Average Error: 4.1 → 1.2
Time: 1.8m
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\beta \le 3.5888856498529296 \cdot 10^{+157}:\\ \;\;\;\;\frac{\frac{\frac{1}{\left(\beta + \alpha\right) + 2}}{\frac{\left(\left(\beta + \alpha\right) + 2\right) + 1.0}{1.0 + \left(\alpha \cdot \beta + \left(\beta + \alpha\right)\right)}}}{\left(\beta + \alpha\right) + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{\left(\beta + \alpha\right) + 2}}{\left(\frac{1}{\beta} + \frac{1}{\alpha}\right) - \frac{1}{{\alpha}^{2}}}}{\left(\beta + \alpha\right) + 2}\\ \end{array}\]
\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}
↓
\begin{array}{l}
\mathbf{if}\;\beta \le 3.5888856498529296 \cdot 10^{+157}:\\
\;\;\;\;\frac{\frac{\frac{1}{\left(\beta + \alpha\right) + 2}}{\frac{\left(\left(\beta + \alpha\right) + 2\right) + 1.0}{1.0 + \left(\alpha \cdot \beta + \left(\beta + \alpha\right)\right)}}}{\left(\beta + \alpha\right) + 2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{\frac{1}{\left(\beta + \alpha\right) + 2}}{\left(\frac{1}{\beta} + \frac{1}{\alpha}\right) - \frac{1}{{\alpha}^{2}}}}{\left(\beta + \alpha\right) + 2}\\

\end{array}
double f(double alpha, double beta) {
        double r4747839 = alpha;
        double r4747840 = beta;
        double r4747841 = r4747839 + r4747840;
        double r4747842 = r4747840 * r4747839;
        double r4747843 = r4747841 + r4747842;
        double r4747844 = 1.0;
        double r4747845 = r4747843 + r4747844;
        double r4747846 = 2.0;
        double r4747847 = 1.0;
        double r4747848 = r4747846 * r4747847;
        double r4747849 = r4747841 + r4747848;
        double r4747850 = r4747845 / r4747849;
        double r4747851 = r4747850 / r4747849;
        double r4747852 = r4747849 + r4747844;
        double r4747853 = r4747851 / r4747852;
        return r4747853;
}
↓
double f(double alpha, double beta) {
        double r4747854 = beta;
        double r4747855 = 3.5888856498529296e+157;
        bool r4747856 = r4747854 <= r4747855;
        double r4747857 = 1.0;
        double r4747858 = alpha;
        double r4747859 = r4747854 + r4747858;
        double r4747860 = 2.0;
        double r4747861 = r4747859 + r4747860;
        double r4747862 = r4747857 / r4747861;
        double r4747863 = 1.0;
        double r4747864 = r4747861 + r4747863;
        double r4747865 = r4747858 * r4747854;
        double r4747866 = r4747865 + r4747859;
        double r4747867 = r4747863 + r4747866;
        double r4747868 = r4747864 / r4747867;
        double r4747869 = r4747862 / r4747868;
        double r4747870 = r4747869 / r4747861;
        double r4747871 = r4747857 / r4747854;
        double r4747872 = r4747857 / r4747858;
        double r4747873 = r4747871 + r4747872;
        double r4747874 = pow(r4747858, r4747860);
        double r4747875 = r4747857 / r4747874;
        double r4747876 = r4747873 - r4747875;
        double r4747877 = r4747862 / r4747876;
        double r4747878 = r4747877 / r4747861;
        double r4747879 = r4747856 ? r4747870 : r4747878;
        return r4747879;
}

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 beta < 3.5888856498529296e+157

    1. Initial program 1.4

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]
    2. Using strategy rm

    3. Applied clear-num1.4

      \[\leadsto \frac{\frac{\color{blue}{\frac{1}{\frac{\left(\alpha + \beta\right) + 2 \cdot 1}{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]
    4. Using strategy rm

    5. Applied *-un-lft-identity1.4

      \[\leadsto \frac{\frac{\frac{1}{\frac{\left(\alpha + \beta\right) + 2 \cdot 1}{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}}}{\color{blue}{1 \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]

    6. Applied associate-/r/1.4

      \[\leadsto \frac{\frac{\color{blue}{\frac{1}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \left(\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0\right)}}{1 \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]

    7. Applied times-frac1.4

      \[\leadsto \frac{\color{blue}{\frac{\frac{1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{1} \cdot \frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]

    8. Applied associate-/l*1.4

      \[\leadsto \color{blue}{\frac{\frac{\frac{1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{1}}{\frac{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}}}\]
    9. Using strategy rm

    10. Applied associate-/r/1.4

      \[\leadsto \frac{\frac{\frac{1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{1}}{\color{blue}{\frac{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0} \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}}\]

    11. Applied associate-/r*1.4

      \[\leadsto \color{blue}{\frac{\frac{\frac{\frac{1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{1}}{\frac{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}\]

    if 3.5888856498529296e+157 < beta

    1. Initial program 17.3

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]
    2. Using strategy rm

    3. Applied clear-num17.3

      \[\leadsto \frac{\frac{\color{blue}{\frac{1}{\frac{\left(\alpha + \beta\right) + 2 \cdot 1}{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]
    4. Using strategy rm

    5. Applied *-un-lft-identity17.3

      \[\leadsto \frac{\frac{\frac{1}{\frac{\left(\alpha + \beta\right) + 2 \cdot 1}{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}}}{\color{blue}{1 \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]

    6. Applied associate-/r/17.3

      \[\leadsto \frac{\frac{\color{blue}{\frac{1}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \left(\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0\right)}}{1 \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]

    7. Applied times-frac17.3

      \[\leadsto \frac{\color{blue}{\frac{\frac{1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{1} \cdot \frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]

    8. Applied associate-/l*17.3

      \[\leadsto \color{blue}{\frac{\frac{\frac{1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{1}}{\frac{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}}}\]
    9. Using strategy rm

    10. Applied associate-/r/17.3

      \[\leadsto \frac{\frac{\frac{1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{1}}{\color{blue}{\frac{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0} \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}}\]

    11. Applied associate-/r*17.3

      \[\leadsto \color{blue}{\frac{\frac{\frac{\frac{1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{1}}{\frac{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}\]

    12. Taylor expanded around inf 0.3

      \[\leadsto \frac{\frac{\frac{\frac{1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{1}}{\color{blue}{\left(\frac{1}{\alpha} + \frac{1}{\beta}\right) - \frac{1}{{\alpha}^{2}}}}}{\left(\alpha + \beta\right) + 2 \cdot 1}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification1.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \le 3.5888856498529296 \cdot 10^{+157}:\\ \;\;\;\;\frac{\frac{\frac{1}{\left(\beta + \alpha\right) + 2}}{\frac{\left(\left(\beta + \alpha\right) + 2\right) + 1.0}{1.0 + \left(\alpha \cdot \beta + \left(\beta + \alpha\right)\right)}}}{\left(\beta + \alpha\right) + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{\left(\beta + \alpha\right) + 2}}{\left(\frac{1}{\beta} + \frac{1}{\alpha}\right) - \frac{1}{{\alpha}^{2}}}}{\left(\beta + \alpha\right) + 2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019155 
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/3"
  :pre (and (> alpha -1) (> beta -1))
  (/ (/ (/ (+ (+ (+ alpha beta) (* beta alpha)) 1.0) (+ (+ alpha beta) (* 2 1))) (+ (+ alpha beta) (* 2 1))) (+ (+ (+ alpha beta) (* 2 1)) 1.0)))