Average Error: 16.5 → 6.1
Time: 14.7s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 8092481.162986398674547672271728515625:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \sqrt[3]{{\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)}^{3}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\left(\frac{4}{{\alpha}^{2}} - \frac{8}{{\alpha}^{3}}\right) - \frac{2}{\alpha}\right)}{2}\\ \end{array}\]
\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \le 8092481.162986398674547672271728515625:\\
\;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \sqrt[3]{{\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)}^{3}}}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\left(\frac{4}{{\alpha}^{2}} - \frac{8}{{\alpha}^{3}}\right) - \frac{2}{\alpha}\right)}{2}\\

\end{array}
double f(double alpha, double beta) {
        double r59627 = beta;
        double r59628 = alpha;
        double r59629 = r59627 - r59628;
        double r59630 = r59628 + r59627;
        double r59631 = 2.0;
        double r59632 = r59630 + r59631;
        double r59633 = r59629 / r59632;
        double r59634 = 1.0;
        double r59635 = r59633 + r59634;
        double r59636 = r59635 / r59631;
        return r59636;
}
↓
double f(double alpha, double beta) {
        double r59637 = alpha;
        double r59638 = 8092481.162986399;
        bool r59639 = r59637 <= r59638;
        double r59640 = beta;
        double r59641 = r59637 + r59640;
        double r59642 = 2.0;
        double r59643 = r59641 + r59642;
        double r59644 = r59640 / r59643;
        double r59645 = r59637 / r59643;
        double r59646 = 1.0;
        double r59647 = r59645 - r59646;
        double r59648 = 3.0;
        double r59649 = pow(r59647, r59648);
        double r59650 = cbrt(r59649);
        double r59651 = r59644 - r59650;
        double r59652 = r59651 / r59642;
        double r59653 = 4.0;
        double r59654 = 2.0;
        double r59655 = pow(r59637, r59654);
        double r59656 = r59653 / r59655;
        double r59657 = 8.0;
        double r59658 = pow(r59637, r59648);
        double r59659 = r59657 / r59658;
        double r59660 = r59656 - r59659;
        double r59661 = r59642 / r59637;
        double r59662 = r59660 - r59661;
        double r59663 = r59644 - r59662;
        double r59664 = r59663 / r59642;
        double r59665 = r59639 ? r59652 : r59664;
        return r59665;
}

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 < 8092481.162986399

    1. Initial program 0.1

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

    3. Applied div-sub0.1

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

    4. Applied associate-+l-0.1

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

    6. Applied add-cbrt-cube0.1

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

    7. Simplified0.1

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

    if 8092481.162986399 < alpha

    1. Initial program 50.0

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

    3. Applied div-sub49.9

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

    4. Applied associate-+l-48.3

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

    6. Applied add-cbrt-cube48.3

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

    7. Simplified48.3

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

    8. Taylor expanded around inf 18.4

      \[\leadsto \frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \color{blue}{\left(4 \cdot \frac{1}{{\alpha}^{2}} - \left(2 \cdot \frac{1}{\alpha} + 8 \cdot \frac{1}{{\alpha}^{3}}\right)\right)}}{2}\]

    9. Simplified18.4

      \[\leadsto \frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \color{blue}{\left(\left(\frac{4}{{\alpha}^{2}} - \frac{8}{{\alpha}^{3}}\right) - \frac{2}{\alpha}\right)}}{2}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification6.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 8092481.162986398674547672271728515625:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \sqrt[3]{{\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)}^{3}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\left(\frac{4}{{\alpha}^{2}} - \frac{8}{{\alpha}^{3}}\right) - \frac{2}{\alpha}\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/1"
  :precision binary64
  :pre (and (> alpha -1) (> beta -1))
  (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2)) 1) 2))