Average Error: 3.7 → 2.9
Time: 29.2s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 1.790144384427998241951969767039972903026 \cdot 10^{55}:\\ \;\;\;\;\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \frac{1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(1 - 1 \cdot \frac{1}{\alpha}\right) + \frac{\frac{2}{\alpha}}{\alpha}\right) \cdot \frac{1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\\ \end{array}\]
\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \le 1.790144384427998241951969767039972903026 \cdot 10^{55}:\\
\;\;\;\;\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \frac{1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\\

\mathbf{else}:\\
\;\;\;\;\frac{\left(\left(1 - 1 \cdot \frac{1}{\alpha}\right) + \frac{\frac{2}{\alpha}}{\alpha}\right) \cdot \frac{1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\\

\end{array}
double f(double alpha, double beta) {
        double r92772 = alpha;
        double r92773 = beta;
        double r92774 = r92772 + r92773;
        double r92775 = r92773 * r92772;
        double r92776 = r92774 + r92775;
        double r92777 = 1.0;
        double r92778 = r92776 + r92777;
        double r92779 = 2.0;
        double r92780 = r92779 * r92777;
        double r92781 = r92774 + r92780;
        double r92782 = r92778 / r92781;
        double r92783 = r92782 / r92781;
        double r92784 = r92781 + r92777;
        double r92785 = r92783 / r92784;
        return r92785;
}
↓
double f(double alpha, double beta) {
        double r92786 = alpha;
        double r92787 = 1.7901443844279982e+55;
        bool r92788 = r92786 <= r92787;
        double r92789 = beta;
        double r92790 = r92786 + r92789;
        double r92791 = r92789 * r92786;
        double r92792 = r92790 + r92791;
        double r92793 = 1.0;
        double r92794 = r92792 + r92793;
        double r92795 = 2.0;
        double r92796 = r92795 * r92793;
        double r92797 = r92790 + r92796;
        double r92798 = r92794 / r92797;
        double r92799 = 1.0;
        double r92800 = r92799 / r92797;
        double r92801 = r92798 * r92800;
        double r92802 = r92797 + r92793;
        double r92803 = r92801 / r92802;
        double r92804 = r92799 / r92786;
        double r92805 = r92793 * r92804;
        double r92806 = r92799 - r92805;
        double r92807 = r92795 / r92786;
        double r92808 = r92807 / r92786;
        double r92809 = r92806 + r92808;
        double r92810 = r92809 * r92800;
        double r92811 = r92810 / r92802;
        double r92812 = r92788 ? r92803 : r92811;
        return r92812;
}

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 < 1.7901443844279982e+55

    1. Initial program 0.3

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

    3. Applied div-inv0.3

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

    if 1.7901443844279982e+55 < alpha

    1. Initial program 12.5

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

    3. Applied div-inv12.5

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

    4. Taylor expanded around inf 9.6

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

    5. Simplified9.6

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

  4. Final simplification2.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 1.790144384427998241951969767039972903026 \cdot 10^{55}:\\ \;\;\;\;\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \frac{1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(1 - 1 \cdot \frac{1}{\alpha}\right) + \frac{\frac{2}{\alpha}}{\alpha}\right) \cdot \frac{1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2019294 
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/3"
  :precision binary64
  :pre (and (> alpha -1) (> beta -1))
  (/ (/ (/ (+ (+ (+ alpha beta) (* beta alpha)) 1) (+ (+ alpha beta) (* 2 1))) (+ (+ alpha beta) (* 2 1))) (+ (+ (+ alpha beta) (* 2 1)) 1)))