Average Error: 15.7 → 6.2
Time: 24.6s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 1634825657092984630335766528:\\ \;\;\;\;\frac{\frac{\beta}{2 + \left(\beta + \alpha\right)} - \mathsf{fma}\left(\frac{\frac{\alpha}{2 + \left(\beta + \alpha\right)}}{2}, 1, \log \left(\sqrt{e^{\frac{\alpha}{2 + \left(\beta + \alpha\right)}}}\right) - 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{2 + \left(\beta + \alpha\right)} - \left(\left(\frac{\frac{4}{\alpha}}{\alpha} - \frac{2}{\alpha}\right) - \frac{8}{\alpha \cdot \left(\alpha \cdot \alpha\right)}\right)}{2}\\ \end{array}\]
\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \le 1634825657092984630335766528:\\
\;\;\;\;\frac{\frac{\beta}{2 + \left(\beta + \alpha\right)} - \mathsf{fma}\left(\frac{\frac{\alpha}{2 + \left(\beta + \alpha\right)}}{2}, 1, \log \left(\sqrt{e^{\frac{\alpha}{2 + \left(\beta + \alpha\right)}}}\right) - 1\right)}{2}\\

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

\end{array}
double f(double alpha, double beta) {
        double r4005864 = beta;
        double r4005865 = alpha;
        double r4005866 = r4005864 - r4005865;
        double r4005867 = r4005865 + r4005864;
        double r4005868 = 2.0;
        double r4005869 = r4005867 + r4005868;
        double r4005870 = r4005866 / r4005869;
        double r4005871 = 1.0;
        double r4005872 = r4005870 + r4005871;
        double r4005873 = r4005872 / r4005868;
        return r4005873;
}
↓
double f(double alpha, double beta) {
        double r4005874 = alpha;
        double r4005875 = 1.6348256570929846e+27;
        bool r4005876 = r4005874 <= r4005875;
        double r4005877 = beta;
        double r4005878 = 2.0;
        double r4005879 = r4005877 + r4005874;
        double r4005880 = r4005878 + r4005879;
        double r4005881 = r4005877 / r4005880;
        double r4005882 = r4005874 / r4005880;
        double r4005883 = 2.0;
        double r4005884 = r4005882 / r4005883;
        double r4005885 = 1.0;
        double r4005886 = exp(r4005882);
        double r4005887 = sqrt(r4005886);
        double r4005888 = log(r4005887);
        double r4005889 = 1.0;
        double r4005890 = r4005888 - r4005889;
        double r4005891 = fma(r4005884, r4005885, r4005890);
        double r4005892 = r4005881 - r4005891;
        double r4005893 = r4005892 / r4005878;
        double r4005894 = 4.0;
        double r4005895 = r4005894 / r4005874;
        double r4005896 = r4005895 / r4005874;
        double r4005897 = r4005878 / r4005874;
        double r4005898 = r4005896 - r4005897;
        double r4005899 = 8.0;
        double r4005900 = r4005874 * r4005874;
        double r4005901 = r4005874 * r4005900;
        double r4005902 = r4005899 / r4005901;
        double r4005903 = r4005898 - r4005902;
        double r4005904 = r4005881 - r4005903;
        double r4005905 = r4005904 / r4005878;
        double r4005906 = r4005876 ? r4005893 : r4005905;
        return r4005906;
}

Error

Bits error versus alpha

Bits error versus beta

Derivation

  1. Split input into 2 regimes

  2. if alpha < 1.6348256570929846e+27

    1. Initial program 1.1

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

    3. Applied div-sub1.1

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

    4. Applied associate-+l-1.1

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

    6. Applied add-log-exp1.1

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

    8. Applied add-sqr-sqrt1.1

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

    9. Applied log-prod1.1

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

    10. Applied associate--l+1.1

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

    12. Applied *-un-lft-identity1.1

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

    13. Applied exp-prod1.1

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

    14. Applied sqrt-pow11.1

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

    15. Applied log-pow1.1

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

    16. Applied fma-def1.1

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

    if 1.6348256570929846e+27 < alpha

    1. Initial program 50.4

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

    3. Applied div-sub50.3

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

    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

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

  4. Final simplification6.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 1634825657092984630335766528:\\ \;\;\;\;\frac{\frac{\beta}{2 + \left(\beta + \alpha\right)} - \mathsf{fma}\left(\frac{\frac{\alpha}{2 + \left(\beta + \alpha\right)}}{2}, 1, \log \left(\sqrt{e^{\frac{\alpha}{2 + \left(\beta + \alpha\right)}}}\right) - 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{2 + \left(\beta + \alpha\right)} - \left(\left(\frac{\frac{4}{\alpha}}{\alpha} - \frac{2}{\alpha}\right) - \frac{8}{\alpha \cdot \left(\alpha \cdot \alpha\right)}\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019179 +o rules:numerics
(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))