Average Error: 3.8 → 2.3
Time: 39.2s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;\beta \le 7.725152532273346988388680017546606101409 \cdot 10^{182}:\\ \;\;\;\;\frac{\frac{\sqrt[3]{\left(\frac{\sqrt{1 + \left(\alpha \cdot \beta + \left(\alpha + \beta\right)\right)}}{\sqrt{\left(\alpha + \beta\right) + 1 \cdot 2}} \cdot \frac{\sqrt{1 + \left(\alpha \cdot \beta + \left(\alpha + \beta\right)\right)}}{\sqrt{\left(\alpha + \beta\right) + 1 \cdot 2}}\right) \cdot \frac{\sqrt{1 + \left(\alpha \cdot \beta + \left(\alpha + \beta\right)\right)}}{\sqrt{\left(\alpha + \beta\right) + 1 \cdot 2}}}}{\frac{\left(\alpha + \beta\right) + 1 \cdot 2}{\frac{\sqrt{1 + \left(\alpha \cdot \beta + \left(\alpha + \beta\right)\right)}}{\sqrt{\left(\alpha + \beta\right) + 1 \cdot 2}}}}}{1 + \left(\left(\alpha + \beta\right) + 1 \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]
\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}
↓
\begin{array}{l}
\mathbf{if}\;\beta \le 7.725152532273346988388680017546606101409 \cdot 10^{182}:\\
\;\;\;\;\frac{\frac{\sqrt[3]{\left(\frac{\sqrt{1 + \left(\alpha \cdot \beta + \left(\alpha + \beta\right)\right)}}{\sqrt{\left(\alpha + \beta\right) + 1 \cdot 2}} \cdot \frac{\sqrt{1 + \left(\alpha \cdot \beta + \left(\alpha + \beta\right)\right)}}{\sqrt{\left(\alpha + \beta\right) + 1 \cdot 2}}\right) \cdot \frac{\sqrt{1 + \left(\alpha \cdot \beta + \left(\alpha + \beta\right)\right)}}{\sqrt{\left(\alpha + \beta\right) + 1 \cdot 2}}}}{\frac{\left(\alpha + \beta\right) + 1 \cdot 2}{\frac{\sqrt{1 + \left(\alpha \cdot \beta + \left(\alpha + \beta\right)\right)}}{\sqrt{\left(\alpha + \beta\right) + 1 \cdot 2}}}}}{1 + \left(\left(\alpha + \beta\right) + 1 \cdot 2\right)}\\

\mathbf{else}:\\
\;\;\;\;0\\

\end{array}
double f(double alpha, double beta) {
        double r4945886 = alpha;
        double r4945887 = beta;
        double r4945888 = r4945886 + r4945887;
        double r4945889 = r4945887 * r4945886;
        double r4945890 = r4945888 + r4945889;
        double r4945891 = 1.0;
        double r4945892 = r4945890 + r4945891;
        double r4945893 = 2.0;
        double r4945894 = r4945893 * r4945891;
        double r4945895 = r4945888 + r4945894;
        double r4945896 = r4945892 / r4945895;
        double r4945897 = r4945896 / r4945895;
        double r4945898 = r4945895 + r4945891;
        double r4945899 = r4945897 / r4945898;
        return r4945899;
}
↓
double f(double alpha, double beta) {
        double r4945900 = beta;
        double r4945901 = 7.725152532273347e+182;
        bool r4945902 = r4945900 <= r4945901;
        double r4945903 = 1.0;
        double r4945904 = alpha;
        double r4945905 = r4945904 * r4945900;
        double r4945906 = r4945904 + r4945900;
        double r4945907 = r4945905 + r4945906;
        double r4945908 = r4945903 + r4945907;
        double r4945909 = sqrt(r4945908);
        double r4945910 = 2.0;
        double r4945911 = r4945903 * r4945910;
        double r4945912 = r4945906 + r4945911;
        double r4945913 = sqrt(r4945912);
        double r4945914 = r4945909 / r4945913;
        double r4945915 = r4945914 * r4945914;
        double r4945916 = r4945915 * r4945914;
        double r4945917 = cbrt(r4945916);
        double r4945918 = r4945912 / r4945914;
        double r4945919 = r4945917 / r4945918;
        double r4945920 = r4945903 + r4945912;
        double r4945921 = r4945919 / r4945920;
        double r4945922 = 0.0;
        double r4945923 = r4945902 ? r4945921 : r4945922;
        return r4945923;
}

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 < 7.725152532273347e+182

    1. Initial program 1.5

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

    3. Applied add-sqr-sqrt2.1

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

    4. Applied add-sqr-sqrt2.0

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

    5. Applied times-frac2.0

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

    6. Applied associate-/l*1.6

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

    8. Applied add-cbrt-cube1.6

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

    if 7.725152532273347e+182 < beta

    1. Initial program 17.5

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

    2. Taylor expanded around inf 6.4

      \[\leadsto \color{blue}{0}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification2.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \le 7.725152532273346988388680017546606101409 \cdot 10^{182}:\\ \;\;\;\;\frac{\frac{\sqrt[3]{\left(\frac{\sqrt{1 + \left(\alpha \cdot \beta + \left(\alpha + \beta\right)\right)}}{\sqrt{\left(\alpha + \beta\right) + 1 \cdot 2}} \cdot \frac{\sqrt{1 + \left(\alpha \cdot \beta + \left(\alpha + \beta\right)\right)}}{\sqrt{\left(\alpha + \beta\right) + 1 \cdot 2}}\right) \cdot \frac{\sqrt{1 + \left(\alpha \cdot \beta + \left(\alpha + \beta\right)\right)}}{\sqrt{\left(\alpha + \beta\right) + 1 \cdot 2}}}}{\frac{\left(\alpha + \beta\right) + 1 \cdot 2}{\frac{\sqrt{1 + \left(\alpha \cdot \beta + \left(\alpha + \beta\right)\right)}}{\sqrt{\left(\alpha + \beta\right) + 1 \cdot 2}}}}}{1 + \left(\left(\alpha + \beta\right) + 1 \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/3"
  :pre (and (> alpha -1.0) (> beta -1.0))
  (/ (/ (/ (+ (+ (+ alpha beta) (* beta alpha)) 1.0) (+ (+ alpha beta) (* 2.0 1.0))) (+ (+ alpha beta) (* 2.0 1.0))) (+ (+ (+ alpha beta) (* 2.0 1.0)) 1.0)))