Average Error: 15.8 → 6.4
Time: 23.4s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 13019.849401606787:\\ \;\;\;\;\frac{1.0 + \frac{\frac{\beta - \alpha}{\sqrt{\left(\beta + \alpha\right) + 2.0}}}{\sqrt{\left(\beta + \alpha\right) + 2.0}}}{2.0}\\ \mathbf{elif}\;\alpha \le 5.286547659356629 \cdot 10^{+135}:\\ \;\;\;\;\frac{\left(\sqrt[3]{\beta} \cdot \sqrt[3]{\beta}\right) \cdot \frac{\sqrt[3]{\sqrt[3]{\beta}} \cdot \left(\sqrt[3]{\sqrt[3]{\beta}} \cdot \sqrt[3]{\sqrt[3]{\beta}}\right)}{\left(\beta + \alpha\right) + 2.0} - \left(\left(\frac{4.0}{\alpha \cdot \alpha} - \frac{2.0}{\alpha}\right) - \frac{\frac{8.0}{\alpha}}{\alpha \cdot \alpha}\right)}{2.0}\\ \mathbf{elif}\;\alpha \le 3.26519642354199 \cdot 10^{+153}:\\ \;\;\;\;\frac{e^{\log \left(\frac{\beta}{\left(\beta + \alpha\right) + 2.0}\right)} - \left(\frac{\alpha}{\left(\beta + \alpha\right) + 2.0} - 1.0\right)}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\sqrt[3]{\beta} \cdot \sqrt[3]{\beta}\right) \cdot \frac{\sqrt[3]{\sqrt[3]{\beta}} \cdot \left(\sqrt[3]{\sqrt[3]{\beta}} \cdot \sqrt[3]{\sqrt[3]{\beta}}\right)}{\left(\beta + \alpha\right) + 2.0} - \left(\left(\frac{4.0}{\alpha \cdot \alpha} - \frac{2.0}{\alpha}\right) - \frac{\frac{8.0}{\alpha}}{\alpha \cdot \alpha}\right)}{2.0}\\ \end{array}\]
\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \le 13019.849401606787:\\
\;\;\;\;\frac{1.0 + \frac{\frac{\beta - \alpha}{\sqrt{\left(\beta + \alpha\right) + 2.0}}}{\sqrt{\left(\beta + \alpha\right) + 2.0}}}{2.0}\\

\mathbf{elif}\;\alpha \le 5.286547659356629 \cdot 10^{+135}:\\
\;\;\;\;\frac{\left(\sqrt[3]{\beta} \cdot \sqrt[3]{\beta}\right) \cdot \frac{\sqrt[3]{\sqrt[3]{\beta}} \cdot \left(\sqrt[3]{\sqrt[3]{\beta}} \cdot \sqrt[3]{\sqrt[3]{\beta}}\right)}{\left(\beta + \alpha\right) + 2.0} - \left(\left(\frac{4.0}{\alpha \cdot \alpha} - \frac{2.0}{\alpha}\right) - \frac{\frac{8.0}{\alpha}}{\alpha \cdot \alpha}\right)}{2.0}\\

\mathbf{elif}\;\alpha \le 3.26519642354199 \cdot 10^{+153}:\\
\;\;\;\;\frac{e^{\log \left(\frac{\beta}{\left(\beta + \alpha\right) + 2.0}\right)} - \left(\frac{\alpha}{\left(\beta + \alpha\right) + 2.0} - 1.0\right)}{2.0}\\

\mathbf{else}:\\
\;\;\;\;\frac{\left(\sqrt[3]{\beta} \cdot \sqrt[3]{\beta}\right) \cdot \frac{\sqrt[3]{\sqrt[3]{\beta}} \cdot \left(\sqrt[3]{\sqrt[3]{\beta}} \cdot \sqrt[3]{\sqrt[3]{\beta}}\right)}{\left(\beta + \alpha\right) + 2.0} - \left(\left(\frac{4.0}{\alpha \cdot \alpha} - \frac{2.0}{\alpha}\right) - \frac{\frac{8.0}{\alpha}}{\alpha \cdot \alpha}\right)}{2.0}\\

