Average Error: 3.6 → 2.3
Time: 39.5s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\beta \le 1.1044946435875943 \cdot 10^{+164}:\\ \;\;\;\;\frac{\sqrt{\frac{1.0 + \left(\alpha \cdot \beta + \left(\alpha + \beta\right)\right)}{\left(\alpha + \beta\right) + 2}} \cdot \sqrt{\frac{\frac{1.0 + \left(\alpha \cdot \beta + \left(\alpha + \beta\right)\right)}{\left(\alpha + \beta\right) + 2}}{\left(\alpha + \beta\right) + 2}}}{\left(1.0 + \left(\left(\alpha + \beta\right) + 2\right)\right) \cdot \sqrt{\left(\alpha + \beta\right) + 2}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]
\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}
↓
\begin{array}{l}
\mathbf{if}\;\beta \le 1.1044946435875943 \cdot 10^{+164}:\\
\;\;\;\;\frac{\sqrt{\frac{1.0 + \left(\alpha \cdot \beta + \left(\alpha + \beta\right)\right)}{\left(\alpha + \beta\right) + 2}} \cdot \sqrt{\frac{\frac{1.0 + \left(\alpha \cdot \beta + \left(\alpha + \beta\right)\right)}{\left(\alpha + \beta\right) + 2}}{\left(\alpha + \beta\right) + 2}}}{\left(1.0 + \left(\left(\alpha + \beta\right) + 2\right)\right) \cdot \sqrt{\left(\alpha + \beta\right) + 2}}\\

\mathbf{else}:\\
\;\;\;\;0\\

\end{array}
double f(double alpha, double beta) {
        double r4090689 = alpha;
        double r4090690 = beta;
        double r4090691 = r4090689 + r4090690;
        double r4090692 = r4090690 * r4090689;
        double r4090693 = r4090691 + r4090692;
        double r4090694 = 1.0;
        double r4090695 = r4090693 + r4090694;
        double r4090696 = 2.0;
        double r4090697 = 1.0;
        double r4090698 = r4090696 * r4090697;
        double r4090699 = r4090691 + r4090698;
        double r4090700 = r4090695 / r4090699;
        double r4090701 = r4090700 / r4090699;
        double r4090702 = r4090699 + r4090694;
        double r4090703 = r4090701 / r4090702;
        return r4090703;
}
↓
double f(double alpha, double beta) {
        double r4090704 = beta;
        double r4090705 = 1.1044946435875943e+164;
        bool r4090706 = r4090704 <= r4090705;
        double r4090707 = 1.0;
        double r4090708 = alpha;
        double r4090709 = r4090708 * r4090704;
        double r4090710 = r4090708 + r4090704;
        double r4090711 = r4090709 + r4090710;
        double r4090712 = r4090707 + r4090711;
        double r4090713 = 2.0;
        double r4090714 = r4090710 + r4090713;
        double r4090715 = r4090712 / r4090714;
        double r4090716 = sqrt(r4090715);
        double r4090717 = r4090715 / r4090714;
        double r4090718 = sqrt(r4090717);
        double r4090719 = r4090716 * r4090718;
        double r4090720 = r4090707 + r4090714;
        double r4090721 = sqrt(r4090714);
        double r4090722 = r4090720 * r4090721;
        double r4090723 = r4090719 / r4090722;
        double r4090724 = 0.0;
        double r4090725 = r4090706 ? r4090723 : r4090724;
        return r4090725;
}

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 < 1.1044946435875943e+164

    1. Initial program 1.3

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]
    2. Using strategy rm

    3. Applied add-sqr-sqrt1.3

      \[\leadsto \frac{\color{blue}{\sqrt{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}} \cdot \sqrt{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]
    4. Using strategy rm

    5. Applied sqrt-div1.3

      \[\leadsto \frac{\color{blue}{\frac{\sqrt{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}}} \cdot \sqrt{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]

    6. Applied associate-*l/1.3

      \[\leadsto \frac{\color{blue}{\frac{\sqrt{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}} \cdot \sqrt{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]

    7. Applied associate-/l/1.3

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}} \cdot \sqrt{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0\right) \cdot \sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}}}\]

    if 1.1044946435875943e+164 < beta

    1. Initial program 16.4

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]

    2. Taylor expanded around inf 7.9

      \[\leadsto \color{blue}{0}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification2.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \le 1.1044946435875943 \cdot 10^{+164}:\\ \;\;\;\;\frac{\sqrt{\frac{1.0 + \left(\alpha \cdot \beta + \left(\alpha + \beta\right)\right)}{\left(\alpha + \beta\right) + 2}} \cdot \sqrt{\frac{\frac{1.0 + \left(\alpha \cdot \beta + \left(\alpha + \beta\right)\right)}{\left(\alpha + \beta\right) + 2}}{\left(\alpha + \beta\right) + 2}}}{\left(1.0 + \left(\left(\alpha + \beta\right) + 2\right)\right) \cdot \sqrt{\left(\alpha + \beta\right) + 2}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Reproduce

herbie shell --seed 2019158 
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/3"
  :pre (and (> alpha -1) (> beta -1))
  (/ (/ (/ (+ (+ (+ alpha beta) (* beta alpha)) 1.0) (+ (+ alpha beta) (* 2 1))) (+ (+ alpha beta) (* 2 1))) (+ (+ (+ alpha beta) (* 2 1)) 1.0)))