Average Error: 16.0 → 6.0
Time: 9.4s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 1015227874.2528594:\\ \;\;\;\;\frac{e^{\log \left({\left(\frac{\beta}{\left(\alpha + \beta\right) + 2}\right)}^{3} - {\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)}^{3}\right) - \log \left(\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right) \cdot \left(\frac{\beta}{\left(\alpha + \beta\right) + 2} + \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)\right) + \frac{\beta}{\left(\alpha + \beta\right) + 2} \cdot \frac{\beta}{\left(\alpha + \beta\right) + 2}\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\left(\frac{4}{\alpha \cdot \alpha} - \frac{2}{\alpha}\right) - \frac{8}{{\alpha}^{3}}\right)}{2}\\ \end{array}\]
\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \le 1015227874.2528594:\\
\;\;\;\;\frac{e^{\log \left({\left(\frac{\beta}{\left(\alpha + \beta\right) + 2}\right)}^{3} - {\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)}^{3}\right) - \log \left(\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right) \cdot \left(\frac{\beta}{\left(\alpha + \beta\right) + 2} + \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)\right) + \frac{\beta}{\left(\alpha + \beta\right) + 2} \cdot \frac{\beta}{\left(\alpha + \beta\right) + 2}\right)}}{2}\\

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

\end{array}
double f(double alpha, double beta) {
        double r133662 = beta;
        double r133663 = alpha;
        double r133664 = r133662 - r133663;
        double r133665 = r133663 + r133662;
        double r133666 = 2.0;
        double r133667 = r133665 + r133666;
        double r133668 = r133664 / r133667;
        double r133669 = 1.0;
        double r133670 = r133668 + r133669;
        double r133671 = r133670 / r133666;
        return r133671;
}
↓
double f(double alpha, double beta) {
        double r133672 = alpha;
        double r133673 = 1015227874.2528594;
        bool r133674 = r133672 <= r133673;
        double r133675 = beta;
        double r133676 = r133672 + r133675;
        double r133677 = 2.0;
        double r133678 = r133676 + r133677;
        double r133679 = r133675 / r133678;
        double r133680 = 3.0;
        double r133681 = pow(r133679, r133680);
        double r133682 = r133672 / r133678;
        double r133683 = 1.0;
        double r133684 = r133682 - r133683;
        double r133685 = pow(r133684, r133680);
        double r133686 = r133681 - r133685;
        double r133687 = log(r133686);
        double r133688 = r133679 + r133684;
        double r133689 = r133684 * r133688;
        double r133690 = r133679 * r133679;
        double r133691 = r133689 + r133690;
        double r133692 = log(r133691);
        double r133693 = r133687 - r133692;
        double r133694 = exp(r133693);
        double r133695 = r133694 / r133677;
        double r133696 = 4.0;
        double r133697 = r133672 * r133672;
        double r133698 = r133696 / r133697;
        double r133699 = r133677 / r133672;
        double r133700 = r133698 - r133699;
        double r133701 = 8.0;
        double r133702 = pow(r133672, r133680);
        double r133703 = r133701 / r133702;
        double r133704 = r133700 - r133703;
        double r133705 = r133679 - r133704;
        double r133706 = r133705 / r133677;
        double r133707 = r133674 ? r133695 : r133706;
        return r133707;
}

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 < 1015227874.2528594

    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-exp-log0.1

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

    8. Applied flip3--0.1

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

    9. Applied log-div0.1

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

    10. Simplified0.1

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

    if 1015227874.2528594 < alpha

    1. Initial program 50.3

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

    3. Applied div-sub50.3

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

    4. Applied associate-+l-48.6

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

    5. Taylor expanded around inf 18.7

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

    6. Simplified18.7

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

  4. Final simplification6.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 1015227874.2528594:\\ \;\;\;\;\frac{e^{\log \left({\left(\frac{\beta}{\left(\alpha + \beta\right) + 2}\right)}^{3} - {\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)}^{3}\right) - \log \left(\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right) \cdot \left(\frac{\beta}{\left(\alpha + \beta\right) + 2} + \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)\right) + \frac{\beta}{\left(\alpha + \beta\right) + 2} \cdot \frac{\beta}{\left(\alpha + \beta\right) + 2}\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\left(\frac{4}{\alpha \cdot \alpha} - \frac{2}{\alpha}\right) - \frac{8}{{\alpha}^{3}}\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/1"
  :precision binary64
  :pre (and (> alpha -1) (> beta -1))
  (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2)) 1) 2))