Average Error: 16.6 → 6.1
Time: 16.3s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 14037838382120007680:\\ \;\;\;\;\frac{\sqrt[3]{\frac{\beta}{2 + \left(\beta + \alpha\right)}} \cdot \left(\sqrt[3]{\frac{\beta}{2 + \left(\beta + \alpha\right)}} \cdot \left(\log \left(\sqrt{e^{\sqrt[3]{\frac{\beta}{\alpha + \left(\beta + 2\right)}}}}\right) + \log \left(\sqrt{e^{\sqrt[3]{\frac{\beta}{\alpha + \left(\beta + 2\right)}}}}\right)\right)\right) - \left(\frac{\alpha}{\alpha + \left(\beta + 2\right)} - 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{2 + \left(\beta + \alpha\right)} - \left(\left(\frac{4}{\alpha \cdot \alpha} - \frac{2}{\alpha}\right) - \frac{8}{{\alpha}^{3}}\right)}{2}\\ \end{array}\]
\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \le 14037838382120007680:\\
\;\;\;\;\frac{\sqrt[3]{\frac{\beta}{2 + \left(\beta + \alpha\right)}} \cdot \left(\sqrt[3]{\frac{\beta}{2 + \left(\beta + \alpha\right)}} \cdot \left(\log \left(\sqrt{e^{\sqrt[3]{\frac{\beta}{\alpha + \left(\beta + 2\right)}}}}\right) + \log \left(\sqrt{e^{\sqrt[3]{\frac{\beta}{\alpha + \left(\beta + 2\right)}}}}\right)\right)\right) - \left(\frac{\alpha}{\alpha + \left(\beta + 2\right)} - 1\right)}{2}\\

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

\end{array}
double f(double alpha, double beta) {
        double r78947 = beta;
        double r78948 = alpha;
        double r78949 = r78947 - r78948;
        double r78950 = r78948 + r78947;
        double r78951 = 2.0;
        double r78952 = r78950 + r78951;
        double r78953 = r78949 / r78952;
        double r78954 = 1.0;
        double r78955 = r78953 + r78954;
        double r78956 = r78955 / r78951;
        return r78956;
}
↓
double f(double alpha, double beta) {
        double r78957 = alpha;
        double r78958 = 1.4037838382120008e+19;
        bool r78959 = r78957 <= r78958;
        double r78960 = beta;
        double r78961 = 2.0;
        double r78962 = r78960 + r78957;
        double r78963 = r78961 + r78962;
        double r78964 = r78960 / r78963;
        double r78965 = cbrt(r78964);
        double r78966 = r78960 + r78961;
        double r78967 = r78957 + r78966;
        double r78968 = r78960 / r78967;
        double r78969 = cbrt(r78968);
        double r78970 = exp(r78969);
        double r78971 = sqrt(r78970);
        double r78972 = log(r78971);
        double r78973 = r78972 + r78972;
        double r78974 = r78965 * r78973;
        double r78975 = r78965 * r78974;
        double r78976 = r78957 / r78967;
        double r78977 = 1.0;
        double r78978 = r78976 - r78977;
        double r78979 = r78975 - r78978;
        double r78980 = r78979 / r78961;
        double r78981 = 4.0;
        double r78982 = r78957 * r78957;
        double r78983 = r78981 / r78982;
        double r78984 = r78961 / r78957;
        double r78985 = r78983 - r78984;
        double r78986 = 8.0;
        double r78987 = 3.0;
        double r78988 = pow(r78957, r78987);
        double r78989 = r78986 / r78988;
        double r78990 = r78985 - r78989;
        double r78991 = r78964 - r78990;
        double r78992 = r78991 / r78961;
        double r78993 = r78959 ? r78980 : r78992;
        return r78993;
}

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 < 1.4037838382120008e+19

    1. Initial program 0.6

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

    3. Applied div-sub0.6

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

    4. Applied associate-+l-0.6

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

    5. Simplified0.6

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

    7. Applied add-cube-cbrt0.6

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

    8. Simplified0.6

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

    9. Simplified0.6

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

    11. Applied add-log-exp0.6

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

    12. Simplified0.6

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

    14. Applied add-sqr-sqrt0.6

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

    15. Applied log-prod0.6

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

    16. Simplified0.6

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

    17. Simplified0.6

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

    if 1.4037838382120008e+19 < alpha

    1. Initial program 50.6

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

    3. Applied div-sub50.6

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

    4. Applied associate-+l-49.1

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

    5. Simplified49.1

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

    6. Taylor expanded around inf 17.9

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

    7. Simplified17.9

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

  4. Final simplification6.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 14037838382120007680:\\ \;\;\;\;\frac{\sqrt[3]{\frac{\beta}{2 + \left(\beta + \alpha\right)}} \cdot \left(\sqrt[3]{\frac{\beta}{2 + \left(\beta + \alpha\right)}} \cdot \left(\log \left(\sqrt{e^{\sqrt[3]{\frac{\beta}{\alpha + \left(\beta + 2\right)}}}}\right) + \log \left(\sqrt{e^{\sqrt[3]{\frac{\beta}{\alpha + \left(\beta + 2\right)}}}}\right)\right)\right) - \left(\frac{\alpha}{\alpha + \left(\beta + 2\right)} - 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{2 + \left(\beta + \alpha\right)} - \left(\left(\frac{4}{\alpha \cdot \alpha} - \frac{2}{\alpha}\right) - \frac{8}{{\alpha}^{3}}\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019195 +o rules:numerics
(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))