Average Error: 16.2 → 6.7
Time: 1.5m
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 2.370766474895455 \cdot 10^{+23}:\\ \;\;\;\;e^{\sqrt[3]{\left(\left(\log \left(\frac{\beta}{2.0 + \left(\beta + \alpha\right)} - \left(\frac{\alpha}{2.0 + \left(\beta + \alpha\right)} - 1.0\right)\right) - \log 2.0\right) \cdot \left(\log \left(\frac{\beta}{2.0 + \left(\beta + \alpha\right)} - \left(\frac{\alpha}{2.0 + \left(\beta + \alpha\right)} - 1.0\right)\right) - \log 2.0\right)\right) \cdot \left(\log \left(\frac{\beta}{2.0 + \left(\beta + \alpha\right)} - \left(\frac{\alpha}{2.0 + \left(\beta + \alpha\right)} - 1.0\right)\right) - \log 2.0\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{2.0 + \left(\beta + \alpha\right)} - \left(\frac{4.0}{\alpha \cdot \alpha} - \left(\frac{2.0}{\alpha} + \frac{\frac{8.0}{\alpha \cdot \alpha}}{\alpha}\right)\right)}{2.0}\\ \end{array}\]
\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \le 2.370766474895455 \cdot 10^{+23}:\\
\;\;\;\;e^{\sqrt[3]{\left(\left(\log \left(\frac{\beta}{2.0 + \left(\beta + \alpha\right)} - \left(\frac{\alpha}{2.0 + \left(\beta + \alpha\right)} - 1.0\right)\right) - \log 2.0\right) \cdot \left(\log \left(\frac{\beta}{2.0 + \left(\beta + \alpha\right)} - \left(\frac{\alpha}{2.0 + \left(\beta + \alpha\right)} - 1.0\right)\right) - \log 2.0\right)\right) \cdot \left(\log \left(\frac{\beta}{2.0 + \left(\beta + \alpha\right)} - \left(\frac{\alpha}{2.0 + \left(\beta + \alpha\right)} - 1.0\right)\right) - \log 2.0\right)}}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{\beta}{2.0 + \left(\beta + \alpha\right)} - \left(\frac{4.0}{\alpha \cdot \alpha} - \left(\frac{2.0}{\alpha} + \frac{\frac{8.0}{\alpha \cdot \alpha}}{\alpha}\right)\right)}{2.0}\\

\end{array}
double f(double alpha, double beta) {
        double r20854950 = beta;
        double r20854951 = alpha;
        double r20854952 = r20854950 - r20854951;
        double r20854953 = r20854951 + r20854950;
        double r20854954 = 2.0;
        double r20854955 = r20854953 + r20854954;
        double r20854956 = r20854952 / r20854955;
        double r20854957 = 1.0;
        double r20854958 = r20854956 + r20854957;
        double r20854959 = r20854958 / r20854954;
        return r20854959;
}
↓
double f(double alpha, double beta) {
        double r20854960 = alpha;
        double r20854961 = 2.370766474895455e+23;
        bool r20854962 = r20854960 <= r20854961;
        double r20854963 = beta;
        double r20854964 = 2.0;
        double r20854965 = r20854963 + r20854960;
        double r20854966 = r20854964 + r20854965;
        double r20854967 = r20854963 / r20854966;
        double r20854968 = r20854960 / r20854966;
        double r20854969 = 1.0;
        double r20854970 = r20854968 - r20854969;
        double r20854971 = r20854967 - r20854970;
        double r20854972 = log(r20854971);
        double r20854973 = log(r20854964);
        double r20854974 = r20854972 - r20854973;
        double r20854975 = r20854974 * r20854974;
        double r20854976 = r20854975 * r20854974;
        double r20854977 = cbrt(r20854976);
        double r20854978 = exp(r20854977);
        double r20854979 = 4.0;
        double r20854980 = r20854960 * r20854960;
        double r20854981 = r20854979 / r20854980;
        double r20854982 = r20854964 / r20854960;
        double r20854983 = 8.0;
        double r20854984 = r20854983 / r20854980;
        double r20854985 = r20854984 / r20854960;
        double r20854986 = r20854982 + r20854985;
        double r20854987 = r20854981 - r20854986;
        double r20854988 = r20854967 - r20854987;
        double r20854989 = r20854988 / r20854964;
        double r20854990 = r20854962 ? r20854978 : r20854989;
        return r20854990;
}

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 < 2.370766474895455e+23

    1. Initial program 0.9

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}\]
    2. Using strategy rm

    3. Applied div-sub0.9

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

    4. Applied associate-+l-0.9

      \[\leadsto \frac{\color{blue}{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0\right)}}{2.0}\]
    5. Using strategy rm

    6. Applied add-exp-log0.9

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

    7. Applied add-exp-log0.9

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

    8. Applied div-exp0.9

      \[\leadsto \color{blue}{e^{\log \left(\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0\right)\right) - \log 2.0}}\]
    9. Using strategy rm

    10. Applied add-cbrt-cube0.9

      \[\leadsto e^{\color{blue}{\sqrt[3]{\left(\left(\log \left(\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0\right)\right) - \log 2.0\right) \cdot \left(\log \left(\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0\right)\right) - \log 2.0\right)\right) \cdot \left(\log \left(\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0\right)\right) - \log 2.0\right)}}}\]

    if 2.370766474895455e+23 < alpha

    1. Initial program 49.9

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}\]
    2. Using strategy rm

    3. Applied div-sub49.9

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

    4. Applied associate-+l-48.2

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

    5. Taylor expanded around inf 19.2

      \[\leadsto \frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \color{blue}{\left(4.0 \cdot \frac{1}{{\alpha}^{2}} - \left(2.0 \cdot \frac{1}{\alpha} + 8.0 \cdot \frac{1}{{\alpha}^{3}}\right)\right)}}{2.0}\]

    6. Simplified19.2

      \[\leadsto \frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \color{blue}{\left(\frac{4.0}{\alpha \cdot \alpha} - \left(\frac{\frac{8.0}{\alpha \cdot \alpha}}{\alpha} + \frac{2.0}{\alpha}\right)\right)}}{2.0}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification6.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 2.370766474895455 \cdot 10^{+23}:\\ \;\;\;\;e^{\sqrt[3]{\left(\left(\log \left(\frac{\beta}{2.0 + \left(\beta + \alpha\right)} - \left(\frac{\alpha}{2.0 + \left(\beta + \alpha\right)} - 1.0\right)\right) - \log 2.0\right) \cdot \left(\log \left(\frac{\beta}{2.0 + \left(\beta + \alpha\right)} - \left(\frac{\alpha}{2.0 + \left(\beta + \alpha\right)} - 1.0\right)\right) - \log 2.0\right)\right) \cdot \left(\log \left(\frac{\beta}{2.0 + \left(\beta + \alpha\right)} - \left(\frac{\alpha}{2.0 + \left(\beta + \alpha\right)} - 1.0\right)\right) - \log 2.0\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{2.0 + \left(\beta + \alpha\right)} - \left(\frac{4.0}{\alpha \cdot \alpha} - \left(\frac{2.0}{\alpha} + \frac{\frac{8.0}{\alpha \cdot \alpha}}{\alpha}\right)\right)}{2.0}\\ \end{array}\]

Reproduce

herbie shell --seed 2019119 +o rules:numerics
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/1"
  :pre (and (> alpha -1) (> beta -1))
  (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0))