beta + \alpha\right)} - 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{2 + \left(\beta + \alpha\right)}} \cdot \frac{\beta}{\sqrt{2 + \left(\beta + \alpha\right)}} - \left(\left(\frac{4}{\alpha \cdot \alpha} - \frac{\frac{8}{\alpha}}{\alpha \cdot \alpha}\right) - \frac{2}{\alpha}\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/1"
  :pre (and (> alpha -1.0) (> beta -1.0))
  (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0))