Average Error: 16.4 → 6.8
Time: 16.6s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 5062494549189153:\\ \;\;\;\;e^{\left(-\log \left(\sqrt{2}\right)\right) + \log \left(\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)}{\sqrt{2}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\left(\frac{4}{\alpha \cdot \alpha} - \frac{2}{\alpha}\right) - \frac{8}{{\alpha}^{3}}\right)}{2}\\ \end{array}\]
\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \le 5062494549189153:\\
\;\;\;\;e^{\left(-\log \left(\sqrt{2}\right)\right) + \log \left(\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)}{\sqrt{2}}\right)}\\

\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 r64685 = beta;
        double r64686 = alpha;
        double r64687 = r64685 - r64686;
        double r64688 = r64686 + r64685;
        double r64689 = 2.0;
        double r64690 = r64688 + r64689;
        double r64691 = r64687 / r64690;
        double r64692 = 1.0;
        double r64693 = r64691 + r64692;
        double r64694 = r64693 / r64689;
        return r64694;
}
↓
double f(double alpha, double beta) {
        double r64695 = alpha;
        double r64696 = 5062494549189153.0;
        bool r64697 = r64695 <= r64696;
        double r64698 = 2.0;
        double r64699 = sqrt(r64698);
        double r64700 = log(r64699);
        double r64701 = -r64700;
        double r64702 = beta;
        double r64703 = r64695 + r64702;
        double r64704 = r64703 + r64698;
        double r64705 = r64702 / r64704;
        double r64706 = r64695 / r64704;
        double r64707 = 1.0;
        double r64708 = r64706 - r64707;
        double r64709 = r64705 - r64708;
        double r64710 = r64709 / r64699;
        double r64711 = log(r64710);
        double r64712 = r64701 + r64711;
        double r64713 = exp(r64712);
        double r64714 = 4.0;
        double r64715 = r64695 * r64695;
        double r64716 = r64714 / r64715;
        double r64717 = r64698 / r64695;
        double r64718 = r64716 - r64717;
        double r64719 = 8.0;
        double r64720 = 3.0;
        double r64721 = pow(r64695, r64720);
        double r64722 = r64719 / r64721;
        double r64723 = r64718 - r64722;
        double r64724 = r64705 - r64723;
        double r64725 = r64724 / r64698;
        double r64726 = r64697 ? r64713 : r64725;
        return r64726;
}

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 < 5062494549189153.0

    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-exp-log0.4

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

    7. Applied add-exp-log0.4

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

    8. Applied div-exp0.4

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

    9. Simplified0.4

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

    11. Applied add-sqr-sqrt1.9

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

    12. Applied *-un-lft-identity1.9

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

    13. Applied times-frac1.9

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

    14. Applied log-prod1.7

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

    15. Simplified1.4

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

    if 5062494549189153.0 < alpha

    1. Initial program 50.4

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

    3. Applied div-sub50.4

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

    4. Applied associate-+l-48.7

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

    5. Taylor expanded around inf 18.4

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

    6. Simplified18.4

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

  4. Final simplification6.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 5062494549189153:\\ \;\;\;\;e^{\left(-\log \left(\sqrt{2}\right)\right) + \log \left(\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)}{\sqrt{2}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\left(\frac{4}{\alpha \cdot \alpha} - \frac{2}{\alpha}\right) - \frac{8}{{\alpha}^{3}}\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019303 
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/1"
  :precision binary64
  :pre (and (> alpha -1) (> beta -1))
  (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2)) 1) 2))