Average Error: 16.1 → 6.0
Time: 19.5s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 417035.8774050206993706524372100830078125:\\ \;\;\;\;e^{\log \left(\frac{\frac{\beta}{2 + \left(\beta + \alpha\right)} - \left(\frac{\alpha}{2 + \left(\beta + \alpha\right)} - 1\right)}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{\frac{\beta}{2 + \left(\beta + \alpha\right)}} \cdot \left(\sqrt[3]{\frac{\beta}{2 + \left(\beta + \alpha\right)}} \cdot \sqrt[3]{\frac{\beta}{2 + \left(\beta + \alpha\right)}}\right) - \left(\left(4 - \frac{8}{\alpha}\right) \cdot \frac{1}{\alpha \cdot \alpha} - \frac{2}{\alpha}\right)}{2}\\ \end{array}\]
\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \le 417035.8774050206993706524372100830078125:\\
\;\;\;\;e^{\log \left(\frac{\frac{\beta}{2 + \left(\beta + \alpha\right)} - \left(\frac{\alpha}{2 + \left(\beta + \alpha\right)} - 1\right)}{2}\right)}\\

\mathbf{else}:\\
\;\;\;\;\frac{\sqrt[3]{\frac{\beta}{2 + \left(\beta + \alpha\right)}} \cdot \left(\sqrt[3]{\frac{\beta}{2 + \left(\beta + \alpha\right)}} \cdot \sqrt[3]{\frac{\beta}{2 + \left(\beta + \alpha\right)}}\right) - \left(\left(4 - \frac{8}{\alpha}\right) \cdot \frac{1}{\alpha \cdot \alpha} - \frac{2}{\alpha}\right)}{2}\\

\end{array}
double f(double alpha, double beta) {
        double r4076756 = beta;
        double r4076757 = alpha;
        double r4076758 = r4076756 - r4076757;
        double r4076759 = r4076757 + r4076756;
        double r4076760 = 2.0;
        double r4076761 = r4076759 + r4076760;
        double r4076762 = r4076758 / r4076761;
        double r4076763 = 1.0;
        double r4076764 = r4076762 + r4076763;
        double r4076765 = r4076764 / r4076760;
        return r4076765;
}
↓
double f(double alpha, double beta) {
        double r4076766 = alpha;
        double r4076767 = 417035.8774050207;
        bool r4076768 = r4076766 <= r4076767;
        double r4076769 = beta;
        double r4076770 = 2.0;
        double r4076771 = r4076769 + r4076766;
        double r4076772 = r4076770 + r4076771;
        double r4076773 = r4076769 / r4076772;
        double r4076774 = r4076766 / r4076772;
        double r4076775 = 1.0;
        double r4076776 = r4076774 - r4076775;
        double r4076777 = r4076773 - r4076776;
        double r4076778 = r4076777 / r4076770;
        double r4076779 = log(r4076778);
        double r4076780 = exp(r4076779);
        double r4076781 = cbrt(r4076773);
        double r4076782 = r4076781 * r4076781;
        double r4076783 = r4076781 * r4076782;
        double r4076784 = 4.0;
        double r4076785 = 8.0;
        double r4076786 = r4076785 / r4076766;
        double r4076787 = r4076784 - r4076786;
        double r4076788 = 1.0;
        double r4076789 = r4076766 * r4076766;
        double r4076790 = r4076788 / r4076789;
        double r4076791 = r4076787 * r4076790;
        double r4076792 = r4076770 / r4076766;
        double r4076793 = r4076791 - r4076792;
        double r4076794 = r4076783 - r4076793;
        double r4076795 = r4076794 / r4076770;
        double r4076796 = r4076768 ? r4076780 : r4076795;
        return r4076796;
}

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 < 417035.8774050207

    1. Initial program 0.0

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

    3. Applied div-sub0.0

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

    4. Applied associate-+l-0.0

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

    6. Applied add-exp-log0.0

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

    if 417035.8774050207 < alpha

    1. Initial program 48.7

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

    3. Applied div-sub48.7

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

    4. Applied associate-+l-47.3

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

    6. Applied add-cube-cbrt47.3

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

    7. Taylor expanded around inf 18.1

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

    8. Simplified18.1

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

  4. Final simplification6.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 417035.8774050206993706524372100830078125:\\ \;\;\;\;e^{\log \left(\frac{\frac{\beta}{2 + \left(\beta + \alpha\right)} - \left(\frac{\alpha}{2 + \left(\beta + \alpha\right)} - 1\right)}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{\frac{\beta}{2 + \left(\beta + \alpha\right)}} \cdot \left(\sqrt[3]{\frac{\beta}{2 + \left(\beta + \alpha\right)}} \cdot \sqrt[3]{\frac{\beta}{2 + \left(\beta + \alpha\right)}}\right) - \left(\left(4 - \frac{8}{\alpha}\right) \cdot \frac{1}{\alpha \cdot \alpha} - \frac{2}{\alpha}\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 
(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))