Average Error: 16.2 → 6.0
Time: 11.4s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 538664.502316254307515919208526611328125:\\ \;\;\;\;\frac{\frac{\sqrt[3]{\beta} \cdot \sqrt[3]{\beta}}{\sqrt[3]{\left(\alpha + \beta\right) + 2} \cdot \sqrt[3]{\left(\alpha + \beta\right) + 2}} \cdot \frac{\sqrt[3]{\beta}}{\sqrt{\sqrt[3]{\left(\alpha + \beta\right) + 2}} \cdot \sqrt{\sqrt[3]{\left(\alpha + \beta\right) + 2}}} - \left(\frac{1}{\frac{\left(\alpha + \beta\right) + 2}{\alpha}} - 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\left(\frac{4}{\alpha \cdot \alpha} - \frac{2}{\alpha}\right) - \frac{8}{{\alpha}^{3}}\right)}{2}\\ \end{array}\]
\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \le 538664.502316254307515919208526611328125:\\
\;\;\;\;\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{\sqrt[3]{\left(\alpha + \beta\right) + 2}} \cdot \sqrt{\sqrt[3]{\left(\alpha + \beta\right) + 2}}} - \left(\frac{1}{\frac{\left(\alpha + \beta\right) + 2}{\alpha}} - 1\right)}{2}\\

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

\end{array}
double f(double alpha, double beta) {
        double r170659 = beta;
        double r170660 = alpha;
        double r170661 = r170659 - r170660;
        double r170662 = r170660 + r170659;
        double r170663 = 2.0;
        double r170664 = r170662 + r170663;
        double r170665 = r170661 / r170664;
        double r170666 = 1.0;
        double r170667 = r170665 + r170666;
        double r170668 = r170667 / r170663;
        return r170668;
}
↓
double f(double alpha, double beta) {
        double r170669 = alpha;
        double r170670 = 538664.5023162543;
        bool r170671 = r170669 <= r170670;
        double r170672 = beta;
        double r170673 = cbrt(r170672);
        double r170674 = r170673 * r170673;
        double r170675 = r170669 + r170672;
        double r170676 = 2.0;
        double r170677 = r170675 + r170676;
        double r170678 = cbrt(r170677);
        double r170679 = r170678 * r170678;
        double r170680 = r170674 / r170679;
        double r170681 = sqrt(r170678);
        double r170682 = r170681 * r170681;
        double r170683 = r170673 / r170682;
        double r170684 = r170680 * r170683;
        double r170685 = 1.0;
        double r170686 = r170677 / r170669;
        double r170687 = r170685 / r170686;
        double r170688 = 1.0;
        double r170689 = r170687 - r170688;
        double r170690 = r170684 - r170689;
        double r170691 = r170690 / r170676;
        double r170692 = r170672 / r170677;
        double r170693 = 4.0;
        double r170694 = r170669 * r170669;
        double r170695 = r170693 / r170694;
        double r170696 = r170676 / r170669;
        double r170697 = r170695 - r170696;
        double r170698 = 8.0;
        double r170699 = 3.0;
        double r170700 = pow(r170669, r170699);
        double r170701 = r170698 / r170700;
        double r170702 = r170697 - r170701;
        double r170703 = r170692 - r170702;
        double r170704 = r170703 / r170676;
        double r170705 = r170671 ? r170691 : r170704;
        return r170705;
}

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 < 538664.5023162543

    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 clear-num0.0

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

    8. Applied add-cube-cbrt0.3

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

    9. Applied add-cube-cbrt0.1

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

    10. Applied times-frac0.1

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

    12. Applied add-sqr-sqrt0.1

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

    if 538664.5023162543 < alpha

    1. Initial program 49.5

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

    3. Applied div-sub49.5

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

    4. Applied associate-+l-48.0

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

    5. Taylor expanded around inf 18.1

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

    6. Simplified18.1

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

  4. Final simplification6.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 538664.502316254307515919208526611328125:\\ \;\;\;\;\frac{\frac{\sqrt[3]{\beta} \cdot \sqrt[3]{\beta}}{\sqrt[3]{\left(\alpha + \beta\right) + 2} \cdot \sqrt[3]{\left(\alpha + \beta\right) + 2}} \cdot \frac{\sqrt[3]{\beta}}{\sqrt{\sqrt[3]{\left(\alpha + \beta\right) + 2}} \cdot \sqrt{\sqrt[3]{\left(\alpha + \beta\right) + 2}}} - \left(\frac{1}{\frac{\left(\alpha + \beta\right) + 2}{\alpha}} - 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\left(\frac{4}{\alpha \cdot \alpha} - \frac{2}{\alpha}\right) - \frac{8}{{\alpha}^{3}}\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019350 
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/1"
  :precision binary64
  :pre (and (> alpha -1) (> beta -1))
  (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2)) 1) 2))