\end{array}
double f(double alpha, double beta) {
        double r5397551 = beta;
        double r5397552 = alpha;
        double r5397553 = r5397551 - r5397552;
        double r5397554 = r5397552 + r5397551;
        double r5397555 = 2.0;
        double r5397556 = r5397554 + r5397555;
        double r5397557 = r5397553 / r5397556;
        double r5397558 = 1.0;
        double r5397559 = r5397557 + r5397558;
        double r5397560 = r5397559 / r5397555;
        return r5397560;
}
↓
double f(double alpha, double beta) {
        double r5397561 = alpha;
        double r5397562 = 13019.849401606787;
        bool r5397563 = r5397561 <= r5397562;
        double r5397564 = 1.0;
        double r5397565 = beta;
        double r5397566 = r5397565 - r5397561;
        double r5397567 = r5397565 + r5397561;
        double r5397568 = 2.0;
        double r5397569 = r5397567 + r5397568;
        double r5397570 = sqrt(r5397569);
        double r5397571 = r5397566 / r5397570;
        double r5397572 = r5397571 / r5397570;
        double r5397573 = r5397564 + r5397572;
        double r5397574 = r5397573 / r5397568;
        double r5397575 = 5.286547659356629e+135;
        bool r5397576 = r5397561 <= r5397575;
        double r5397577 = cbrt(r5397565);
        double r5397578 = r5397577 * r5397577;
        double r5397579 = cbrt(r5397577);
        double r5397580 = r5397579 * r5397579;
        double r5397581 = r5397579 * r5397580;
        double r5397582 = r5397581 / r5397569;
        double r5397583 = r5397578 * r5397582;
        double r5397584 = 4.0;
        double r5397585 = r5397561 * r5397561;
        double r5397586 = r5397584 / r5397585;
        double r5397587 = r5397568 / r5397561;
        double r5397588 = r5397586 - r5397587;
        double r5397589 = 8.0;
        double r5397590 = r5397589 / r5397561;
        double r5397591 = r5397590 / r5397585;
        double r5397592 = r5397588 - r5397591;
        double r5397593 = r5397583 - r5397592;
        double r5397594 = r5397593 / r5397568;
        double r5397595 = 3.26519642354199e+153;
        bool r5397596 = r5397561 <= r5397595;
        double r5397597 = r5397565 / r5397569;
        double r5397598 = log(r5397597);
        double r5397599 = exp(r5397598);
        double r5397600 = r5397561 / r5397569;
        double r5397601 = r5397600 - r5397564;
        double r5397602 = r5397599 - r5397601;
        double r5397603 = r5397602 / r5397568;
        double r5397604 = r5397596 ? r5397603 : r5397594;
        double r5397605 = r5397576 ? r5397594 : r5397604;
        double r5397606 = r5397563 ? r5397574 : r5397605;
        return r5397606;
}

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 3 regimes

  2. if alpha < 13019.849401606787

    1. Initial program 0.0

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}\]
    2. Using strategy rm

    3. Applied add-sqr-sqrt0.1

      \[\leadsto \frac{\frac{\beta - \alpha}{\color{blue}{\sqrt{\left(\alpha + \beta\right) + 2.0} \cdot \sqrt{\left(\alpha + \beta\right) + 2.0}}} + 1.0}{2.0}\]

    4. Applied associate-/r*0.1

      \[\leadsto \frac{\color{blue}{\frac{\frac{\beta - \alpha}{\sqrt{\left(\alpha + \beta\right) + 2.0}}}{\sqrt{\left(\alpha + \beta\right) + 2.0}}} + 1.0}{2.0}\]

    if 13019.849401606787 < alpha < 5.286547659356629e+135 or 3.26519642354199e+153 < alpha

    1. Initial program 49.0

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}\]
    2. Using strategy rm

    3. Applied div-sub49.0

      \[\leadsto \frac{\color{blue}{\left(\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \frac{\alpha}{\left(\alpha + \beta\right) + 2.0}\right)} + 1.0}{2.0}\]

    4. Applied associate-+l-47.5

      \[\leadsto \frac{\color{blue}{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0\right)}}{2.0}\]
    5. Using strategy rm

    6. Applied *-un-lft-identity47.5

      \[\leadsto \frac{\frac{\beta}{\color{blue}{1 \cdot \left(\left(\alpha + \beta\right) + 2.0\right)}} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0\right)}{2.0}\]

    7. Applied add-cube-cbrt47.6

      \[\leadsto \frac{\frac{\color{blue}{\left(\sqrt[3]{\beta} \cdot \sqrt[3]{\beta}\right) \cdot \sqrt[3]{\beta}}}{1 \cdot \left(\left(\alpha + \beta\right) + 2.0\right)} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0\right)}{2.0}\]

    8. Applied times-frac47.6

      \[\leadsto \frac{\color{blue}{\frac{\sqrt[3]{\beta} \cdot \sqrt[3]{\beta}}{1} \cdot \frac{\sqrt[3]{\beta}}{\left(\alpha + \beta\right) + 2.0}} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0\right)}{2.0}\]

    9. Simplified47.6

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\beta} \cdot \sqrt[3]{\beta}\right)} \cdot \frac{\sqrt[3]{\beta}}{\left(\alpha + \beta\right) + 2.0} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0\right)}{2.0}\]
    10. Using strategy rm

