Average Error: 15.7 → 3.0
Time: 18.8s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} \le -0.9999999999999637:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \left(\frac{4.0}{\alpha \cdot \alpha} - \left(\frac{8.0}{\left(\alpha \cdot \alpha\right) \cdot \alpha} + \frac{2.0}{\alpha}\right)\right)}{2.0}\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0\right)}{2.0}\right)}\\ \end{array}\]
\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}
↓
\begin{array}{l}
\mathbf{if}\;\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} \le -0.9999999999999637:\\
\;\;\;\;\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}\\

\mathbf{else}:\\
\;\;\;\;e^{\log \left(\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0\right)}{2.0}\right)}\\

\end{array}
double f(double alpha, double beta) {
        double r1644918 = beta;
        double r1644919 = alpha;
        double r1644920 = r1644918 - r1644919;
        double r1644921 = r1644919 + r1644918;
        double r1644922 = 2.0;
        double r1644923 = r1644921 + r1644922;
        double r1644924 = r1644920 / r1644923;
        double r1644925 = 1.0;
        double r1644926 = r1644924 + r1644925;
        double r1644927 = r1644926 / r1644922;
        return r1644927;
}
↓
double f(double alpha, double beta) {
        double r1644928 = beta;
        double r1644929 = alpha;
        double r1644930 = r1644928 - r1644929;
        double r1644931 = r1644929 + r1644928;
        double r1644932 = 2.0;
        double r1644933 = r1644931 + r1644932;
        double r1644934 = r1644930 / r1644933;
        double r1644935 = -0.9999999999999637;
        bool r1644936 = r1644934 <= r1644935;
        double r1644937 = r1644928 / r1644933;
        double r1644938 = 4.0;
        double r1644939 = r1644929 * r1644929;
        double r1644940 = r1644938 / r1644939;
        double r1644941 = 8.0;
        double r1644942 = r1644939 * r1644929;
        double r1644943 = r1644941 / r1644942;
        double r1644944 = r1644932 / r1644929;
        double r1644945 = r1644943 + r1644944;
        double r1644946 = r1644940 - r1644945;
        double r1644947 = r1644937 - r1644946;
        double r1644948 = r1644947 / r1644932;
        double r1644949 = r1644929 / r1644933;
        double r1644950 = 1.0;
        double r1644951 = r1644949 - r1644950;
        double r1644952 = r1644937 - r1644951;
        double r1644953 = r1644952 / r1644932;
        double r1644954 = log(r1644953);
        double r1644955 = exp(r1644954);
        double r1644956 = r1644936 ? r1644948 : r1644955;
        return r1644956;
}

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 alpha) (+ (+ alpha beta) 2.0)) < -0.9999999999999637

    1. Initial program 60.4

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

    3. Applied div-sub60.4

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

    4. Applied associate-+l-58.7

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

    6. Applied add-log-exp58.7

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

    7. Taylor expanded around -inf 10.6

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

    8. Simplified10.6

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

    if -0.9999999999999637 < (/ (- beta alpha) (+ (+ alpha beta) 2.0))

    1. Initial program 0.4

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

    3. Applied div-sub0.4

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

    4. Applied associate-+l-0.4

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

    6. Applied add-exp-log0.4

      \[\leadsto \color{blue}{e^{\log \left(\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0\right)}{2.0}\right)}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification3.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} \le -0.9999999999999637:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \left(\frac{4.0}{\alpha \cdot \alpha} - \left(\frac{8.0}{\left(\alpha \cdot \alpha\right) \cdot \alpha} + \frac{2.0}{\alpha}\right)\right)}{2.0}\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0\right)}{2.0}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019153 
(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))