Average Error: 3.8 → 3.7
Time: 42.0s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 2593180110544.6416:\\ \;\;\;\;\frac{\frac{\frac{\sqrt[3]{1.0 + \left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right)}}{2 + \left(\alpha + \beta\right)} \cdot \left(\sqrt[3]{1.0 + \left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right)} \cdot \sqrt[3]{1.0 + \left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right)}\right)}{2 + \left(\alpha + \beta\right)}}{\left(2 + \left(\alpha + \beta\right)\right) + 1.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(1 - \frac{1.0}{\alpha}\right) + \frac{2.0}{\alpha \cdot \alpha}}{2 + \left(\alpha + \beta\right)}}{\left(2 + \left(\alpha + \beta\right)\right) + 1.0}\\ \end{array}\]
\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}\;\alpha \le 2593180110544.6416:\\
\;\;\;\;\frac{\frac{\frac{\sqrt[3]{1.0 + \left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right)}}{2 + \left(\alpha + \beta\right)} \cdot \left(\sqrt[3]{1.0 + \left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right)} \cdot \sqrt[3]{1.0 + \left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right)}\right)}{2 + \left(\alpha + \beta\right)}}{\left(2 + \left(\alpha + \beta\right)\right) + 1.0}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{\left(1 - \frac{1.0}{\alpha}\right) + \frac{2.0}{\alpha \cdot \alpha}}{2 + \left(\alpha + \beta\right)}}{\left(2 + \left(\alpha + \beta\right)\right) + 1.0}\\

\end{array}
double f(double alpha, double beta) {
        double r3911874 = alpha;
        double r3911875 = beta;
        double r3911876 = r3911874 + r3911875;
        double r3911877 = r3911875 * r3911874;
        double r3911878 = r3911876 + r3911877;
        double r3911879 = 1.0;
        double r3911880 = r3911878 + r3911879;
        double r3911881 = 2.0;
        double r3911882 = 1.0;
        double r3911883 = r3911881 * r3911882;
        double r3911884 = r3911876 + r3911883;
        double r3911885 = r3911880 / r3911884;
        double r3911886 = r3911885 / r3911884;
        double r3911887 = r3911884 + r3911879;
        double r3911888 = r3911886 / r3911887;
        return r3911888;
}
↓
double f(double alpha, double beta) {
        double r3911889 = alpha;
        double r3911890 = 2593180110544.6416;
        bool r3911891 = r3911889 <= r3911890;
        double r3911892 = 1.0;
        double r3911893 = beta;
        double r3911894 = r3911889 + r3911893;
        double r3911895 = r3911893 * r3911889;
        double r3911896 = r3911894 + r3911895;
        double r3911897 = r3911892 + r3911896;
        double r3911898 = cbrt(r3911897);
        double r3911899 = 2.0;
        double r3911900 = r3911899 + r3911894;
        double r3911901 = r3911898 / r3911900;
        double r3911902 = r3911898 * r3911898;
        double r3911903 = r3911901 * r3911902;
        double r3911904 = r3911903 / r3911900;
        double r3911905 = r3911900 + r3911892;
        double r3911906 = r3911904 / r3911905;
        double r3911907 = 1.0;
        double r3911908 = r3911892 / r3911889;
        double r3911909 = r3911907 - r3911908;
        double r3911910 = 2.0;
        double r3911911 = r3911889 * r3911889;
        double r3911912 = r3911910 / r3911911;
        double r3911913 = r3911909 + r3911912;
        double r3911914 = r3911913 / r3911900;
        double r3911915 = r3911914 / r3911905;
        double r3911916 = r3911891 ? r3911906 : r3911915;
        return r3911916;
}

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 < 2593180110544.6416

    1. Initial program 0.1

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

    2. Simplified0.1

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

    4. Applied +-commutative0.1

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

    6. Applied *-un-lft-identity0.1

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

    7. Applied add-cube-cbrt0.3

      \[\leadsto \frac{\frac{\frac{\color{blue}{\left(\sqrt[3]{1.0 + \left(\left(\beta + \alpha\right) + \alpha \cdot \beta\right)} \cdot \sqrt[3]{1.0 + \left(\left(\beta + \alpha\right) + \alpha \cdot \beta\right)}\right) \cdot \sqrt[3]{1.0 + \left(\left(\beta + \alpha\right) + \alpha \cdot \beta\right)}}}{1 \cdot \left(2 + \left(\beta + \alpha\right)\right)}}{2 + \left(\beta + \alpha\right)}}{\left(2 + \left(\beta + \alpha\right)\right) + 1.0}\]

    8. Applied times-frac0.3

      \[\leadsto \frac{\frac{\color{blue}{\frac{\sqrt[3]{1.0 + \left(\left(\beta + \alpha\right) + \alpha \cdot \beta\right)} \cdot \sqrt[3]{1.0 + \left(\left(\beta + \alpha\right) + \alpha \cdot \beta\right)}}{1} \cdot \frac{\sqrt[3]{1.0 + \left(\left(\beta + \alpha\right) + \alpha \cdot \beta\right)}}{2 + \left(\beta + \alpha\right)}}}{2 + \left(\beta + \alpha\right)}}{\left(2 + \left(\beta + \alpha\right)\right) + 1.0}\]

    if 2593180110544.6416 < alpha

    1. Initial program 11.5

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

    2. Simplified11.5

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

    3. Taylor expanded around inf 10.7

      \[\leadsto \frac{\frac{\color{blue}{\left(2.0 \cdot \frac{1}{{\alpha}^{2}} + 1\right) - 1.0 \cdot \frac{1}{\alpha}}}{2 + \left(\beta + \alpha\right)}}{\left(2 + \left(\beta + \alpha\right)\right) + 1.0}\]

    4. Simplified10.7

      \[\leadsto \frac{\frac{\color{blue}{\left(1 - \frac{1.0}{\alpha}\right) + \frac{2.0}{\alpha \cdot \alpha}}}{2 + \left(\beta + \alpha\right)}}{\left(2 + \left(\beta + \alpha\right)\right) + 1.0}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification3.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 2593180110544.6416:\\ \;\;\;\;\frac{\frac{\frac{\sqrt[3]{1.0 + \left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right)}}{2 + \left(\alpha + \beta\right)} \cdot \left(\sqrt[3]{1.0 + \left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right)} \cdot \sqrt[3]{1.0 + \left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right)}\right)}{2 + \left(\alpha + \beta\right)}}{\left(2 + \left(\alpha + \beta\right)\right) + 1.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(1 - \frac{1.0}{\alpha}\right) + \frac{2.0}{\alpha \cdot \alpha}}{2 + \left(\alpha + \beta\right)}}{\left(2 + \left(\alpha + \beta\right)\right) + 1.0}\\ \end{array}\]

Reproduce

herbie shell --seed 2019152 
(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)))