Average Error: 3.7 → 2.2
Time: 3.9m
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\beta \le 3.990720137325503 \cdot 10^{+164}:\\ \;\;\;\;\frac{\frac{\frac{1.0 + \left(\alpha \cdot \beta + \left(\beta + \alpha\right)\right)}{\left(\beta + \alpha\right) + 2}}{\left(\beta + \alpha\right) + 2}}{\left(1.0 + \left(\beta + \alpha\right)\right) + 2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]
\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 3.990720137325503 \cdot 10^{+164}:\\
\;\;\;\;\frac{\frac{\frac{1.0 + \left(\alpha \cdot \beta + \left(\beta + \alpha\right)\right)}{\left(\beta + \alpha\right) + 2}}{\left(\beta + \alpha\right) + 2}}{\left(1.0 + \left(\beta + \alpha\right)\right) + 2}\\

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

\end{array}
double f(double alpha, double beta) {
        double r7047525 = alpha;
        double r7047526 = beta;
        double r7047527 = r7047525 + r7047526;
        double r7047528 = r7047526 * r7047525;
        double r7047529 = r7047527 + r7047528;
        double r7047530 = 1.0;
        double r7047531 = r7047529 + r7047530;
        double r7047532 = 2.0;
        double r7047533 = 1.0;
        double r7047534 = r7047532 * r7047533;
        double r7047535 = r7047527 + r7047534;
        double r7047536 = r7047531 / r7047535;
        double r7047537 = r7047536 / r7047535;
        double r7047538 = r7047535 + r7047530;
        double r7047539 = r7047537 / r7047538;
        return r7047539;
}
↓
double f(double alpha, double beta) {
        double r7047540 = beta;
        double r7047541 = 3.990720137325503e+164;
        bool r7047542 = r7047540 <= r7047541;
        double r7047543 = 1.0;
        double r7047544 = alpha;
        double r7047545 = r7047544 * r7047540;
        double r7047546 = r7047540 + r7047544;
        double r7047547 = r7047545 + r7047546;
        double r7047548 = r7047543 + r7047547;
        double r7047549 = 2.0;
        double r7047550 = r7047546 + r7047549;
        double r7047551 = r7047548 / r7047550;
        double r7047552 = r7047551 / r7047550;
        double r7047553 = r7047543 + r7047546;
        double r7047554 = r7047553 + r7047549;
        double r7047555 = r7047552 / r7047554;
        double r7047556 = 0.0;
        double r7047557 = r7047542 ? r7047555 : r7047556;
        return r7047557;
}

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 < 3.990720137325503e+164

    1. Initial program 1.2

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

    2. Simplified1.2

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

    if 3.990720137325503e+164 < beta

    1. Initial program 16.7

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

    2. Simplified16.7

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

    3. Taylor expanded around inf 7.4

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

  4. Final simplification2.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \le 3.990720137325503 \cdot 10^{+164}:\\ \;\;\;\;\frac{\frac{\frac{1.0 + \left(\alpha \cdot \beta + \left(\beta + \alpha\right)\right)}{\left(\beta + \alpha\right) + 2}}{\left(\beta + \alpha\right) + 2}}{\left(1.0 + \left(\beta + \alpha\right)\right) + 2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Reproduce

herbie shell --seed 2019135 
(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)))