Average Error: 16.0 → 5.9
Time: 29.5s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 3.630174269413438 \cdot 10^{+21}:\\ \;\;\;\;\frac{e^{\log \left(\frac{\beta}{2.0 + \left(\beta + \alpha\right)} - \left(\frac{\alpha}{2.0 + \left(\beta + \alpha\right)} - 1.0\right)\right)}}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{2.0 + \left(\beta + \alpha\right)} - \left(\left(\frac{4.0}{\alpha \cdot \alpha} - \frac{2.0}{\alpha}\right) - \frac{8.0}{\alpha \cdot \left(\alpha \cdot \alpha\right)}\right)}{2.0}\\ \end{array}\]
\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \le 3.630174269413438 \cdot 10^{+21}:\\
\;\;\;\;\frac{e^{\log \left(\frac{\beta}{2.0 + \left(\beta + \alpha\right)} - \left(\frac{\alpha}{2.0 + \left(\beta + \alpha\right)} - 1.0\right)\right)}}{2.0}\\

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

\end{array}
double f(double alpha, double beta) {
        double r3090822 = beta;
        double r3090823 = alpha;
        double r3090824 = r3090822 - r3090823;
        double r3090825 = r3090823 + r3090822;
        double r3090826 = 2.0;
        double r3090827 = r3090825 + r3090826;
        double r3090828 = r3090824 / r3090827;
        double r3090829 = 1.0;
        double r3090830 = r3090828 + r3090829;
        double r3090831 = r3090830 / r3090826;
        return r3090831;
}
↓
double f(double alpha, double beta) {
        double r3090832 = alpha;
        double r3090833 = 3.630174269413438e+21;
        bool r3090834 = r3090832 <= r3090833;
        double r3090835 = beta;
        double r3090836 = 2.0;
        double r3090837 = r3090835 + r3090832;
        double r3090838 = r3090836 + r3090837;
        double r3090839 = r3090835 / r3090838;
        double r3090840 = r3090832 / r3090838;
        double r3090841 = 1.0;
        double r3090842 = r3090840 - r3090841;
        double r3090843 = r3090839 - r3090842;
        double r3090844 = log(r3090843);
        double r3090845 = exp(r3090844);
        double r3090846 = r3090845 / r3090836;
        double r3090847 = 4.0;
        double r3090848 = r3090832 * r3090832;
        double r3090849 = r3090847 / r3090848;
        double r3090850 = r3090836 / r3090832;
        double r3090851 = r3090849 - r3090850;
        double r3090852 = 8.0;
        double r3090853 = r3090832 * r3090848;
        double r3090854 = r3090852 / r3090853;
        double r3090855 = r3090851 - r3090854;
        double r3090856 = r3090839 - r3090855;
        double r3090857 = r3090856 / r3090836;
        double r3090858 = r3090834 ? r3090846 : r3090857;
        return r3090858;
}

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 < 3.630174269413438e+21

    1. Initial program 0.7

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

    3. Applied div-sub0.7

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

    4. Applied associate-+l-0.7

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

    6. Applied add-exp-log0.7

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

    if 3.630174269413438e+21 < alpha

    1. Initial program 50.6

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

    3. Applied div-sub50.6

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

    4. Applied associate-+l-49.0

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

    5. Taylor expanded around -inf 17.9

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

    6. Simplified17.9

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

  4. Final simplification5.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 3.630174269413438 \cdot 10^{+21}:\\ \;\;\;\;\frac{e^{\log \left(\frac{\beta}{2.0 + \left(\beta + \alpha\right)} - \left(\frac{\alpha}{2.0 + \left(\beta + \alpha\right)} - 1.0\right)\right)}}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{2.0 + \left(\beta + \alpha\right)} - \left(\left(\frac{4.0}{\alpha \cdot \alpha} - \frac{2.0}{\alpha}\right) - \frac{8.0}{\alpha \cdot \left(\alpha \cdot \alpha\right)}\right)}{2.0}\\ \end{array}\]

Reproduce

herbie shell --seed 2019141 
(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))