Average Error: 3.6 → 2.7
Time: 54.4s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 3.9275252152036123 \cdot 10^{+114}:\\ \;\;\;\;\frac{\frac{\frac{1.0 + \left(\beta \cdot \alpha + \left(\beta + \alpha\right)\right)}{\left(\beta + \alpha\right) + 2}}{\left(\beta + \alpha\right) + 2}}{1.0 + \left(\left(\beta + \alpha\right) + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(1 - \frac{1.0}{\alpha}\right) + \frac{2.0}{\alpha \cdot \alpha}}{\left(\beta + \alpha\right) + 2}}{1.0 + \left(\left(\beta + \alpha\right) + 2\right)}\\ \end{array}\]
\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 3.9275252152036123 \cdot 10^{+114}:\\
\;\;\;\;\frac{\frac{\frac{1.0 + \left(\beta \cdot \alpha + \left(\beta + \alpha\right)\right)}{\left(\beta + \alpha\right) + 2}}{\left(\beta + \alpha\right) + 2}}{1.0 + \left(\left(\beta + \alpha\right) + 2\right)}\\

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

\end{array}
double f(double alpha, double beta) {
        double r2468570 = alpha;
        double r2468571 = beta;
        double r2468572 = r2468570 + r2468571;
        double r2468573 = r2468571 * r2468570;
        double r2468574 = r2468572 + r2468573;
        double r2468575 = 1.0;
        double r2468576 = r2468574 + r2468575;
        double r2468577 = 2.0;
        double r2468578 = 1.0;
        double r2468579 = r2468577 * r2468578;
        double r2468580 = r2468572 + r2468579;
        double r2468581 = r2468576 / r2468580;
        double r2468582 = r2468581 / r2468580;
        double r2468583 = r2468580 + r2468575;
        double r2468584 = r2468582 / r2468583;
        return r2468584;
}
↓
double f(double alpha, double beta) {
        double r2468585 = alpha;
        double r2468586 = 3.9275252152036123e+114;
        bool r2468587 = r2468585 <= r2468586;
        double r2468588 = 1.0;
        double r2468589 = beta;
        double r2468590 = r2468589 * r2468585;
        double r2468591 = r2468589 + r2468585;
        double r2468592 = r2468590 + r2468591;
        double r2468593 = r2468588 + r2468592;
        double r2468594 = 2.0;
        double r2468595 = r2468591 + r2468594;
        double r2468596 = r2468593 / r2468595;
        double r2468597 = r2468596 / r2468595;
        double r2468598 = r2468588 + r2468595;
        double r2468599 = r2468597 / r2468598;
        double r2468600 = 1.0;
        double r2468601 = r2468588 / r2468585;
        double r2468602 = r2468600 - r2468601;
        double r2468603 = 2.0;
        double r2468604 = r2468585 * r2468585;
        double r2468605 = r2468603 / r2468604;
        double r2468606 = r2468602 + r2468605;
        double r2468607 = r2468606 / r2468595;
        double r2468608 = r2468607 / r2468598;
        double r2468609 = r2468587 ? r2468599 : r2468608;
        return r2468609;
}

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 < 3.9275252152036123e+114

    1. Initial program 0.8

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

    3. Applied +-commutative0.8

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

    if 3.9275252152036123e+114 < alpha

    1. Initial program 14.1

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

    2. Taylor expanded around inf 9.6

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

    3. Simplified9.6

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

  4. Final simplification2.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 3.9275252152036123 \cdot 10^{+114}:\\ \;\;\;\;\frac{\frac{\frac{1.0 + \left(\beta \cdot \alpha + \left(\beta + \alpha\right)\right)}{\left(\beta + \alpha\right) + 2}}{\left(\beta + \alpha\right) + 2}}{1.0 + \left(\left(\beta + \alpha\right) + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(1 - \frac{1.0}{\alpha}\right) + \frac{2.0}{\alpha \cdot \alpha}}{\left(\beta + \alpha\right) + 2}}{1.0 + \left(\left(\beta + \alpha\right) + 2\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019153 
(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)))