Average Error: 3.8 → 2.3
Time: 39.6s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;\beta \le 7.725152532273346988388680017546606101409 \cdot 10^{182}:\\ \;\;\;\;\frac{\frac{\sqrt[3]{\left(\frac{\sqrt{1 + \left(\alpha \cdot \beta + \left(\alpha + \beta\right)\right)}}{\sqrt{\left(\alpha + \beta\right) + 1 \cdot 2}} \cdot \frac{\sqrt{1 + \left(\alpha \cdot \beta + \left(\alpha + \beta\right)\right)}}{\sqrt{\left(\alpha + \beta\right) + 1 \cdot 2}}\right) \cdot \frac{\sqrt{1 + \left(\alpha \cdot \beta + \left(\alpha + \beta\right)\right)}}{\sqrt{\left(\alpha + \beta\right) + 1 \cdot 2}}}}{\frac{\left(\alpha + \beta\right) + 1 \cdot 2}{\frac{\sqrt{1 + \left(\alpha \cdot \beta + \left(\alpha + \beta\right)\right)}}{\sqrt{\left(\alpha + \beta\right) + 1 \cdot 2}}}}}{1 + \left(\left(\alpha + \beta\right) + 1 \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]
\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}
↓
\begin{array}{l}
\mathbf{if}\;\beta \le 7.725152532273346988388680017546606101409 \cdot 10^{182}:\\
\;\;\;\;\frac{\frac{\sqrt[3]{\left(\frac{\sqrt{1 + \left(\alpha \cdot \beta + \left(\alpha + \beta\right)\right)}}{\sqrt{\left(\alpha + \beta\right) + 1 \cdot 2}} \cdot \frac{\sqrt{1 + \left(\alpha \cdot \beta + \left(\alpha + \beta\right)\right)}}{\sqrt{\left(\alpha + \beta\right) + 1 \cdot 2}}\right) \cdot \frac{\sqrt{1 + \left(\alpha \cdot \beta + \left(\alpha + \beta\right)\right)}}{\sqrt{\left(\alpha + \beta\right) + 1 \cdot 2}}}}{\frac{\left(\alpha + \beta\right) + 1 \cdot 2}{\frac{\sqrt{1 + \left(\alpha \cdot \beta + \left(\alpha + \beta\right)\right)}}{\sqrt{\left(\alpha + \beta\right) + 1 \cdot 2}}}}}{1 + \left(\left(\alpha + \beta\right) + 1 \cdot 2\right)}\\

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

\end{array}
double f(double alpha, double beta) {
        double r4679623 = alpha;
        double r4679624 = beta;
        double r4679625 = r4679623 + r4679624;
        double r4679626 = r4679624 * r4679623;
        double r4679627 = r4679625 + r4679626;
        double r4679628 = 1.0;
        double r4679629 = r4679627 + r4679628;
        double r4679630 = 2.0;
        double r4679631 = r4679630 * r4679628;
        double r4679632 = r4679625 + r4679631;
        double r4679633 = r4679629 / r4679632;
        double r4679634 = r4679633 / r4679632;
        double r4679635 = r4679632 + r4679628;
        double r4679636 = r4679634 / r4679635;
        return r4679636;
}
↓
double f(double alpha, double beta) {
        double r4679637 = beta;
        double r4679638 = 7.725152532273347e+182;
        bool r4679639 = r4679637 <= r4679638;
        double r4679640 = 1.0;
        double r4679641 = alpha;
        double r4679642 = r4679641 * r4679637;
        double r4679643 = r4679641 + r4679637;
        double r4679644 = r4679642 + r4679643;
        double r4679645 = r4679640 + r4679644;
        double r4679646 = sqrt(r4679645);
        double r4679647 = 2.0;
        double r4679648 = r4679640 * r4679647;
        double r4679649 = r4679643 + r4679648;
        double r4679650 = sqrt(r4679649);
        double r4679651 = r4679646 / r4679650;
        double r4679652 = r4679651 * r4679651;
        double r4679653 = r4679652 * r4679651;
        double r4679654 = cbrt(r4679653);
        double r4679655 = r4679649 / r4679651;
        double r4679656 = r4679654 / r4679655;
        double r4679657 = r4679640 + r4679649;
        double r4679658 = r4679656 / r4679657;
        double r4679659 = 0.0;
        double r4679660 = r4679639 ? r4679658 : r4679659;
        return r4679660;
}

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 < 7.725152532273347e+182

    1. Initial program 1.5

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

    3. Applied add-sqr-sqrt2.1

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

    4. Applied add-sqr-sqrt2.0

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

    5. Applied times-frac2.0

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

    6. Applied associate-/l*1.6

      \[\leadsto \frac{\color{blue}{\frac{\frac{\sqrt{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\frac{\left(\alpha + \beta\right) + 2 \cdot 1}{\frac{\sqrt{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}}}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\]
    7. Using strategy rm

    8. Applied add-cbrt-cube1.6

      \[\leadsto \frac{\frac{\color{blue}{\sqrt[3]{\left(\frac{\sqrt{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}} \cdot \frac{\sqrt{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}}\right) \cdot \frac{\sqrt{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}}}}}{\frac{\left(\alpha + \beta\right) + 2 \cdot 1}{\frac{\sqrt{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\]

    if 7.725152532273347e+182 < beta

    1. Initial program 17.5

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

    2. Taylor expanded around inf 6.4

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

  4. Final simplification2.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \le 7.725152532273346988388680017546606101409 \cdot 10^{182}:\\ \;\;\;\;\frac{\frac{\sqrt[3]{\left(\frac{\sqrt{1 + \left(\alpha \cdot \beta + \left(\alpha + \beta\right)\right)}}{\sqrt{\left(\alpha + \beta\right) + 1 \cdot 2}} \cdot \frac{\sqrt{1 + \left(\alpha \cdot \beta + \left(\alpha + \beta\right)\right)}}{\sqrt{\left(\alpha + \beta\right) + 1 \cdot 2}}\right) \cdot \frac{\sqrt{1 + \left(\alpha \cdot \beta + \left(\alpha + \beta\right)\right)}}{\sqrt{\left(\alpha + \beta\right) + 1 \cdot 2}}}}{\frac{\left(\alpha + \beta\right) + 1 \cdot 2}{\frac{\sqrt{1 + \left(\alpha \cdot \beta + \left(\alpha + \beta\right)\right)}}{\sqrt{\left(\alpha + \beta\right) + 1 \cdot 2}}}}}{1 + \left(\left(\alpha + \beta\right) + 1 \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/3"
  :pre (and (> alpha -1.0) (> beta -1.0))
  (/ (/ (/ (+ (+ (+ alpha beta) (* beta alpha)) 1.0) (+ (+ alpha beta) (* 2.0 1.0))) (+ (+ alpha beta) (* 2.0 1.0))) (+ (+ (+ alpha beta) (* 2.0 1.0)) 1.0)))