Average Error: 16.5 → 6.2
Time: 3.7s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 759519.7231227644:\\ \;\;\;\;\frac{e^{\log \left(\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\sqrt[3]{\beta} \cdot \sqrt[3]{\beta}\right) \cdot \frac{\sqrt[3]{\beta}}{\left(\alpha + \beta\right) + 2} - \frac{1}{\alpha} \cdot \left(\left(\frac{4}{\alpha} - 2\right) - \frac{8}{{\alpha}^{2}}\right)}{2}\\ \end{array}\]
\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \le 759519.7231227644:\\
\;\;\;\;\frac{e^{\log \left(\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)\right)}}{2}\\

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

\end{array}
double f(double alpha, double beta) {
        double r123639 = beta;
        double r123640 = alpha;
        double r123641 = r123639 - r123640;
        double r123642 = r123640 + r123639;
        double r123643 = 2.0;
        double r123644 = r123642 + r123643;
        double r123645 = r123641 / r123644;
        double r123646 = 1.0;
        double r123647 = r123645 + r123646;
        double r123648 = r123647 / r123643;
        return r123648;
}
↓
double f(double alpha, double beta) {
        double r123649 = alpha;
        double r123650 = 759519.7231227644;
        bool r123651 = r123649 <= r123650;
        double r123652 = beta;
        double r123653 = r123649 + r123652;
        double r123654 = 2.0;
        double r123655 = r123653 + r123654;
        double r123656 = r123652 / r123655;
        double r123657 = r123649 / r123655;
        double r123658 = 1.0;
        double r123659 = r123657 - r123658;
        double r123660 = r123656 - r123659;
        double r123661 = log(r123660);
        double r123662 = exp(r123661);
        double r123663 = r123662 / r123654;
        double r123664 = cbrt(r123652);
        double r123665 = r123664 * r123664;
        double r123666 = r123664 / r123655;
        double r123667 = r123665 * r123666;
        double r123668 = 1.0;
        double r123669 = r123668 / r123649;
        double r123670 = 4.0;
        double r123671 = r123670 / r123649;
        double r123672 = r123671 - r123654;
        double r123673 = 8.0;
        double r123674 = 2.0;
        double r123675 = pow(r123649, r123674);
        double r123676 = r123673 / r123675;
        double r123677 = r123672 - r123676;
        double r123678 = r123669 * r123677;
        double r123679 = r123667 - r123678;
        double r123680 = r123679 / r123654;
        double r123681 = r123651 ? r123663 : r123680;
        return r123681;
}

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 < 759519.7231227644

    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

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

    if 759519.7231227644 < alpha

    1. Initial program 49.2

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

    3. Applied div-sub49.2

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

    4. Applied associate-+l-47.6

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

    6. Applied *-un-lft-identity47.6

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

    7. Applied add-cube-cbrt47.7

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

    8. Applied times-frac47.7

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

    9. Simplified47.7

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

    10. Taylor expanded around inf 18.5

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

    11. Simplified18.5

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

  4. Final simplification6.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 759519.7231227644:\\ \;\;\;\;\frac{e^{\log \left(\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\sqrt[3]{\beta} \cdot \sqrt[3]{\beta}\right) \cdot \frac{\sqrt[3]{\beta}}{\left(\alpha + \beta\right) + 2} - \frac{1}{\alpha} \cdot \left(\left(\frac{4}{\alpha} - 2\right) - \frac{8}{{\alpha}^{2}}\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020059 
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/1"
  :precision binary64
  :pre (and (> alpha -1) (> beta -1))
  (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2)) 1) 2))