Average Error: 16.2 → 6.4
Time: 11.4s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 2494813499756589053116416:\\ \;\;\;\;\frac{\left(\sqrt[3]{\frac{\beta}{\left(\alpha + \beta\right) + 2}} \cdot \log \left(e^{\sqrt[3]{\frac{\beta}{\left(\alpha + \beta\right) + 2}}}\right)\right) \cdot \log \left(e^{\sqrt[3]{\frac{\beta}{\left(\alpha + \beta\right) + 2}}}\right) - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\left(\frac{4}{\alpha \cdot \alpha} - \frac{8}{{\alpha}^{3}}\right) - \frac{2}{\alpha}\right)}{2}\\ \end{array}\]
\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \le 2494813499756589053116416:\\
\;\;\;\;\frac{\left(\sqrt[3]{\frac{\beta}{\left(\alpha + \beta\right) + 2}} \cdot \log \left(e^{\sqrt[3]{\frac{\beta}{\left(\alpha + \beta\right) + 2}}}\right)\right) \cdot \log \left(e^{\sqrt[3]{\frac{\beta}{\left(\alpha + \beta\right) + 2}}}\right) - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\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 r103590 = beta;
        double r103591 = alpha;
        double r103592 = r103590 - r103591;
        double r103593 = r103591 + r103590;
        double r103594 = 2.0;
        double r103595 = r103593 + r103594;
        double r103596 = r103592 / r103595;
        double r103597 = 1.0;
        double r103598 = r103596 + r103597;
        double r103599 = r103598 / r103594;
        return r103599;
}
↓
double f(double alpha, double beta) {
        double r103600 = alpha;
        double r103601 = 2.494813499756589e+24;
        bool r103602 = r103600 <= r103601;
        double r103603 = beta;
        double r103604 = r103600 + r103603;
        double r103605 = 2.0;
        double r103606 = r103604 + r103605;
        double r103607 = r103603 / r103606;
        double r103608 = cbrt(r103607);
        double r103609 = exp(r103608);
        double r103610 = log(r103609);
        double r103611 = r103608 * r103610;
        double r103612 = r103611 * r103610;
        double r103613 = r103600 / r103606;
        double r103614 = 1.0;
        double r103615 = r103613 - r103614;
        double r103616 = r103612 - r103615;
        double r103617 = r103616 / r103605;
        double r103618 = 4.0;
        double r103619 = r103600 * r103600;
        double r103620 = r103618 / r103619;
        double r103621 = 8.0;
        double r103622 = 3.0;
        double r103623 = pow(r103600, r103622);
        double r103624 = r103621 / r103623;
        double r103625 = r103620 - r103624;
        double r103626 = r103605 / r103600;
        double r103627 = r103625 - r103626;
        double r103628 = r103607 - r103627;
        double r103629 = r103628 / r103605;
        double r103630 = r103602 ? r103617 : r103629;
        return r103630;
}

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 < 2.494813499756589e+24

    1. Initial program 0.9

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

    3. Applied div-sub0.9

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

    4. Applied associate-+l-0.9

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

    6. Applied add-cube-cbrt0.9

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

    8. Applied add-log-exp0.9

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

    10. Applied add-log-exp0.9

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

    if 2.494813499756589e+24 < 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-48.7

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

    5. Taylor expanded around inf 18.5

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

    6. Simplified18.5

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

  4. Final simplification6.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 2494813499756589053116416:\\ \;\;\;\;\frac{\left(\sqrt[3]{\frac{\beta}{\left(\alpha + \beta\right) + 2}} \cdot \log \left(e^{\sqrt[3]{\frac{\beta}{\left(\alpha + \beta\right) + 2}}}\right)\right) \cdot \log \left(e^{\sqrt[3]{\frac{\beta}{\left(\alpha + \beta\right) + 2}}}\right) - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\left(\frac{4}{\alpha \cdot \alpha} - \frac{8}{{\alpha}^{3}}\right) - \frac{2}{\alpha}\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019351 +o rules:numerics
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/1"
  :precision binary64
  :pre (and (> alpha -1) (> beta -1))
  (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2)) 1) 2))