Average Error: 3.6 → 2.3
Time: 9.8m
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 3.3940725538369387 \cdot 10^{+163}:\\ \;\;\;\;\frac{\frac{\frac{1.0 + \left(\beta \cdot \alpha + \left(\alpha + \beta\right)\right)}{\left(\alpha + \beta\right) + 2}}{\left(\alpha + \beta\right) + 2}}{\left(1.0 + \left(\alpha + \beta\right)\right) + 2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]
\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 3.3940725538369387 \cdot 10^{+163}:\\
\;\;\;\;\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 r52492441 = alpha;
        double r52492442 = beta;
        double r52492443 = r52492441 + r52492442;
        double r52492444 = r52492442 * r52492441;
        double r52492445 = r52492443 + r52492444;
        double r52492446 = 1.0;
        double r52492447 = r52492445 + r52492446;
        double r52492448 = 2.0;
        double r52492449 = 1.0;
        double r52492450 = r52492448 * r52492449;
        double r52492451 = r52492443 + r52492450;
        double r52492452 = r52492447 / r52492451;
        double r52492453 = r52492452 / r52492451;
        double r52492454 = r52492451 + r52492446;
        double r52492455 = r52492453 / r52492454;
        return r52492455;
}
↓
double f(double alpha, double beta) {
        double r52492456 = alpha;
        double r52492457 = 3.3940725538369387e+163;
        bool r52492458 = r52492456 <= r52492457;
        double r52492459 = 1.0;
        double r52492460 = beta;
        double r52492461 = r52492460 * r52492456;
        double r52492462 = r52492456 + r52492460;
        double r52492463 = r52492461 + r52492462;
        double r52492464 = r52492459 + r52492463;
        double r52492465 = 2.0;
        double r52492466 = r52492462 + r52492465;
        double r52492467 = r52492464 / r52492466;
        double r52492468 = r52492467 / r52492466;
        double r52492469 = r52492459 + r52492462;
        double r52492470 = r52492469 + r52492465;
        double r52492471 = r52492468 / r52492470;
        double r52492472 = 0.0;
        double r52492473 = r52492458 ? r52492471 : r52492472;
        return r52492473;
}

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 < 3.3940725538369387e+163

    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. Simplified1.3

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

    if 3.3940725538369387e+163 < alpha

    1. Initial program 15.9

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

    2. Simplified15.9

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

    3. Taylor expanded around inf 7.8

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

  4. Final simplification2.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 3.3940725538369387 \cdot 10^{+163}:\\ \;\;\;\;\frac{\frac{\frac{1.0 + \left(\beta \cdot \alpha + \left(\alpha + \beta\right)\right)}{\left(\alpha + \beta\right) + 2}}{\left(\alpha + \beta\right) + 2}}{\left(1.0 + \left(\alpha + \beta\right)\right) + 2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Reproduce

herbie shell --seed 2019121 
(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)))