Average Error: 16.6 → 6.9
Time: 19.6s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 9249179198057380398250710513657839616:\\ \;\;\;\;\frac{\frac{1}{\frac{\left(\alpha + \beta\right) + 2}{\beta}} - \mathsf{fma}\left(\alpha, \frac{1}{\left(\alpha + \beta\right) + 2}, -1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{\sqrt[3]{\frac{\left(\alpha + \beta\right) + 2}{\beta}} \cdot \sqrt[3]{\frac{\left(\alpha + \beta\right) + 2}{\beta}}}}{\sqrt[3]{\frac{\left(\alpha + \beta\right) + 2}{\beta}}} - \left(\left(\frac{4}{{\alpha}^{2}} - \frac{2}{\alpha}\right) - \frac{8}{{\alpha}^{3}}\right)}{2}\\ \end{array}\]
\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \le 9249179198057380398250710513657839616:\\
\;\;\;\;\frac{\frac{1}{\frac{\left(\alpha + \beta\right) + 2}{\beta}} - \mathsf{fma}\left(\alpha, \frac{1}{\left(\alpha + \beta\right) + 2}, -1\right)}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{\frac{1}{\sqrt[3]{\frac{\left(\alpha + \beta\right) + 2}{\beta}} \cdot \sqrt[3]{\frac{\left(\alpha + \beta\right) + 2}{\beta}}}}{\sqrt[3]{\frac{\left(\alpha + \beta\right) + 2}{\beta}}} - \left(\left(\frac{4}{{\alpha}^{2}} - \frac{2}{\alpha}\right) - \frac{8}{{\alpha}^{3}}\right)}{2}\\

\end{array}
double f(double alpha, double beta) {
        double r72897 = beta;
        double r72898 = alpha;
        double r72899 = r72897 - r72898;
        double r72900 = r72898 + r72897;
        double r72901 = 2.0;
        double r72902 = r72900 + r72901;
        double r72903 = r72899 / r72902;
        double r72904 = 1.0;
        double r72905 = r72903 + r72904;
        double r72906 = r72905 / r72901;
        return r72906;
}
↓
double f(double alpha, double beta) {
        double r72907 = alpha;
        double r72908 = 9.24917919805738e+36;
        bool r72909 = r72907 <= r72908;
        double r72910 = 1.0;
        double r72911 = beta;
        double r72912 = r72907 + r72911;
        double r72913 = 2.0;
        double r72914 = r72912 + r72913;
        double r72915 = r72914 / r72911;
        double r72916 = r72910 / r72915;
        double r72917 = r72910 / r72914;
        double r72918 = 1.0;
        double r72919 = -r72918;
        double r72920 = fma(r72907, r72917, r72919);
        double r72921 = r72916 - r72920;
        double r72922 = r72921 / r72913;
        double r72923 = cbrt(r72915);
        double r72924 = r72923 * r72923;
        double r72925 = r72910 / r72924;
        double r72926 = r72925 / r72923;
        double r72927 = 4.0;
        double r72928 = 2.0;
        double r72929 = pow(r72907, r72928);
        double r72930 = r72927 / r72929;
        double r72931 = r72913 / r72907;
        double r72932 = r72930 - r72931;
        double r72933 = 8.0;
        double r72934 = 3.0;
        double r72935 = pow(r72907, r72934);
        double r72936 = r72933 / r72935;
        double r72937 = r72932 - r72936;
        double r72938 = r72926 - r72937;
        double r72939 = r72938 / r72913;
        double r72940 = r72909 ? r72922 : r72939;
        return r72940;
}

Error

Bits error versus alpha

Bits error versus beta

Derivation

  1. Split input into 2 regimes

  2. if alpha < 9.24917919805738e+36

    1. Initial program 2.0

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

    3. Applied div-sub2.0

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

    4. Applied associate-+l-2.0

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

    6. Applied clear-num2.0

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

    8. Applied div-inv2.0

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

    9. Applied fma-neg1.9

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

    if 9.24917919805738e+36 < alpha

    1. Initial program 50.5

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

    3. Applied div-sub50.5

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

    4. Applied associate-+l-49.0

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

    6. Applied clear-num49.0

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

    8. Applied add-cube-cbrt49.0

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

    9. Applied associate-/r*49.0

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

    10. Taylor expanded around inf 18.3

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

    11. Simplified18.3

      \[\leadsto \frac{\frac{\frac{1}{\sqrt[3]{\frac{\left(\alpha + \beta\right) + 2}{\beta}} \cdot \sqrt[3]{\frac{\left(\alpha + \beta\right) + 2}{\beta}}}}{\sqrt[3]{\frac{\left(\alpha + \beta\right) + 2}{\beta}}} - \color{blue}{\left(\left(\frac{4}{{\alpha}^{2}} - \frac{2}{\alpha}\right) - \frac{8}{{\alpha}^{3}}\right)}}{2}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification6.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 9249179198057380398250710513657839616:\\ \;\;\;\;\frac{\frac{1}{\frac{\left(\alpha + \beta\right) + 2}{\beta}} - \mathsf{fma}\left(\alpha, \frac{1}{\left(\alpha + \beta\right) + 2}, -1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{\sqrt[3]{\frac{\left(\alpha + \beta\right) + 2}{\beta}} \cdot \sqrt[3]{\frac{\left(\alpha + \beta\right) + 2}{\beta}}}}{\sqrt[3]{\frac{\left(\alpha + \beta\right) + 2}{\beta}}} - \left(\left(\frac{4}{{\alpha}^{2}} - \frac{2}{\alpha}\right) - \frac{8}{{\alpha}^{3}}\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019326 +o rules:numerics
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/1"
  :precision binary64
  :pre (and (> alpha -1) (> beta -1))
  (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2)) 1) 2))