Average Error: 16.0 → 6.4
Time: 26.4s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 2.3969048622229074 \cdot 10^{+30}:\\ \;\;\;\;\frac{\frac{e^{\log \left(\frac{\left(1.0 \cdot 1.0 + \frac{\alpha}{\left(\alpha + \beta\right) + 2.0} \cdot \left(1.0 + \frac{\alpha}{\left(\alpha + \beta\right) + 2.0}\right)\right) \cdot \beta - \left(\left(\alpha + \beta\right) + 2.0\right) \cdot \left(\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} \cdot \frac{\alpha}{\left(\alpha + \beta\right) + 2.0}\right) \cdot \frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0 \cdot \left(1.0 \cdot 1.0\right)\right)}{\left(\alpha + \beta\right) + 2.0}\right)}}{1.0 \cdot 1.0 + \frac{\alpha}{\left(\alpha + \beta\right) + 2.0} \cdot \left(1.0 + \frac{\alpha}{\left(\alpha + \beta\right) + 2.0}\right)}}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \left(\frac{4.0}{\alpha \cdot \alpha} - \left(\frac{8.0}{\left(\alpha \cdot \alpha\right) \cdot \alpha} + \frac{2.0}{\alpha}\right)\right)}{2.0}\\ \end{array}\]
\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \le 2.3969048622229074 \cdot 10^{+30}:\\
\;\;\;\;\frac{\frac{e^{\log \left(\frac{\left(1.0 \cdot 1.0 + \frac{\alpha}{\left(\alpha + \beta\right) + 2.0} \cdot \left(1.0 + \frac{\alpha}{\left(\alpha + \beta\right) + 2.0}\right)\right) \cdot \beta - \left(\left(\alpha + \beta\right) + 2.0\right) \cdot \left(\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} \cdot \frac{\alpha}{\left(\alpha + \beta\right) + 2.0}\right) \cdot \frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0 \cdot \left(1.0 \cdot 1.0\right)\right)}{\left(\alpha + \beta\right) + 2.0}\right)}}{1.0 \cdot 1.0 + \frac{\alpha}{\left(\alpha + \beta\right) + 2.0} \cdot \left(1.0 + \frac{\alpha}{\left(\alpha + \beta\right) + 2.0}\right)}}{2.0}\\

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

\end{array}
double f(double alpha, double beta) {
        double r5270893 = beta;
        double r5270894 = alpha;
        double r5270895 = r5270893 - r5270894;
        double r5270896 = r5270894 + r5270893;
        double r5270897 = 2.0;
        double r5270898 = r5270896 + r5270897;
        double r5270899 = r5270895 / r5270898;
        double r5270900 = 1.0;
        double r5270901 = r5270899 + r5270900;
        double r5270902 = r5270901 / r5270897;
        return r5270902;
}
↓
double f(double alpha, double beta) {
        double r5270903 = alpha;
        double r5270904 = 2.3969048622229074e+30;
        bool r5270905 = r5270903 <= r5270904;
        double r5270906 = 1.0;
        double r5270907 = r5270906 * r5270906;
        double r5270908 = beta;
        double r5270909 = r5270903 + r5270908;
        double r5270910 = 2.0;
        double r5270911 = r5270909 + r5270910;
        double r5270912 = r5270903 / r5270911;
        double r5270913 = r5270906 + r5270912;
        double r5270914 = r5270912 * r5270913;
        double r5270915 = r5270907 + r5270914;
        double r5270916 = r5270915 * r5270908;
        double r5270917 = r5270912 * r5270912;
        double r5270918 = r5270917 * r5270912;
        double r5270919 = r5270906 * r5270907;
        double r5270920 = r5270918 - r5270919;
        double r5270921 = r5270911 * r5270920;
        double r5270922 = r5270916 - r5270921;
        double r5270923 = r5270922 / r5270911;
        double r5270924 = log(r5270923);
        double r5270925 = exp(r5270924);
        double r5270926 = r5270925 / r5270915;
        double r5270927 = r5270926 / r5270910;
        double r5270928 = r5270908 / r5270911;
        double r5270929 = 4.0;
        double r5270930 = r5270903 * r5270903;
        double r5270931 = r5270929 / r5270930;
        double r5270932 = 8.0;
        double r5270933 = r5270930 * r5270903;
        double r5270934 = r5270932 / r5270933;
        double r5270935 = r5270910 / r5270903;
        double r5270936 = r5270934 + r5270935;
        double r5270937 = r5270931 - r5270936;
        double r5270938 = r5270928 - r5270937;
        double r5270939 = r5270938 / r5270910;
        double r5270940 = r5270905 ? r5270927 : r5270939;
        return r5270940;
}

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 < 2.3969048622229074e+30

    1. Initial program 1.2

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

    3. Applied div-sub1.2

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

    4. Applied associate-+l-1.2

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

    6. Applied flip3--1.2

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

    7. Applied frac-sub1.2

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

    8. Simplified1.2

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

    9. Simplified1.2

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

    11. Applied associate-/r*1.2

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

    13. Applied add-exp-log1.2

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

    if 2.3969048622229074e+30 < alpha

    1. Initial program 50.2

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

    3. Applied div-sub50.2

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

    4. Applied associate-+l-48.5

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

    5. Taylor expanded around -inf 18.5

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

    6. Simplified18.5

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

  4. Final simplification6.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 2.3969048622229074 \cdot 10^{+30}:\\ \;\;\;\;\frac{\frac{e^{\log \left(\frac{\left(1.0 \cdot 1.0 + \frac{\alpha}{\left(\alpha + \beta\right) + 2.0} \cdot \left(1.0 + \frac{\alpha}{\left(\alpha + \beta\right) + 2.0}\right)\right) \cdot \beta - \left(\left(\alpha + \beta\right) + 2.0\right) \cdot \left(\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} \cdot \frac{\alpha}{\left(\alpha + \beta\right) + 2.0}\right) \cdot \frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0 \cdot \left(1.0 \cdot 1.0\right)\right)}{\left(\alpha + \beta\right) + 2.0}\right)}}{1.0 \cdot 1.0 + \frac{\alpha}{\left(\alpha + \beta\right) + 2.0} \cdot \left(1.0 + \frac{\alpha}{\left(\alpha + \beta\right) + 2.0}\right)}}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \left(\frac{4.0}{\alpha \cdot \alpha} - \left(\frac{8.0}{\left(\alpha \cdot \alpha\right) \cdot \alpha} + \frac{2.0}{\alpha}\right)\right)}{2.0}\\ \end{array}\]

Reproduce

herbie shell --seed 2019132 
(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))