Average Error: 3.6 → 2.1
Time: 55.1s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 1.6790194235791028 \cdot 10^{+189}:\\ \;\;\;\;\frac{\frac{\frac{1.0 + \left(\beta \cdot \alpha + \left(\alpha + \beta\right)\right)}{\left(\alpha + \beta\right) + 2}}{\left(\alpha + \beta\right) + 2}}{\left(1.0 + \left(\alpha + \beta\right)\right) + 2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]
\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}\;\alpha \le 1.6790194235791028 \cdot 10^{+189}:\\
\;\;\;\;\frac{\frac{\frac{1.0 + \left(\beta \cdot \alpha + \left(\alpha + \beta\right)\right)}{\left(\alpha + \beta\right) + 2}}{\left(\alpha + \beta\right) + 2}}{\left(1.0 + \left(\alpha + \beta\right)\right) + 2}\\

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

\end{array}
double f(double alpha, double beta) {
        double r3189456 = alpha;
        double r3189457 = beta;
        double r3189458 = r3189456 + r3189457;
        double r3189459 = r3189457 * r3189456;
        double r3189460 = r3189458 + r3189459;
        double r3189461 = 1.0;
        double r3189462 = r3189460 + r3189461;
        double r3189463 = 2.0;
        double r3189464 = 1.0;
        double r3189465 = r3189463 * r3189464;
        double r3189466 = r3189458 + r3189465;
        double r3189467 = r3189462 / r3189466;
        double r3189468 = r3189467 / r3189466;
        double r3189469 = r3189466 + r3189461;
        double r3189470 = r3189468 / r3189469;
        return r3189470;
}
↓
double f(double alpha, double beta) {
        double r3189471 = alpha;
        double r3189472 = 1.6790194235791028e+189;
        bool r3189473 = r3189471 <= r3189472;
        double r3189474 = 1.0;
        double r3189475 = beta;
        double r3189476 = r3189475 * r3189471;
        double r3189477 = r3189471 + r3189475;
        double r3189478 = r3189476 + r3189477;
        double r3189479 = r3189474 + r3189478;
        double r3189480 = 2.0;
        double r3189481 = r3189477 + r3189480;
        double r3189482 = r3189479 / r3189481;
        double r3189483 = r3189482 / r3189481;
        double r3189484 = r3189474 + r3189477;
        double r3189485 = r3189484 + r3189480;
        double r3189486 = r3189483 / r3189485;
        double r3189487 = 0.0;
        double r3189488 = r3189473 ? r3189486 : r3189487;
        return r3189488;
}

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 < 1.6790194235791028e+189

    1. Initial program 1.7

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

    2. Simplified1.7

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

    if 1.6790194235791028e+189 < alpha

    1. Initial program 16.2

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

    2. Simplified16.2

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

    3. Taylor expanded around -inf 5.1

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

  4. Final simplification2.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 1.6790194235791028 \cdot 10^{+189}:\\ \;\;\;\;\frac{\frac{\frac{1.0 + \left(\beta \cdot \alpha + \left(\alpha + \beta\right)\right)}{\left(\alpha + \beta\right) + 2}}{\left(\alpha + \beta\right) + 2}}{\left(1.0 + \left(\alpha + \beta\right)\right) + 2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Reproduce

herbie shell --seed 2019134 
(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)))