Average Error: 16.4 → 6.3
Time: 14.7s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 76510312.3131744563579559326171875:\\ \;\;\;\;\frac{e^{\log \left(\frac{\beta}{2 + \left(\beta + \alpha\right)} - \left(\frac{\alpha}{\beta + \left(2 + \alpha\right)} - 1\right)\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt[3]{\beta}}{\sqrt[3]{2 + \left(\beta + \alpha\right)}} \cdot \frac{\frac{\sqrt[3]{\beta} \cdot \sqrt[3]{\beta}}{\sqrt[3]{2 + \left(\beta + \alpha\right)}}}{\sqrt[3]{2 + \left(\beta + \alpha\right)}} - \left(\left(\frac{4}{{\alpha}^{2}} - \frac{2}{\alpha}\right) - \frac{8}{{\alpha}^{3}}\right)}{2}\\ \end{array}\]
\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \le 76510312.3131744563579559326171875:\\
\;\;\;\;\frac{e^{\log \left(\frac{\beta}{2 + \left(\beta + \alpha\right)} - \left(\frac{\alpha}{\beta + \left(2 + \alpha\right)} - 1\right)\right)}}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{\sqrt[3]{\beta}}{\sqrt[3]{2 + \left(\beta + \alpha\right)}} \cdot \frac{\frac{\sqrt[3]{\beta} \cdot \sqrt[3]{\beta}}{\sqrt[3]{2 + \left(\beta + \alpha\right)}}}{\sqrt[3]{2 + \left(\beta + \alpha\right)}} - \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 r64524 = beta;
        double r64525 = alpha;
        double r64526 = r64524 - r64525;
        double r64527 = r64525 + r64524;
        double r64528 = 2.0;
        double r64529 = r64527 + r64528;
        double r64530 = r64526 / r64529;
        double r64531 = 1.0;
        double r64532 = r64530 + r64531;
        double r64533 = r64532 / r64528;
        return r64533;
}
↓
double f(double alpha, double beta) {
        double r64534 = alpha;
        double r64535 = 76510312.31317446;
        bool r64536 = r64534 <= r64535;
        double r64537 = beta;
        double r64538 = 2.0;
        double r64539 = r64537 + r64534;
        double r64540 = r64538 + r64539;
        double r64541 = r64537 / r64540;
        double r64542 = r64538 + r64534;
        double r64543 = r64537 + r64542;
        double r64544 = r64534 / r64543;
        double r64545 = 1.0;
        double r64546 = r64544 - r64545;
        double r64547 = r64541 - r64546;
        double r64548 = log(r64547);
        double r64549 = exp(r64548);
        double r64550 = r64549 / r64538;
        double r64551 = cbrt(r64537);
        double r64552 = cbrt(r64540);
        double r64553 = r64551 / r64552;
        double r64554 = r64551 * r64551;
        double r64555 = r64554 / r64552;
        double r64556 = r64555 / r64552;
        double r64557 = r64553 * r64556;
        double r64558 = 4.0;
        double r64559 = 2.0;
        double r64560 = pow(r64534, r64559);
        double r64561 = r64558 / r64560;
        double r64562 = r64538 / r64534;
        double r64563 = r64561 - r64562;
        double r64564 = 8.0;
        double r64565 = 3.0;
        double r64566 = pow(r64534, r64565);
        double r64567 = r64564 / r64566;
        double r64568 = r64563 - r64567;
        double r64569 = r64557 - r64568;
        double r64570 = r64569 / r64538;
        double r64571 = r64536 ? r64550 : r64570;
        return r64571;
}

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 < 76510312.31317446

    1. Initial program 0.1

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

    2. Simplified0.1

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

    4. Applied div-sub0.1

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

    5. Applied associate-+l-0.1

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

    6. Simplified0.1

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

    8. Applied add-exp-log0.1

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

    9. Simplified0.1

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

    if 76510312.31317446 < alpha

    1. Initial program 49.6

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

    2. Simplified49.6

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

    4. Applied div-sub49.6

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

    5. Applied associate-+l-48.0

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

    6. Simplified48.0

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

    8. Applied add-log-exp48.0

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

    9. Applied add-log-exp48.0

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

    10. Applied diff-log48.0

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

    11. Simplified48.0

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

    13. Applied add-cube-cbrt48.1

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

    14. Applied add-cube-cbrt48.0

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

    15. Applied times-frac48.0

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

    16. Simplified48.0

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

    17. Simplified48.0

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

    18. Taylor expanded around inf 19.0

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

    19. Simplified19.0

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

  4. Final simplification6.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 76510312.3131744563579559326171875:\\ \;\;\;\;\frac{e^{\log \left(\frac{\beta}{2 + \left(\beta + \alpha\right)} - \left(\frac{\alpha}{\beta + \left(2 + \alpha\right)} - 1\right)\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt[3]{\beta}}{\sqrt[3]{2 + \left(\beta + \alpha\right)}} \cdot \frac{\frac{\sqrt[3]{\beta} \cdot \sqrt[3]{\beta}}{\sqrt[3]{2 + \left(\beta + \alpha\right)}}}{\sqrt[3]{2 + \left(\beta + \alpha\right)}} - \left(\left(\frac{4}{{\alpha}^{2}} - \frac{2}{\alpha}\right) - \frac{8}{{\alpha}^{3}}\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019194 
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/1"
  :pre (and (> alpha -1.0) (> beta -1.0))
  (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0))