Average Error: 16.0 → 6.2
Time: 17.4s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 12650120878.39988:\\ \;\;\;\;\frac{\frac{\beta \cdot \left(\frac{\alpha}{2.0 + \left(\beta + \alpha\right)} + 1.0\right) - \left(2.0 + \left(\beta + \alpha\right)\right) \cdot \sqrt[3]{\left(\left(\frac{\alpha}{2.0 + \left(\beta + \alpha\right)} \cdot \frac{\alpha}{2.0 + \left(\beta + \alpha\right)} - 1.0 \cdot 1.0\right) \cdot \left(\frac{\alpha}{2.0 + \left(\beta + \alpha\right)} \cdot \frac{\alpha}{2.0 + \left(\beta + \alpha\right)} - 1.0 \cdot 1.0\right)\right) \cdot \left(\frac{\alpha}{2.0 + \left(\beta + \alpha\right)} \cdot \frac{\alpha}{2.0 + \left(\beta + \alpha\right)} - 1.0 \cdot 1.0\right)}}{\left(\frac{\alpha}{2.0 + \left(\beta + \alpha\right)} + 1.0\right) \cdot \left(2.0 + \left(\beta + \alpha\right)\right)}}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{2.0 + \left(\beta + \alpha\right)} - \left(\left(\frac{\frac{4.0}{\alpha}}{\alpha} - \frac{2.0}{\alpha}\right) - \frac{\frac{8.0}{\alpha}}{\alpha \cdot \alpha}\right)}{2.0}\\ \end{array}\]
\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \le 12650120878.39988:\\
\;\;\;\;\frac{\frac{\beta \cdot \left(\frac{\alpha}{2.0 + \left(\beta + \alpha\right)} + 1.0\right) - \left(2.0 + \left(\beta + \alpha\right)\right) \cdot \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)}}{\left(\frac{\alpha}{2.0 + \left(\beta + \alpha\right)} + 1.0\right) \cdot \left(2.0 + \left(\beta + \alpha\right)\right)}}{2.0}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{\beta}{2.0 + \left(\beta + \alpha\right)} - \left(\left(\frac{\frac{4.0}{\alpha}}{\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 r2169501 = beta;
        double r2169502 = alpha;
        double r2169503 = r2169501 - r2169502;
        double r2169504 = r2169502 + r2169501;
        double r2169505 = 2.0;
        double r2169506 = r2169504 + r2169505;
        double r2169507 = r2169503 / r2169506;
        double r2169508 = 1.0;
        double r2169509 = r2169507 + r2169508;
        double r2169510 = r2169509 / r2169505;
        return r2169510;
}
↓
double f(double alpha, double beta) {
        double r2169511 = alpha;
        double r2169512 = 12650120878.39988;
        bool r2169513 = r2169511 <= r2169512;
        double r2169514 = beta;
        double r2169515 = 2.0;
        double r2169516 = r2169514 + r2169511;
        double r2169517 = r2169515 + r2169516;
        double r2169518 = r2169511 / r2169517;
        double r2169519 = 1.0;
        double r2169520 = r2169518 + r2169519;
        double r2169521 = r2169514 * r2169520;
        double r2169522 = r2169518 * r2169518;
        double r2169523 = r2169519 * r2169519;
        double r2169524 = r2169522 - r2169523;
        double r2169525 = r2169524 * r2169524;
        double r2169526 = r2169525 * r2169524;
        double r2169527 = cbrt(r2169526);
        double r2169528 = r2169517 * r2169527;
        double r2169529 = r2169521 - r2169528;
        double r2169530 = r2169520 * r2169517;
        double r2169531 = r2169529 / r2169530;
        double r2169532 = r2169531 / r2169515;
        double r2169533 = r2169514 / r2169517;
        double r2169534 = 4.0;
        double r2169535 = r2169534 / r2169511;
        double r2169536 = r2169535 / r2169511;
        double r2169537 = r2169515 / r2169511;
        double r2169538 = r2169536 - r2169537;
        double r2169539 = 8.0;
        double r2169540 = r2169539 / r2169511;
        double r2169541 = r2169511 * r2169511;
        double r2169542 = r2169540 / r2169541;
        double r2169543 = r2169538 - r2169542;
        double r2169544 = r2169533 - r2169543;
        double r2169545 = r2169544 / r2169515;
        double r2169546 = r2169513 ? r2169532 : r2169545;
        return r2169546;
}

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 < 12650120878.39988

    1. Initial program 0.2

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

    3. Applied div-sub0.2

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

    4. Applied associate-+l-0.2

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

    6. Applied flip--0.2

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

    7. Applied frac-sub0.2

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

    9. Applied add-cbrt-cube0.2

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

    if 12650120878.39988 < alpha

    1. Initial program 49.7

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

    3. Applied div-sub49.6

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

    4. Applied associate-+l-48.0

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

    5. Taylor expanded around inf 18.9

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

    6. Simplified18.9

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

  4. Final simplification6.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 12650120878.39988:\\ \;\;\;\;\frac{\frac{\beta \cdot \left(\frac{\alpha}{2.0 + \left(\beta + \alpha\right)} + 1.0\right) - \left(2.0 + \left(\beta + \alpha\right)\right) \cdot \sqrt[3]{\left(\left(\frac{\alpha}{2.0 + \left(\beta + \alpha\right)} \cdot \frac{\alpha}{2.0 + \left(\beta + \alpha\right)} - 1.0 \cdot 1.0\right) \cdot \left(\frac{\alpha}{2.0 + \left(\beta + \alpha\right)} \cdot \frac{\alpha}{2.0 + \left(\beta + \alpha\right)} - 1.0 \cdot 1.0\right)\right) \cdot \left(\frac{\alpha}{2.0 + \left(\beta + \alpha\right)} \cdot \frac{\alpha}{2.0 + \left(\beta + \alpha\right)} - 1.0 \cdot 1.0\right)}}{\left(\frac{\alpha}{2.0 + \left(\beta + \alpha\right)} + 1.0\right) \cdot \left(2.0 + \left(\beta + \alpha\right)\right)}}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{2.0 + \left(\beta + \alpha\right)} - \left(\left(\frac{\frac{4.0}{\alpha}}{\alpha} - \frac{2.0}{\alpha}\right) - \frac{\frac{8.0}{\alpha}}{\alpha \cdot \alpha}\right)}{2.0}\\ \end{array}\]

Reproduce

herbie shell --seed 2019155 
(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))