Average Error: 16.2 → 6.0
Time: 4.8s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 6665734284871154:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \frac{\left(-1 \cdot 1\right) + \frac{\frac{\alpha}{\left(\alpha + \beta\right) + 2} \cdot \alpha}{\left(\alpha + \beta\right) + 2}}{\alpha \cdot \frac{1}{\left(\alpha + \beta\right) + 2} + 1}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt[3]{\beta} \cdot \sqrt[3]{\beta}}{\sqrt[3]{\left(\alpha + \beta\right) + 2} \cdot \sqrt[3]{\left(\alpha + \beta\right) + 2}} \cdot \frac{\sqrt[3]{\beta}}{\sqrt[3]{\left(\alpha + \beta\right) + 2}} - \left(\frac{1}{{\alpha}^{2}} \cdot \left(4 - \frac{8}{\alpha}\right) + \frac{-2}{\alpha}\right)}{2}\\ \end{array}\]
\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \le 6665734284871154:\\
\;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \frac{\left(-1 \cdot 1\right) + \frac{\frac{\alpha}{\left(\alpha + \beta\right) + 2} \cdot \alpha}{\left(\alpha + \beta\right) + 2}}{\alpha \cdot \frac{1}{\left(\alpha + \beta\right) + 2} + 1}}{2}\\

\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(\frac{1}{{\alpha}^{2}} \cdot \left(4 - \frac{8}{\alpha}\right) + \frac{-2}{\alpha}\right)}{2}\\

\end{array}
double f(double alpha, double beta) {
        double r88608 = beta;
        double r88609 = alpha;
        double r88610 = r88608 - r88609;
        double r88611 = r88609 + r88608;
        double r88612 = 2.0;
        double r88613 = r88611 + r88612;
        double r88614 = r88610 / r88613;
        double r88615 = 1.0;
        double r88616 = r88614 + r88615;
        double r88617 = r88616 / r88612;
        return r88617;
}
↓
double f(double alpha, double beta) {
        double r88618 = alpha;
        double r88619 = 6665734284871154.0;
        bool r88620 = r88618 <= r88619;
        double r88621 = beta;
        double r88622 = r88618 + r88621;
        double r88623 = 2.0;
        double r88624 = r88622 + r88623;
        double r88625 = r88621 / r88624;
        double r88626 = 1.0;
        double r88627 = r88626 * r88626;
        double r88628 = -r88627;
        double r88629 = r88618 / r88624;
        double r88630 = r88629 * r88618;
        double r88631 = r88630 / r88624;
        double r88632 = r88628 + r88631;
        double r88633 = 1.0;
        double r88634 = r88633 / r88624;
        double r88635 = r88618 * r88634;
        double r88636 = r88635 + r88626;
        double r88637 = r88632 / r88636;
        double r88638 = r88625 - r88637;
        double r88639 = r88638 / r88623;
        double r88640 = cbrt(r88621);
        double r88641 = r88640 * r88640;
        double r88642 = cbrt(r88624);
        double r88643 = r88642 * r88642;
        double r88644 = r88641 / r88643;
        double r88645 = r88640 / r88642;
        double r88646 = r88644 * r88645;
        double r88647 = 2.0;
        double r88648 = pow(r88618, r88647);
        double r88649 = r88633 / r88648;
        double r88650 = 4.0;
        double r88651 = 8.0;
        double r88652 = r88651 / r88618;
        double r88653 = r88650 - r88652;
        double r88654 = r88649 * r88653;
        double r88655 = -r88623;
        double r88656 = r88655 / r88618;
        double r88657 = r88654 + r88656;
        double r88658 = r88646 - r88657;
        double r88659 = r88658 / r88623;
        double r88660 = r88620 ? r88639 : r88659;
        return r88660;
}

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 < 6665734284871154.0

    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 div-inv0.4

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

    8. Applied flip--0.4

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

    9. Simplified0.4

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

    if 6665734284871154.0 < alpha

    1. Initial program 50.8

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

    3. Applied div-sub50.7

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

    4. Applied associate-+l-49.1

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

    6. Applied add-cube-cbrt49.2

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

    7. Applied add-cube-cbrt49.1

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

    8. Applied times-frac49.1

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

    9. Taylor expanded around inf 18.2

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

    10. Simplified18.2

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

  4. Final simplification6.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 6665734284871154:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \frac{\left(-1 \cdot 1\right) + \frac{\frac{\alpha}{\left(\alpha + \beta\right) + 2} \cdot \alpha}{\left(\alpha + \beta\right) + 2}}{\alpha \cdot \frac{1}{\left(\alpha + \beta\right) + 2} + 1}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt[3]{\beta} \cdot \sqrt[3]{\beta}}{\sqrt[3]{\left(\alpha + \beta\right) + 2} \cdot \sqrt[3]{\left(\alpha + \beta\right) + 2}} \cdot \frac{\sqrt[3]{\beta}}{\sqrt[3]{\left(\alpha + \beta\right) + 2}} - \left(\frac{1}{{\alpha}^{2}} \cdot \left(4 - \frac{8}{\alpha}\right) + \frac{-2}{\alpha}\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019353 
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/1"
  :precision binary64
  :pre (and (> alpha -1) (> beta -1))
  (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2)) 1) 2))