Average Error: 3.8 → 2.6
Time: 1.1m
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 2.8836479899208303 \cdot 10^{+124}:\\ \;\;\;\;\frac{\sqrt{\frac{\frac{1.0 + \left(\beta \cdot \alpha + \left(\alpha + \beta\right)\right)}{\left(\alpha + \beta\right) + 2}}{\left(\alpha + \beta\right) + 2}} \cdot \sqrt{\frac{\frac{1.0 + \left(\beta \cdot \alpha + \left(\alpha + \beta\right)\right)}{\left(\alpha + \beta\right) + 2}}{\left(\alpha + \beta\right) + 2}}}{2 + \left(1.0 + \left(\alpha + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(1 - 1.0 \cdot \frac{1}{\alpha}\right) + \frac{1}{\alpha \cdot \alpha} \cdot 2.0}{\left(\alpha + \beta\right) + 2}}{2 + \left(1.0 + \left(\alpha + \beta\right)\right)}\\ \end{array}\]
\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}\;\alpha \le 2.8836479899208303 \cdot 10^{+124}:\\
\;\;\;\;\frac{\sqrt{\frac{\frac{1.0 + \left(\beta \cdot \alpha + \left(\alpha + \beta\right)\right)}{\left(\alpha + \beta\right) + 2}}{\left(\alpha + \beta\right) + 2}} \cdot \sqrt{\frac{\frac{1.0 + \left(\beta \cdot \alpha + \left(\alpha + \beta\right)\right)}{\left(\alpha + \beta\right) + 2}}{\left(\alpha + \beta\right) + 2}}}{2 + \left(1.0 + \left(\alpha + \beta\right)\right)}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{\left(1 - 1.0 \cdot \frac{1}{\alpha}\right) + \frac{1}{\alpha \cdot \alpha} \cdot 2.0}{\left(\alpha + \beta\right) + 2}}{2 + \left(1.0 + \left(\alpha + \beta\right)\right)}\\

\end{array}
double f(double alpha, double beta) {
        double r3480694 = alpha;
        double r3480695 = beta;
        double r3480696 = r3480694 + r3480695;
        double r3480697 = r3480695 * r3480694;
        double r3480698 = r3480696 + r3480697;
        double r3480699 = 1.0;
        double r3480700 = r3480698 + r3480699;
        double r3480701 = 2.0;
        double r3480702 = 1.0;
        double r3480703 = r3480701 * r3480702;
        double r3480704 = r3480696 + r3480703;
        double r3480705 = r3480700 / r3480704;
        double r3480706 = r3480705 / r3480704;
        double r3480707 = r3480704 + r3480699;
        double r3480708 = r3480706 / r3480707;
        return r3480708;
}
↓
double f(double alpha, double beta) {
        double r3480709 = alpha;
        double r3480710 = 2.8836479899208303e+124;
        bool r3480711 = r3480709 <= r3480710;
        double r3480712 = 1.0;
        double r3480713 = beta;
        double r3480714 = r3480713 * r3480709;
        double r3480715 = r3480709 + r3480713;
        double r3480716 = r3480714 + r3480715;
        double r3480717 = r3480712 + r3480716;
        double r3480718 = 2.0;
        double r3480719 = r3480715 + r3480718;
        double r3480720 = r3480717 / r3480719;
        double r3480721 = r3480720 / r3480719;
        double r3480722 = sqrt(r3480721);
        double r3480723 = r3480722 * r3480722;
        double r3480724 = r3480712 + r3480715;
        double r3480725 = r3480718 + r3480724;
        double r3480726 = r3480723 / r3480725;
        double r3480727 = 1.0;
        double r3480728 = r3480727 / r3480709;
        double r3480729 = r3480712 * r3480728;
        double r3480730 = r3480727 - r3480729;
        double r3480731 = r3480709 * r3480709;
        double r3480732 = r3480727 / r3480731;
        double r3480733 = 2.0;
        double r3480734 = r3480732 * r3480733;
        double r3480735 = r3480730 + r3480734;
        double r3480736 = r3480735 / r3480719;
        double r3480737 = r3480736 / r3480725;
        double r3480738 = r3480711 ? r3480726 : r3480737;
        return r3480738;
}

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 < 2.8836479899208303e+124

    1. Initial program 0.9

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

    2. Simplified0.9

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

    4. Applied add-sqr-sqrt0.9

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

    if 2.8836479899208303e+124 < alpha

    1. Initial program 16.0

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

    2. Simplified16.0

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

    3. Taylor expanded around inf 9.6

      \[\leadsto \frac{\frac{\color{blue}{\left(2.0 \cdot \frac{1}{{\alpha}^{2}} + 1\right) - 1.0 \cdot \frac{1}{\alpha}}}{2 + \left(\beta + \alpha\right)}}{\left(1.0 + \left(\beta + \alpha\right)\right) + 2}\]

    4. Simplified9.6

      \[\leadsto \frac{\frac{\color{blue}{\frac{1}{\alpha \cdot \alpha} \cdot 2.0 + \left(1 - 1.0 \cdot \frac{1}{\alpha}\right)}}{2 + \left(\beta + \alpha\right)}}{\left(1.0 + \left(\beta + \alpha\right)\right) + 2}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification2.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 2.8836479899208303 \cdot 10^{+124}:\\ \;\;\;\;\frac{\sqrt{\frac{\frac{1.0 + \left(\beta \cdot \alpha + \left(\alpha + \beta\right)\right)}{\left(\alpha + \beta\right) + 2}}{\left(\alpha + \beta\right) + 2}} \cdot \sqrt{\frac{\frac{1.0 + \left(\beta \cdot \alpha + \left(\alpha + \beta\right)\right)}{\left(\alpha + \beta\right) + 2}}{\left(\alpha + \beta\right) + 2}}}{2 + \left(1.0 + \left(\alpha + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(1 - 1.0 \cdot \frac{1}{\alpha}\right) + \frac{1}{\alpha \cdot \alpha} \cdot 2.0}{\left(\alpha + \beta\right) + 2}}{2 + \left(1.0 + \left(\alpha + \beta\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019137 
(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)))