Average Error: 15.9 → 6.2
Time: 22.3s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 5.669431488829545 \cdot 10^{+27}:\\ \;\;\;\;{e}^{\left(\log \left(\frac{\frac{\beta}{\left(2.0 + \alpha\right) + \beta} - \left(\frac{\alpha}{\left(2.0 + \alpha\right) + \beta} - 1.0\right)}{2.0}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \left(\frac{4.0}{\alpha \cdot \alpha} - \left(\frac{2.0}{\alpha} + \frac{\frac{8.0}{\alpha}}{\alpha \cdot \alpha}\right)\right)}{2.0}\\ \end{array}\]
\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \le 5.669431488829545 \cdot 10^{+27}:\\
\;\;\;\;{e}^{\left(\log \left(\frac{\frac{\beta}{\left(2.0 + \alpha\right) + \beta} - \left(\frac{\alpha}{\left(2.0 + \alpha\right) + \beta} - 1.0\right)}{2.0}\right)\right)}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \left(\frac{4.0}{\alpha \cdot \alpha} - \left(\frac{2.0}{\alpha} + \frac{\frac{8.0}{\alpha}}{\alpha \cdot \alpha}\right)\right)}{2.0}\\

\end{array}
double f(double alpha, double beta) {
        double r3566998 = beta;
        double r3566999 = alpha;
        double r3567000 = r3566998 - r3566999;
        double r3567001 = r3566999 + r3566998;
        double r3567002 = 2.0;
        double r3567003 = r3567001 + r3567002;
        double r3567004 = r3567000 / r3567003;
        double r3567005 = 1.0;
        double r3567006 = r3567004 + r3567005;
        double r3567007 = r3567006 / r3567002;
        return r3567007;
}
↓
double f(double alpha, double beta) {
        double r3567008 = alpha;
        double r3567009 = 5.669431488829545e+27;
        bool r3567010 = r3567008 <= r3567009;
        double r3567011 = exp(1.0);
        double r3567012 = beta;
        double r3567013 = 2.0;
        double r3567014 = r3567013 + r3567008;
        double r3567015 = r3567014 + r3567012;
        double r3567016 = r3567012 / r3567015;
        double r3567017 = r3567008 / r3567015;
        double r3567018 = 1.0;
        double r3567019 = r3567017 - r3567018;
        double r3567020 = r3567016 - r3567019;
        double r3567021 = r3567020 / r3567013;
        double r3567022 = log(r3567021);
        double r3567023 = pow(r3567011, r3567022);
        double r3567024 = r3567008 + r3567012;
        double r3567025 = r3567024 + r3567013;
        double r3567026 = r3567012 / r3567025;
        double r3567027 = 4.0;
        double r3567028 = r3567008 * r3567008;
        double r3567029 = r3567027 / r3567028;
        double r3567030 = r3567013 / r3567008;
        double r3567031 = 8.0;
        double r3567032 = r3567031 / r3567008;
        double r3567033 = r3567032 / r3567028;
        double r3567034 = r3567030 + r3567033;
        double r3567035 = r3567029 - r3567034;
        double r3567036 = r3567026 - r3567035;
        double r3567037 = r3567036 / r3567013;
        double r3567038 = r3567010 ? r3567023 : r3567037;
        return r3567038;
}

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 < 5.669431488829545e+27

    1. Initial program 1.2

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}\]
    2. Using strategy rm

    3. Applied div-sub1.2

      \[\leadsto \frac{\color{blue}{\left(\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \frac{\alpha}{\left(\alpha + \beta\right) + 2.0}\right)} + 1.0}{2.0}\]

    4. Applied associate-+l-1.2

      \[\leadsto \frac{\color{blue}{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0\right)}}{2.0}\]
    5. Using strategy rm

    6. Applied add-exp-log1.2

      \[\leadsto \frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0\right)}{\color{blue}{e^{\log 2.0}}}\]

    7. Applied add-exp-log1.2

      \[\leadsto \frac{\color{blue}{e^{\log \left(\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0\right)\right)}}}{e^{\log 2.0}}\]

    8. Applied div-exp1.2

      \[\leadsto \color{blue}{e^{\log \left(\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0\right)\right) - \log 2.0}}\]

    9. Simplified1.2

      \[\leadsto e^{\color{blue}{\log \left(\frac{\frac{\beta}{\beta + \left(\alpha + 2.0\right)} - \left(\frac{\alpha}{\beta + \left(\alpha + 2.0\right)} - 1.0\right)}{2.0}\right)}}\]
    10. Using strategy rm

