Average Error: 15.9 → 6.1
Time: 18.4s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 2.081929906225012 \cdot 10^{+19}:\\ \;\;\;\;\frac{1.0 + \left(\beta - \alpha\right) \cdot \frac{1}{2.0 + \left(\beta + \alpha\right)}}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{2.0 + \left(\beta + \alpha\right)} - \left(\left(\frac{4.0}{\alpha \cdot \alpha} - \frac{2.0}{\alpha}\right) - \frac{8.0}{\alpha \cdot \left(\alpha \cdot \alpha\right)}\right)}{2.0}\\ \end{array}\]
\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \le 2.081929906225012 \cdot 10^{+19}:\\
\;\;\;\;\frac{1.0 + \left(\beta - \alpha\right) \cdot \frac{1}{2.0 + \left(\beta + \alpha\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 r2566748 = beta;
        double r2566749 = alpha;
        double r2566750 = r2566748 - r2566749;
        double r2566751 = r2566749 + r2566748;
        double r2566752 = 2.0;
        double r2566753 = r2566751 + r2566752;
        double r2566754 = r2566750 / r2566753;
        double r2566755 = 1.0;
        double r2566756 = r2566754 + r2566755;
        double r2566757 = r2566756 / r2566752;
        return r2566757;
}
↓
double f(double alpha, double beta) {
        double r2566758 = alpha;
        double r2566759 = 2.081929906225012e+19;
        bool r2566760 = r2566758 <= r2566759;
        double r2566761 = 1.0;
        double r2566762 = beta;
        double r2566763 = r2566762 - r2566758;
        double r2566764 = 1.0;
        double r2566765 = 2.0;
        double r2566766 = r2566762 + r2566758;
        double r2566767 = r2566765 + r2566766;
        double r2566768 = r2566764 / r2566767;
        double r2566769 = r2566763 * r2566768;
        double r2566770 = r2566761 + r2566769;
        double r2566771 = r2566770 / r2566765;
        double r2566772 = r2566762 / r2566767;
        double r2566773 = 4.0;
        double r2566774 = r2566758 * r2566758;
        double r2566775 = r2566773 / r2566774;
        double r2566776 = r2566765 / r2566758;
        double r2566777 = r2566775 - r2566776;
        double r2566778 = 8.0;
        double r2566779 = r2566758 * r2566774;
        double r2566780 = r2566778 / r2566779;
        double r2566781 = r2566777 - r2566780;
        double r2566782 = r2566772 - r2566781;
        double r2566783 = r2566782 / r2566765;
        double r2566784 = r2566760 ? r2566771 : r2566783;
        return r2566784;
}

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 < 2.081929906225012e+19

    1. Initial program 0.6

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

    3. Applied div-inv0.6

      \[\leadsto \frac{\color{blue}{\left(\beta - \alpha\right) \cdot \frac{1}{\left(\alpha + \beta\right) + 2.0}} + 1.0}{2.0}\]

    if 2.081929906225012e+19 < alpha

    1. Initial program 50.1

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

    3. Applied div-sub50.1

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

    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 18.4

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

    6. Simplified18.4

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

  4. Final simplification6.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 2.081929906225012 \cdot 10^{+19}:\\ \;\;\;\;\frac{1.0 + \left(\beta - \alpha\right) \cdot \frac{1}{2.0 + \left(\beta + \alpha\right)}}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{2.0 + \left(\beta + \alpha\right)} - \left(\left(\frac{4.0}{\alpha \cdot \alpha} - \frac{2.0}{\alpha}\right) - \frac{8.0}{\alpha \cdot \left(\alpha \cdot \alpha\right)}\right)}{2.0}\\ \end{array}\]

Reproduce

herbie shell --seed 2019135 
(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))