Average Error: 16.1 → 6.1
Time: 20.7s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 7715362331151764:\\ \;\;\;\;\frac{\frac{\beta}{2 + \left(\beta + \alpha\right)} - \frac{\sqrt[3]{\left(\left(\frac{\alpha}{2 + \left(\beta + \alpha\right)} \cdot \frac{\alpha}{2 + \left(\beta + \alpha\right)} - 1 \cdot 1\right) \cdot \left(\frac{\alpha}{2 + \left(\beta + \alpha\right)} \cdot \frac{\alpha}{2 + \left(\beta + \alpha\right)} - 1 \cdot 1\right)\right) \cdot \left(\frac{\alpha}{2 + \left(\beta + \alpha\right)} \cdot \frac{\alpha}{2 + \left(\beta + \alpha\right)} - 1 \cdot 1\right)}}{1 + \frac{\alpha}{2 + \left(\beta + \alpha\right)}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{2 + \left(\beta + \alpha\right)} - \left(\left(\frac{4}{\alpha \cdot \alpha} - \frac{2}{\alpha}\right) - \frac{\frac{8}{\alpha}}{\alpha \cdot \alpha}\right)}{2}\\ \end{array}\]
\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \le 7715362331151764:\\
\;\;\;\;\frac{\frac{\beta}{2 + \left(\beta + \alpha\right)} - \frac{\sqrt[3]{\left(\left(\frac{\alpha}{2 + \left(\beta + \alpha\right)} \cdot \frac{\alpha}{2 + \left(\beta + \alpha\right)} - 1 \cdot 1\right) \cdot \left(\frac{\alpha}{2 + \left(\beta + \alpha\right)} \cdot \frac{\alpha}{2 + \left(\beta + \alpha\right)} - 1 \cdot 1\right)\right) \cdot \left(\frac{\alpha}{2 + \left(\beta + \alpha\right)} \cdot \frac{\alpha}{2 + \left(\beta + \alpha\right)} - 1 \cdot 1\right)}}{1 + \frac{\alpha}{2 + \left(\beta + \alpha\right)}}}{2}\\

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

\end{array}
double f(double alpha, double beta) {
        double r4157602 = beta;
        double r4157603 = alpha;
        double r4157604 = r4157602 - r4157603;
        double r4157605 = r4157603 + r4157602;
        double r4157606 = 2.0;
        double r4157607 = r4157605 + r4157606;
        double r4157608 = r4157604 / r4157607;
        double r4157609 = 1.0;
        double r4157610 = r4157608 + r4157609;
        double r4157611 = r4157610 / r4157606;
        return r4157611;
}
↓
double f(double alpha, double beta) {
        double r4157612 = alpha;
        double r4157613 = 7715362331151764.0;
        bool r4157614 = r4157612 <= r4157613;
        double r4157615 = beta;
        double r4157616 = 2.0;
        double r4157617 = r4157615 + r4157612;
        double r4157618 = r4157616 + r4157617;
        double r4157619 = r4157615 / r4157618;
        double r4157620 = r4157612 / r4157618;
        double r4157621 = r4157620 * r4157620;
        double r4157622 = 1.0;
        double r4157623 = r4157622 * r4157622;
        double r4157624 = r4157621 - r4157623;
        double r4157625 = r4157624 * r4157624;
        double r4157626 = r4157625 * r4157624;
        double r4157627 = cbrt(r4157626);
        double r4157628 = r4157622 + r4157620;
        double r4157629 = r4157627 / r4157628;
        double r4157630 = r4157619 - r4157629;
        double r4157631 = r4157630 / r4157616;
        double r4157632 = 4.0;
        double r4157633 = r4157612 * r4157612;
        double r4157634 = r4157632 / r4157633;
        double r4157635 = r4157616 / r4157612;
        double r4157636 = r4157634 - r4157635;
        double r4157637 = 8.0;
        double r4157638 = r4157637 / r4157612;
        double r4157639 = r4157638 / r4157633;
        double r4157640 = r4157636 - r4157639;
        double r4157641 = r4157619 - r4157640;
        double r4157642 = r4157641 / r4157616;
        double r4157643 = r4157614 ? r4157631 : r4157642;
        return r4157643;
}

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 < 7715362331151764.0

    1. Initial program 0.4

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

    3. Applied div-sub0.4

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

    4. Applied associate-+l-0.4

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

    6. Applied flip--0.5

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

    8. Applied add-cbrt-cube0.5

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

    if 7715362331151764.0 < alpha

    1. Initial program 49.9

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
    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. Taylor expanded around inf 18.4

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

    6. Simplified18.4

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

  4. Final simplification6.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 7715362331151764:\\ \;\;\;\;\frac{\frac{\beta}{2 + \left(\beta + \alpha\right)} - \frac{\sqrt[3]{\left(\left(\frac{\alpha}{2 + \left(\beta + \alpha\right)} \cdot \frac{\alpha}{2 + \left(\beta + \alpha\right)} - 1 \cdot 1\right) \cdot \left(\frac{\alpha}{2 + \left(\beta + \alpha\right)} \cdot \frac{\alpha}{2 + \left(\beta + \alpha\right)} - 1 \cdot 1\right)\right) \cdot \left(\frac{\alpha}{2 + \left(\beta + \alpha\right)} \cdot \frac{\alpha}{2 + \left(\beta + \alpha\right)} - 1 \cdot 1\right)}}{1 + \frac{\alpha}{2 + \left(\beta + \alpha\right)}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{2 + \left(\beta + \alpha\right)} - \left(\left(\frac{4}{\alpha \cdot \alpha} - \frac{2}{\alpha}\right) - \frac{\frac{8}{\alpha}}{\alpha \cdot \alpha}\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/1"
  :pre (and (> alpha -1.0) (> beta -1.0))
  (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0))