Average Error: 16.2 → 8.5
Time: 40.3s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 7863989155318432839393331734927900672:\\ \;\;\;\;\frac{\frac{\beta}{\left(\beta + \alpha\right) + 2} - \left(\frac{\alpha}{\left(\beta + \alpha\right) + 2} - 1\right)}{2}\\ \mathbf{elif}\;\alpha \le 1.526450071771057118621392850450502817566 \cdot 10^{113}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\beta + \alpha\right) + 2} - \left(\frac{4}{\alpha \cdot \alpha} - \left(\frac{2}{\alpha} + \frac{\frac{8}{\alpha}}{\alpha \cdot \alpha}\right)\right)}{2}\\ \mathbf{elif}\;\alpha \le 1.143721273478083214922895355787158286969 \cdot 10^{167}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\beta + \alpha\right) + 2} - \left(\frac{\alpha}{\left(\beta + \alpha\right) + 2} - 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\beta + \alpha\right) + 2} - \left(\frac{4}{\alpha \cdot \alpha} - \left(\frac{2}{\alpha} + \frac{\frac{8}{\alpha}}{\alpha \cdot \alpha}\right)\right)}{2}\\ \end{array}\]
\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \le 7863989155318432839393331734927900672:\\
\;\;\;\;\frac{\frac{\beta}{\left(\beta + \alpha\right) + 2} - \left(\frac{\alpha}{\left(\beta + \alpha\right) + 2} - 1\right)}{2}\\

\mathbf{elif}\;\alpha \le 1.526450071771057118621392850450502817566 \cdot 10^{113}:\\
\;\;\;\;\frac{\frac{\beta}{\left(\beta + \alpha\right) + 2} - \left(\frac{4}{\alpha \cdot \alpha} - \left(\frac{2}{\alpha} + \frac{\frac{8}{\alpha}}{\alpha \cdot \alpha}\right)\right)}{2}\\

\mathbf{elif}\;\alpha \le 1.143721273478083214922895355787158286969 \cdot 10^{167}:\\
\;\;\;\;\frac{\frac{\beta}{\left(\beta + \alpha\right) + 2} - \left(\frac{\alpha}{\left(\beta + \alpha\right) + 2} - 1\right)}{2}\\

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

\end{array}
double f(double alpha, double beta) {
        double r4000717 = beta;
        double r4000718 = alpha;
        double r4000719 = r4000717 - r4000718;
        double r4000720 = r4000718 + r4000717;
        double r4000721 = 2.0;
        double r4000722 = r4000720 + r4000721;
        double r4000723 = r4000719 / r4000722;
        double r4000724 = 1.0;
        double r4000725 = r4000723 + r4000724;
        double r4000726 = r4000725 / r4000721;
        return r4000726;
}
↓
double f(double alpha, double beta) {
        double r4000727 = alpha;
        double r4000728 = 7.863989155318433e+36;
        bool r4000729 = r4000727 <= r4000728;
        double r4000730 = beta;
        double r4000731 = r4000730 + r4000727;
        double r4000732 = 2.0;
        double r4000733 = r4000731 + r4000732;
        double r4000734 = r4000730 / r4000733;
        double r4000735 = r4000727 / r4000733;
        double r4000736 = 1.0;
        double r4000737 = r4000735 - r4000736;
        double r4000738 = r4000734 - r4000737;
        double r4000739 = r4000738 / r4000732;
        double r4000740 = 1.5264500717710571e+113;
        bool r4000741 = r4000727 <= r4000740;
        double r4000742 = 4.0;
        double r4000743 = r4000727 * r4000727;
        double r4000744 = r4000742 / r4000743;
        double r4000745 = r4000732 / r4000727;
        double r4000746 = 8.0;
        double r4000747 = r4000746 / r4000727;
        double r4000748 = r4000747 / r4000743;
        double r4000749 = r4000745 + r4000748;
        double r4000750 = r4000744 - r4000749;
        double r4000751 = r4000734 - r4000750;
        double r4000752 = r4000751 / r4000732;
        double r4000753 = 1.1437212734780832e+167;
        bool r4000754 = r4000727 <= r4000753;
        double r4000755 = r4000754 ? r4000739 : r4000752;
        double r4000756 = r4000741 ? r4000752 : r4000755;
        double r4000757 = r4000729 ? r4000739 : r4000756;
        return r4000757;
}

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 < 7.863989155318433e+36 or 1.5264500717710571e+113 < alpha < 1.1437212734780832e+167

    1. Initial program 5.4

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

    3. Applied div-sub5.4

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

    4. Applied associate-+l-5.3

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

    6. Applied add-exp-log5.3

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

    8. Applied pow15.3

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

    9. Applied log-pow5.3

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

    10. Applied exp-prod5.4

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

    11. Simplified5.4

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

    13. Applied *-un-lft-identity5.4

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

    14. Applied *-un-lft-identity5.4

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

    15. Applied times-frac5.4

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

    16. Applied log-prod5.4

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

    17. Applied unpow-prod-up5.4

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

    18. Simplified5.4

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

    19. Simplified5.3

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

    if 7.863989155318433e+36 < alpha < 1.5264500717710571e+113 or 1.1437212734780832e+167 < alpha

    1. Initial program 51.0

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

    3. Applied div-sub51.0

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

    4. Applied associate-+l-49.2

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

    5. Taylor expanded around inf 18.7

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

    6. Simplified18.7

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

  4. Final simplification8.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 7863989155318432839393331734927900672:\\ \;\;\;\;\frac{\frac{\beta}{\left(\beta + \alpha\right) + 2} - \left(\frac{\alpha}{\left(\beta + \alpha\right) + 2} - 1\right)}{2}\\ \mathbf{elif}\;\alpha \le 1.526450071771057118621392850450502817566 \cdot 10^{113}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\beta + \alpha\right) + 2} - \left(\frac{4}{\alpha \cdot \alpha} - \left(\frac{2}{\alpha} + \frac{\frac{8}{\alpha}}{\alpha \cdot \alpha}\right)\right)}{2}\\ \mathbf{elif}\;\alpha \le 1.143721273478083214922895355787158286969 \cdot 10^{167}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\beta + \alpha\right) + 2} - \left(\frac{\alpha}{\left(\beta + \alpha\right) + 2} - 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\beta + \alpha\right) + 2} - \left(\frac{4}{\alpha \cdot \alpha} - \left(\frac{2}{\alpha} + \frac{\frac{8}{\alpha}}{\alpha \cdot \alpha}\right)\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019200 +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))