Average Error: 16.6 → 6.9
Time: 15.4s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 4960190917426963756410657614805860352:\\ \;\;\;\;\frac{\frac{\frac{1}{\sqrt[3]{\frac{\left(\alpha + \beta\right) + 2}{\beta}} \cdot \sqrt[3]{\frac{\left(\alpha + \beta\right) + 2}{\beta}}}}{\sqrt[3]{\frac{\left(\alpha + \beta\right) + 2}{\beta}}} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{\sqrt[3]{\frac{\left(\alpha + \beta\right) + 2}{\beta}} \cdot \sqrt[3]{\frac{\left(\alpha + \beta\right) + 2}{\beta}}}}{\sqrt[3]{\frac{\left(\alpha + \beta\right) + 2}{\beta}}} - \left(\left(\frac{4}{{\alpha}^{2}} - \frac{2}{\alpha}\right) - \frac{8}{{\alpha}^{3}}\right)}{2}\\ \end{array}\]
\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \le 4960190917426963756410657614805860352:\\
\;\;\;\;\frac{\frac{\frac{1}{\sqrt[3]{\frac{\left(\alpha + \beta\right) + 2}{\beta}} \cdot \sqrt[3]{\frac{\left(\alpha + \beta\right) + 2}{\beta}}}}{\sqrt[3]{\frac{\left(\alpha + \beta\right) + 2}{\beta}}} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{\frac{1}{\sqrt[3]{\frac{\left(\alpha + \beta\right) + 2}{\beta}} \cdot \sqrt[3]{\frac{\left(\alpha + \beta\right) + 2}{\beta}}}}{\sqrt[3]{\frac{\left(\alpha + \beta\right) + 2}{\beta}}} - \left(\left(\frac{4}{{\alpha}^{2}} - \frac{2}{\alpha}\right) - \frac{8}{{\alpha}^{3}}\right)}{2}\\

\end{array}
double f(double alpha, double beta) {
        double r54522 = beta;
        double r54523 = alpha;
        double r54524 = r54522 - r54523;
        double r54525 = r54523 + r54522;
        double r54526 = 2.0;
        double r54527 = r54525 + r54526;
        double r54528 = r54524 / r54527;
        double r54529 = 1.0;
        double r54530 = r54528 + r54529;
        double r54531 = r54530 / r54526;
        return r54531;
}
↓
double f(double alpha, double beta) {
        double r54532 = alpha;
        double r54533 = 4.960190917426964e+36;
        bool r54534 = r54532 <= r54533;
        double r54535 = 1.0;
        double r54536 = beta;
        double r54537 = r54532 + r54536;
        double r54538 = 2.0;
        double r54539 = r54537 + r54538;
        double r54540 = r54539 / r54536;
        double r54541 = cbrt(r54540);
        double r54542 = r54541 * r54541;
        double r54543 = r54535 / r54542;
        double r54544 = r54543 / r54541;
        double r54545 = r54532 / r54539;
        double r54546 = 1.0;
        double r54547 = r54545 - r54546;
        double r54548 = r54544 - r54547;
        double r54549 = r54548 / r54538;
        double r54550 = 4.0;
        double r54551 = 2.0;
        double r54552 = pow(r54532, r54551);
        double r54553 = r54550 / r54552;
        double r54554 = r54538 / r54532;
        double r54555 = r54553 - r54554;
        double r54556 = 8.0;
        double r54557 = 3.0;
        double r54558 = pow(r54532, r54557);
        double r54559 = r54556 / r54558;
        double r54560 = r54555 - r54559;
        double r54561 = r54544 - r54560;
        double r54562 = r54561 / r54538;
        double r54563 = r54534 ? r54549 : r54562;
        return r54563;
}

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 < 4.960190917426964e+36

    1. Initial program 2.0

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

    3. Applied div-sub2.0

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

    4. Applied associate-+l-2.0

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

    6. Applied clear-num2.0

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

    8. Applied add-cube-cbrt2.0

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

    9. Applied associate-/r*2.0

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

    if 4.960190917426964e+36 < alpha

    1. Initial program 50.6

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

    3. Applied div-sub50.5

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

    4. Applied associate-+l-49.0

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

    6. Applied clear-num49.0

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

    8. Applied add-cube-cbrt49.0

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

    9. Applied associate-/r*49.0

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

    10. Taylor expanded around inf 18.2

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

    11. Simplified18.2

      \[\leadsto \frac{\frac{\frac{1}{\sqrt[3]{\frac{\left(\alpha + \beta\right) + 2}{\beta}} \cdot \sqrt[3]{\frac{\left(\alpha + \beta\right) + 2}{\beta}}}}{\sqrt[3]{\frac{\left(\alpha + \beta\right) + 2}{\beta}}} - \color{blue}{\left(\left(\frac{4}{{\alpha}^{2}} - \frac{2}{\alpha}\right) - \frac{8}{{\alpha}^{3}}\right)}}{2}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification6.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 4960190917426963756410657614805860352:\\ \;\;\;\;\frac{\frac{\frac{1}{\sqrt[3]{\frac{\left(\alpha + \beta\right) + 2}{\beta}} \cdot \sqrt[3]{\frac{\left(\alpha + \beta\right) + 2}{\beta}}}}{\sqrt[3]{\frac{\left(\alpha + \beta\right) + 2}{\beta}}} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{\sqrt[3]{\frac{\left(\alpha + \beta\right) + 2}{\beta}} \cdot \sqrt[3]{\frac{\left(\alpha + \beta\right) + 2}{\beta}}}}{\sqrt[3]{\frac{\left(\alpha + \beta\right) + 2}{\beta}}} - \left(\left(\frac{4}{{\alpha}^{2}} - \frac{2}{\alpha}\right) - \frac{8}{{\alpha}^{3}}\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019326 
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/1"
  :precision binary64
  :pre (and (> alpha -1) (> beta -1))
  (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2)) 1) 2))