Average Error: 3.8 → 2.3
Time: 35.4s
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 r4532553 = alpha;
        double r4532554 = beta;
        double r4532555 = r4532553 + r4532554;
        double r4532556 = r4532554 * r4532553;
        double r4532557 = r4532555 + r4532556;
        double r4532558 = 1.0;
        double r4532559 = r4532557 + r4532558;
        double r4532560 = 2.0;
        double r4532561 = r4532560 * r4532558;
        double r4532562 = r4532555 + r4532561;
        double r4532563 = r4532559 / r4532562;
        double r4532564 = r4532563 / r4532562;
        double r4532565 = r4532562 + r4532558;
        double r4532566 = r4532564 / r4532565;
        return r4532566;
}
↓
double f(double alpha, double beta) {
        double r4532567 = beta;
        double r4532568 = 7.725152532273347e+182;
        bool r4532569 = r4532567 <= r4532568;
        double r4532570 = 1.0;
        double r4532571 = alpha;
        double r4532572 = r4532571 * r4532567;
        double r4532573 = r4532571 + r4532567;
        double r4532574 = r4532572 + r4532573;
        double r4532575 = r4532570 + r4532574;
        double r4532576 = sqrt(r4532575);
        double r4532577 = 2.0;
        double r4532578 = r4532570 * r4532577;
        double r4532579 = r4532573 + r4532578;
        double r4532580 = sqrt(r4532579);
        double r4532581 = r4532576 / r4532580;
        double r4532582 = r4532581 * r4532581;
        double r4532583 = r4532582 * r4532581;
        double r4532584 = cbrt(r4532583);
        double r4532585 = r4532579 / r4532581;
        double r4532586 = r4532584 / r4532585;
        double r4532587 = r4532570 + r4532579;
        double r4532588 = r4532586 / r4532587;
        double r4532589 = 0.0;
        double r4532590 = r4532569 ? r4532588 : r4532589;
        return r4532590;
}

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)))