Average Error: 16.6 → 6.9
Time: 15.5s
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 r61547 = beta;
        double r61548 = alpha;
        double r61549 = r61547 - r61548;
        double r61550 = r61548 + r61547;
        double r61551 = 2.0;
        double r61552 = r61550 + r61551;
        double r61553 = r61549 / r61552;
        double r61554 = 1.0;
        double r61555 = r61553 + r61554;
        double r61556 = r61555 / r61551;
        return r61556;
}
↓
double f(double alpha, double beta) {
        double r61557 = alpha;
        double r61558 = 4.960190917426964e+36;
        bool r61559 = r61557 <= r61558;
        double r61560 = 1.0;
        double r61561 = beta;
        double r61562 = r61557 + r61561;
        double r61563 = 2.0;
        double r61564 = r61562 + r61563;
        double r61565 = r61564 / r61561;
        double r61566 = cbrt(r61565);
        double r61567 = r61566 * r61566;
        double r61568 = r61560 / r61567;
        double r61569 = r61568 / r61566;
        double r61570 = r61557 / r61564;
        double r61571 = 1.0;
        double r61572 = r61570 - r61571;
        double r61573 = r61569 - r61572;
        double r61574 = r61573 / r61563;
        double r61575 = 4.0;
        double r61576 = 2.0;
        double r61577 = pow(r61557, r61576);
        double r61578 = r61575 / r61577;
        double r61579 = r61563 / r61557;
        double r61580 = r61578 - r61579;
        double r61581 = 8.0;
        double r61582 = 3.0;
        double r61583 = pow(r61557, r61582);
        double r61584 = r61581 / r61583;
        double r61585 = r61580 - r61584;
        double r61586 = r61569 - r61585;
        double r61587 = r61586 / r61563;
        double r61588 = r61559 ? r61574 : r61587;
        return r61588;
}

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))