    11. Applied pow11.2

      \[\leadsto e^{\log \color{blue}{\left({\left(\frac{\frac{\beta}{\beta + \left(\alpha + 2.0\right)} - \left(\frac{\alpha}{\beta + \left(\alpha + 2.0\right)} - 1.0\right)}{2.0}\right)}^{1}\right)}}\]

    12. Applied log-pow1.2

      \[\leadsto e^{\color{blue}{1 \cdot \log \left(\frac{\frac{\beta}{\beta + \left(\alpha + 2.0\right)} - \left(\frac{\alpha}{\beta + \left(\alpha + 2.0\right)} - 1.0\right)}{2.0}\right)}}\]

    13. Applied exp-prod1.2

      \[\leadsto \color{blue}{{\left(e^{1}\right)}^{\left(\log \left(\frac{\frac{\beta}{\beta + \left(\alpha + 2.0\right)} - \left(\frac{\alpha}{\beta + \left(\alpha + 2.0\right)} - 1.0\right)}{2.0}\right)\right)}}\]

    14. Simplified1.2

      \[\leadsto {\color{blue}{e}}^{\left(\log \left(\frac{\frac{\beta}{\beta + \left(\alpha + 2.0\right)} - \left(\frac{\alpha}{\beta + \left(\alpha + 2.0\right)} - 1.0\right)}{2.0}\right)\right)}\]
    15. Using strategy rm

    16. Applied *-un-lft-identity1.2

      \[\leadsto {e}^{\left(\log \left(\frac{\frac{\beta}{\beta + \left(\alpha + 2.0\right)} - \left(\frac{\alpha}{\color{blue}{1 \cdot \left(\beta + \left(\alpha + 2.0\right)\right)}} - 1.0\right)}{2.0}\right)\right)}\]

    17. Applied *-un-lft-identity1.2

      \[\leadsto {e}^{\left(\log \left(\frac{\frac{\beta}{\beta + \left(\alpha + 2.0\right)} - \left(\frac{\color{blue}{1 \cdot \alpha}}{1 \cdot \left(\beta + \left(\alpha + 2.0\right)\right)} - 1.0\right)}{2.0}\right)\right)}\]

    18. Applied times-frac1.2

      \[\leadsto {e}^{\left(\log \left(\frac{\frac{\beta}{\beta + \left(\alpha + 2.0\right)} - \left(\color{blue}{\frac{1}{1} \cdot \frac{\alpha}{\beta + \left(\alpha + 2.0\right)}} - 1.0\right)}{2.0}\right)\right)}\]

    19. Simplified1.2

      \[\leadsto {e}^{\left(\log \left(\frac{\frac{\beta}{\beta + \left(\alpha + 2.0\right)} - \left(\color{blue}{1} \cdot \frac{\alpha}{\beta + \left(\alpha + 2.0\right)} - 1.0\right)}{2.0}\right)\right)}\]

    if 5.669431488829545e+27 < alpha

    1. Initial program 50.6

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}\]
    2. Using strategy rm

    3. Applied div-sub50.6

      \[\leadsto \frac{\color{blue}{\left(\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \frac{\alpha}{\left(\alpha + \beta\right) + 2.0}\right)} + 1.0}{2.0}\]

    4. Applied associate-+l-49.0

      \[\leadsto \frac{\color{blue}{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0\right)}}{2.0}\]

    5. Taylor expanded around inf 17.8

      \[\leadsto \frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \color{blue}{\left(4.0 \cdot \frac{1}{{\alpha}^{2}} - \left(2.0 \cdot \frac{1}{\alpha} + 8.0 \cdot \frac{1}{{\alpha}^{3}}\right)\right)}}{2.0}\]

    6. Simplified17.8

      \[\leadsto \frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \color{blue}{\left(\frac{4.0}{\alpha \cdot \alpha} - \left(\frac{\frac{8.0}{\alpha}}{\alpha \cdot \alpha} + \frac{2.0}{\alpha}\right)\right)}}{2.0}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification6.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 5.669431488829545 \cdot 10^{+27}:\\ \;\;\;\;{e}^{\left(\log \left(\frac{\frac{\beta}{\left(2.0 + \alpha\right) + \beta} - \left(\frac{\alpha}{\left(2.0 + \alpha\right) + \beta} - 1.0\right)}{2.0}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \left(\frac{4.0}{\alpha \cdot \alpha} - \left(\frac{2.0}{\alpha} + \frac{\frac{8.0}{\alpha}}{\alpha \cdot \alpha}\right)\right)}{2.0}\\ \end{array}\]

Reproduce

herbie shell --seed 2019165 +o rules:numerics
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/1"
  :pre (and (> alpha -1) (> beta -1))
  (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0))