    11. Applied add-cube-cbrt47.6

      \[\leadsto \frac{\left(\sqrt[3]{\beta} \cdot \sqrt[3]{\beta}\right) \cdot \frac{\color{blue}{\left(\sqrt[3]{\sqrt[3]{\beta}} \cdot \sqrt[3]{\sqrt[3]{\beta}}\right) \cdot \sqrt[3]{\sqrt[3]{\beta}}}}{\left(\alpha + \beta\right) + 2.0} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0\right)}{2.0}\]

    12. Taylor expanded around inf 18.1

      \[\leadsto \frac{\left(\sqrt[3]{\beta} \cdot \sqrt[3]{\beta}\right) \cdot \frac{\left(\sqrt[3]{\sqrt[3]{\beta}} \cdot \sqrt[3]{\sqrt[3]{\beta}}\right) \cdot \sqrt[3]{\sqrt[3]{\beta}}}{\left(\alpha + \beta\right) + 2.0} - \color{blue}{\left(4.0 \cdot \frac{1}{{\alpha}^{2}} - \left(2.0 \cdot \frac{1}{\alpha} + 8.0 \cdot \frac{1}{{\alpha}^{3}}\right)\right)}}{2.0}\]

    13. Simplified18.1

      \[\leadsto \frac{\left(\sqrt[3]{\beta} \cdot \sqrt[3]{\beta}\right) \cdot \frac{\left(\sqrt[3]{\sqrt[3]{\beta}} \cdot \sqrt[3]{\sqrt[3]{\beta}}\right) \cdot \sqrt[3]{\sqrt[3]{\beta}}}{\left(\alpha + \beta\right) + 2.0} - \color{blue}{\left(\left(\frac{4.0}{\alpha \cdot \alpha} - \frac{2.0}{\alpha}\right) - \frac{\frac{8.0}{\alpha}}{\alpha \cdot \alpha}\right)}}{2.0}\]

    if 5.286547659356629e+135 < alpha < 3.26519642354199e+153

    1. Initial program 46.1

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}\]
    2. Using strategy rm

    3. Applied div-sub46.1

      \[\leadsto \frac{\color{blue}{\left(\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \frac{\alpha}{\left(\alpha + \beta\right) + 2.0}\right)} + 1.0}{2.0}\]

    4. Applied associate-+l-45.0

      \[\leadsto \frac{\color{blue}{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0\right)}}{2.0}\]
    5. Using strategy rm

    6. Applied add-exp-log45.2

      \[\leadsto \frac{\color{blue}{e^{\log \left(\frac{\beta}{\left(\alpha + \beta\right) + 2.0}\right)}} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0\right)}{2.0}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification6.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 13019.849401606787:\\ \;\;\;\;\frac{1.0 + \frac{\frac{\beta - \alpha}{\sqrt{\left(\beta + \alpha\right) + 2.0}}}{\sqrt{\left(\beta + \alpha\right) + 2.0}}}{2.0}\\ \mathbf{elif}\;\alpha \le 5.286547659356629 \cdot 10^{+135}:\\ \;\;\;\;\frac{\left(\sqrt[3]{\beta} \cdot \sqrt[3]{\beta}\right) \cdot \frac{\sqrt[3]{\sqrt[3]{\beta}} \cdot \left(\sqrt[3]{\sqrt[3]{\beta}} \cdot \sqrt[3]{\sqrt[3]{\beta}}\right)}{\left(\beta + \alpha\right) + 2.0} - \left(\left(\frac{4.0}{\alpha \cdot \alpha} - \frac{2.0}{\alpha}\right) - \frac{\frac{8.0}{\alpha}}{\alpha \cdot \alpha}\right)}{2.0}\\ \mathbf{elif}\;\alpha \le 3.26519642354199 \cdot 10^{+153}:\\ \;\;\;\;\frac{e^{\log \left(\frac{\beta}{\left(\beta + \alpha\right) + 2.0}\right)} - \left(\frac{\alpha}{\left(\beta + \alpha\right) + 2.0} - 1.0\right)}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\sqrt[3]{\beta} \cdot \sqrt[3]{\beta}\right) \cdot \frac{\sqrt[3]{\sqrt[3]{\beta}} \cdot \left(\sqrt[3]{\sqrt[3]{\beta}} \cdot \sqrt[3]{\sqrt[3]{\beta}}\right)}{\left(\beta + \alpha\right) + 2.0} - \left(\left(\frac{4.0}{\alpha \cdot \alpha} - \frac{2.0}{\alpha}\right) - \frac{\frac{8.0}{\alpha}}{\alpha \cdot \alpha}\right)}{2.0}\\ \end{array}\]

Reproduce

herbie shell --seed 2019124 
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/1"
  :pre (and (> alpha -1) (> beta -1))
  (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0))