Average Error: 16.2 → 6.4
Time: 15.2s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 9.338107713205004 \cdot 10^{+21}:\\ \;\;\;\;\frac{\frac{\beta}{2.0 + \left(\beta + \alpha\right)} - \frac{\sqrt[3]{\left(\left(\frac{\alpha}{2.0 + \left(\beta + \alpha\right)} \cdot \frac{\alpha}{2.0 + \left(\beta + \alpha\right)} - 1.0 \cdot 1.0\right) \cdot \left(\frac{\alpha}{2.0 + \left(\beta + \alpha\right)} \cdot \frac{\alpha}{2.0 + \left(\beta + \alpha\right)} - 1.0 \cdot 1.0\right)\right) \cdot \left(\frac{\alpha}{2.0 + \left(\beta + \alpha\right)} \cdot \frac{\alpha}{2.0 + \left(\beta + \alpha\right)} - 1.0 \cdot 1.0\right)}}{\frac{\sqrt[3]{\alpha}}{2.0 + \left(\beta + \alpha\right)} \cdot \left(\sqrt[3]{\alpha} \cdot \sqrt[3]{\alpha}\right) + 1.0}}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{2.0 + \left(\beta + \alpha\right)} - \left(\left(\frac{4.0}{\alpha \cdot \alpha} - \frac{8.0}{\alpha \cdot \left(\alpha \cdot \alpha\right)}\right) - \frac{2.0}{\alpha}\right)}{2.0}\\ \end{array}\]
\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \le 9.338107713205004 \cdot 10^{+21}:\\
\;\;\;\;\frac{\frac{\beta}{2.0 + \left(\beta + \alpha\right)} - \frac{\sqrt[3]{\left(\left(\frac{\alpha}{2.0 + \left(\beta + \alpha\right)} \cdot \frac{\alpha}{2.0 + \left(\beta + \alpha\right)} - 1.0 \cdot 1.0\right) \cdot \left(\frac{\alpha}{2.0 + \left(\beta + \alpha\right)} \cdot \frac{\alpha}{2.0 + \left(\beta + \alpha\right)} - 1.0 \cdot 1.0\right)\right) \cdot \left(\frac{\alpha}{2.0 + \left(\beta + \alpha\right)} \cdot \frac{\alpha}{2.0 + \left(\beta + \alpha\right)} - 1.0 \cdot 1.0\right)}}{\frac{\sqrt[3]{\alpha}}{2.0 + \left(\beta + \alpha\right)} \cdot \left(\sqrt[3]{\alpha} \cdot \sqrt[3]{\alpha}\right) + 1.0}}{2.0}\\

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

\end{array}
double f(double alpha, double beta) {
        double r1854485 = beta;
        double r1854486 = alpha;
        double r1854487 = r1854485 - r1854486;
        double r1854488 = r1854486 + r1854485;
        double r1854489 = 2.0;
        double r1854490 = r1854488 + r1854489;
        double r1854491 = r1854487 / r1854490;
        double r1854492 = 1.0;
        double r1854493 = r1854491 + r1854492;
        double r1854494 = r1854493 / r1854489;
        return r1854494;
}
↓
double f(double alpha, double beta) {
        double r1854495 = alpha;
        double r1854496 = 9.338107713205004e+21;
        bool r1854497 = r1854495 <= r1854496;
        double r1854498 = beta;
        double r1854499 = 2.0;
        double r1854500 = r1854498 + r1854495;
        double r1854501 = r1854499 + r1854500;
        double r1854502 = r1854498 / r1854501;
        double r1854503 = r1854495 / r1854501;
        double r1854504 = r1854503 * r1854503;
        double r1854505 = 1.0;
        double r1854506 = r1854505 * r1854505;
        double r1854507 = r1854504 - r1854506;
        double r1854508 = r1854507 * r1854507;
        double r1854509 = r1854508 * r1854507;
        double r1854510 = cbrt(r1854509);
        double r1854511 = cbrt(r1854495);
        double r1854512 = r1854511 / r1854501;
        double r1854513 = r1854511 * r1854511;
        double r1854514 = r1854512 * r1854513;
        double r1854515 = r1854514 + r1854505;
        double r1854516 = r1854510 / r1854515;
        double r1854517 = r1854502 - r1854516;
        double r1854518 = r1854517 / r1854499;
        double r1854519 = 4.0;
        double r1854520 = r1854495 * r1854495;
        double r1854521 = r1854519 / r1854520;
        double r1854522 = 8.0;
        double r1854523 = r1854495 * r1854520;
        double r1854524 = r1854522 / r1854523;
        double r1854525 = r1854521 - r1854524;
        double r1854526 = r1854499 / r1854495;
        double r1854527 = r1854525 - r1854526;
        double r1854528 = r1854502 - r1854527;
        double r1854529 = r1854528 / r1854499;
        double r1854530 = r1854497 ? r1854518 : r1854529;
        return r1854530;
}

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 < 9.338107713205004e+21

    1. Initial program 0.7

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

    3. Applied div-sub0.7

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

    4. Applied associate-+l-0.7

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

    6. Applied flip--0.7

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

    8. Applied *-un-lft-identity0.7

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

    9. Applied add-cube-cbrt0.7

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

    10. Applied times-frac0.7

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

    11. Simplified0.7

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

    13. Applied add-cbrt-cube0.7

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

    if 9.338107713205004e+21 < alpha

    1. Initial program 50.4

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

    3. Applied div-sub50.3

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

    4. Applied associate-+l-48.6

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

    6. Applied flip--48.6

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

    7. Taylor expanded around inf 18.9

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

    8. Simplified18.9

      \[\leadsto \frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \color{blue}{\left(\left(\frac{4.0}{\alpha \cdot \alpha} - \frac{8.0}{\left(\alpha \cdot \alpha\right) \cdot \alpha}\right) - \frac{2.0}{\alpha}\right)}}{2.0}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification6.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 9.338107713205004 \cdot 10^{+21}:\\ \;\;\;\;\frac{\frac{\beta}{2.0 + \left(\beta + \alpha\right)} - \frac{\sqrt[3]{\left(\left(\frac{\alpha}{2.0 + \left(\beta + \alpha\right)} \cdot \frac{\alpha}{2.0 + \left(\beta + \alpha\right)} - 1.0 \cdot 1.0\right) \cdot \left(\frac{\alpha}{2.0 + \left(\beta + \alpha\right)} \cdot \frac{\alpha}{2.0 + \left(\beta + \alpha\right)} - 1.0 \cdot 1.0\right)\right) \cdot \left(\frac{\alpha}{2.0 + \left(\beta + \alpha\right)} \cdot \frac{\alpha}{2.0 + \left(\beta + \alpha\right)} - 1.0 \cdot 1.0\right)}}{\frac{\sqrt[3]{\alpha}}{2.0 + \left(\beta + \alpha\right)} \cdot \left(\sqrt[3]{\alpha} \cdot \sqrt[3]{\alpha}\right) + 1.0}}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{2.0 + \left(\beta + \alpha\right)} - \left(\left(\frac{4.0}{\alpha \cdot \alpha} - \frac{8.0}{\alpha \cdot \left(\alpha \cdot \alpha\right)}\right) - \frac{2.0}{\alpha}\right)}{2.0}\\ \end{array}\]

Reproduce

herbie shell --seed 2019156 +o rules:numerics
(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))