Average Error: 16.1 → 6.0
Time: 17.9s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 417035.8774050206993706524372100830078125:\\ \;\;\;\;{e}^{\left(\log \left(\frac{\frac{\beta}{\left(2 + \alpha\right) + \beta} - \left(\frac{\alpha}{\left(2 + \alpha\right) + \beta} - 1\right)}{2}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(2 + \alpha\right) + \beta} - \left(\frac{\frac{4}{\alpha}}{\alpha} - \left(\frac{8}{{\alpha}^{3}} + \frac{2}{\alpha}\right)\right)}{2}\\ \end{array}\]
\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \le 417035.8774050206993706524372100830078125:\\
\;\;\;\;{e}^{\left(\log \left(\frac{\frac{\beta}{\left(2 + \alpha\right) + \beta} - \left(\frac{\alpha}{\left(2 + \alpha\right) + \beta} - 1\right)}{2}\right)\right)}\\

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

\end{array}
double f(double alpha, double beta) {
        double r65772 = beta;
        double r65773 = alpha;
        double r65774 = r65772 - r65773;
        double r65775 = r65773 + r65772;
        double r65776 = 2.0;
        double r65777 = r65775 + r65776;
        double r65778 = r65774 / r65777;
        double r65779 = 1.0;
        double r65780 = r65778 + r65779;
        double r65781 = r65780 / r65776;
        return r65781;
}
↓
double f(double alpha, double beta) {
        double r65782 = alpha;
        double r65783 = 417035.8774050207;
        bool r65784 = r65782 <= r65783;
        double r65785 = exp(1.0);
        double r65786 = beta;
        double r65787 = 2.0;
        double r65788 = r65787 + r65782;
        double r65789 = r65788 + r65786;
        double r65790 = r65786 / r65789;
        double r65791 = r65782 / r65789;
        double r65792 = 1.0;
        double r65793 = r65791 - r65792;
        double r65794 = r65790 - r65793;
        double r65795 = r65794 / r65787;
        double r65796 = log(r65795);
        double r65797 = pow(r65785, r65796);
        double r65798 = 4.0;
        double r65799 = r65798 / r65782;
        double r65800 = r65799 / r65782;
        double r65801 = 8.0;
        double r65802 = 3.0;
        double r65803 = pow(r65782, r65802);
        double r65804 = r65801 / r65803;
        double r65805 = r65787 / r65782;
        double r65806 = r65804 + r65805;
        double r65807 = r65800 - r65806;
        double r65808 = r65790 - r65807;
        double r65809 = r65808 / r65787;
        double r65810 = r65784 ? r65797 : r65809;
        return r65810;
}

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 < 417035.8774050207

    1. Initial program 0.0

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

    2. Simplified0.0

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

    4. Applied div-sub0.0

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

    5. Applied associate-+l-0.0

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

    6. Simplified0.0

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

    8. Applied add-exp-log0.0

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

    9. Applied add-exp-log0.0

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

    10. Applied div-exp0.0

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

    11. Simplified0.0

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

    13. Applied pow10.0

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

    14. Applied log-pow0.0

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

    15. Applied exp-prod0.1

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

    16. Simplified0.1

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

    if 417035.8774050207 < alpha

    1. Initial program 48.7

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

    2. Simplified48.7

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

    4. Applied div-sub48.7

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

    5. Applied associate-+l-47.3

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

    6. Simplified47.3

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

    7. Taylor expanded around inf 18.1

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

    8. Simplified18.1

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

  4. Final simplification6.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 417035.8774050206993706524372100830078125:\\ \;\;\;\;{e}^{\left(\log \left(\frac{\frac{\beta}{\left(2 + \alpha\right) + \beta} - \left(\frac{\alpha}{\left(2 + \alpha\right) + \beta} - 1\right)}{2}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(2 + \alpha\right) + \beta} - \left(\frac{\frac{4}{\alpha}}{\alpha} - \left(\frac{8}{{\alpha}^{3}} + \frac{2}{\alpha}\right)\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 
(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))