Average Error: 16.0 → 6.0
Time: 8.9s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 1015227874.2528594:\\ \;\;\;\;\frac{e^{\log \left({\left(\frac{\beta}{\left(\alpha + \beta\right) + 2}\right)}^{3} - {\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)}^{3}\right) - \log \left(\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right) \cdot \left(\frac{\beta}{\left(\alpha + \beta\right) + 2} + \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)\right) + \frac{\beta}{\left(\alpha + \beta\right) + 2} \cdot \frac{\beta}{\left(\alpha + \beta\right) + 2}\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\left(\frac{4}{\alpha \cdot \alpha} - \frac{8}{{\alpha}^{3}}\right) - \frac{2}{\alpha}\right)}{2}\\ \end{array}\]
\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \le 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{8}{{\alpha}^{3}}\right) - \frac{2}{\alpha}\right)}{2}\\

\end{array}
double f(double alpha, double beta) {
        double r163608 = beta;
        double r163609 = alpha;
        double r163610 = r163608 - r163609;
        double r163611 = r163609 + r163608;
        double r163612 = 2.0;
        double r163613 = r163611 + r163612;
        double r163614 = r163610 / r163613;
        double r163615 = 1.0;
        double r163616 = r163614 + r163615;
        double r163617 = r163616 / r163612;
        return r163617;
}
↓
double f(double alpha, double beta) {
        double r163618 = alpha;
        double r163619 = 1015227874.2528594;
        bool r163620 = r163618 <= r163619;
        double r163621 = beta;
        double r163622 = r163618 + r163621;
        double r163623 = 2.0;
        double r163624 = r163622 + r163623;
        double r163625 = r163621 / r163624;
        double r163626 = 3.0;
        double r163627 = pow(r163625, r163626);
        double r163628 = r163618 / r163624;
        double r163629 = 1.0;
        double r163630 = r163628 - r163629;
        double r163631 = pow(r163630, r163626);
        double r163632 = r163627 - r163631;
        double r163633 = log(r163632);
        double r163634 = r163625 + r163630;
        double r163635 = r163630 * r163634;
        double r163636 = r163625 * r163625;
        double r163637 = r163635 + r163636;
        double r163638 = log(r163637);
        double r163639 = r163633 - r163638;
        double r163640 = exp(r163639);
        double r163641 = r163640 / r163623;
        double r163642 = 4.0;
        double r163643 = r163618 * r163618;
        double r163644 = r163642 / r163643;
        double r163645 = 8.0;
        double r163646 = pow(r163618, r163626);
        double r163647 = r163645 / r163646;
        double r163648 = r163644 - r163647;
        double r163649 = r163623 / r163618;
        double r163650 = r163648 - r163649;
        double r163651 = r163625 - r163650;
        double r163652 = r163651 / r163623;
        double r163653 = r163620 ? r163641 : r163652;
        return r163653;
}

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

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

  4. Final simplification6.0

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

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))