Average Error: 16.3 → 6.0
Time: 18.2s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 1898265.8597097537:\\ \;\;\;\;\frac{\frac{\beta}{2.0 + \left(\beta + \alpha\right)} - \left(\frac{\frac{\alpha}{2.0 + \left(\beta + \alpha\right)} - 1.0}{2} + \log \left(\sqrt{e^{\frac{\alpha}{2.0 + \left(\beta + \alpha\right)} - 1.0}}\right)\right)}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{2.0 + \left(\beta + \alpha\right)} - \left(\left(\frac{4.0}{\alpha \cdot \alpha} - \frac{8.0}{\alpha \cdot \left(\alpha \cdot \alpha\right)}\right) - \frac{2.0}{\alpha}\right)}{2.0}\\ \end{array}\]
\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \le 1898265.8597097537:\\
\;\;\;\;\frac{\frac{\beta}{2.0 + \left(\beta + \alpha\right)} - \left(\frac{\frac{\alpha}{2.0 + \left(\beta + \alpha\right)} - 1.0}{2} + \log \left(\sqrt{e^{\frac{\alpha}{2.0 + \left(\beta + \alpha\right)} - 1.0}}\right)\right)}{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 r2785409 = beta;
        double r2785410 = alpha;
        double r2785411 = r2785409 - r2785410;
        double r2785412 = r2785410 + r2785409;
        double r2785413 = 2.0;
        double r2785414 = r2785412 + r2785413;
        double r2785415 = r2785411 / r2785414;
        double r2785416 = 1.0;
        double r2785417 = r2785415 + r2785416;
        double r2785418 = r2785417 / r2785413;
        return r2785418;
}
↓
double f(double alpha, double beta) {
        double r2785419 = alpha;
        double r2785420 = 1898265.8597097537;
        bool r2785421 = r2785419 <= r2785420;
        double r2785422 = beta;
        double r2785423 = 2.0;
        double r2785424 = r2785422 + r2785419;
        double r2785425 = r2785423 + r2785424;
        double r2785426 = r2785422 / r2785425;
        double r2785427 = r2785419 / r2785425;
        double r2785428 = 1.0;
        double r2785429 = r2785427 - r2785428;
        double r2785430 = 2.0;
        double r2785431 = r2785429 / r2785430;
        double r2785432 = exp(r2785429);
        double r2785433 = sqrt(r2785432);
        double r2785434 = log(r2785433);
        double r2785435 = r2785431 + r2785434;
        double r2785436 = r2785426 - r2785435;
        double r2785437 = r2785436 / r2785423;
        double r2785438 = 4.0;
        double r2785439 = r2785419 * r2785419;
        double r2785440 = r2785438 / r2785439;
        double r2785441 = 8.0;
        double r2785442 = r2785419 * r2785439;
        double r2785443 = r2785441 / r2785442;
        double r2785444 = r2785440 - r2785443;
        double r2785445 = r2785423 / r2785419;
        double r2785446 = r2785444 - r2785445;
        double r2785447 = r2785426 - r2785446;
        double r2785448 = r2785447 / r2785423;
        double r2785449 = r2785421 ? r2785437 : r2785448;
        return r2785449;
}

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 < 1898265.8597097537

    1. Initial program 0.1

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

    3. Applied div-sub0.1

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

    4. Applied associate-+l-0.1

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

    6. Applied add-log-exp0.1

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

    8. Applied add-sqr-sqrt0.1

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

    9. Applied log-prod0.1

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

    11. Applied *-un-lft-identity0.1

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

    12. Applied *-un-lft-identity0.1

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

    13. Applied distribute-lft-out--0.1

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

    14. Applied exp-prod0.1

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

    15. Applied sqrt-pow10.1

      \[\leadsto \frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \left(\log \left(\sqrt{e^{\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0}}\right) + \log \color{blue}{\left({\left(e^{1}\right)}^{\left(\frac{\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0}{2}\right)}\right)}\right)}{2.0}\]

    16. Applied log-pow0.1

      \[\leadsto \frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \left(\log \left(\sqrt{e^{\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0}}\right) + \color{blue}{\frac{\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0}{2} \cdot \log \left(e^{1}\right)}\right)}{2.0}\]

    17. Simplified0.1

      \[\leadsto \frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \left(\log \left(\sqrt{e^{\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0}}\right) + \frac{\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0}{2} \cdot \color{blue}{1}\right)}{2.0}\]

    if 1898265.8597097537 < alpha

    1. Initial program 50.0

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

    3. Applied div-sub49.9

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

    4. Applied associate-+l-48.3

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

    5. Taylor expanded around -inf 18.2

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

    6. Simplified18.2

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

  4. Final simplification6.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 1898265.8597097537:\\ \;\;\;\;\frac{\frac{\beta}{2.0 + \left(\beta + \alpha\right)} - \left(\frac{\frac{\alpha}{2.0 + \left(\beta + \alpha\right)} - 1.0}{2} + \log \left(\sqrt{e^{\frac{\alpha}{2.0 + \left(\beta + \alpha\right)} - 1.0}}\right)\right)}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{2.0 + \left(\beta + \alpha\right)} - \left(\left(\frac{4.0}{\alpha \cdot \alpha} - \frac{8.0}{\alpha \cdot \left(\alpha \cdot \alpha\right)}\right) - \frac{2.0}{\alpha}\right)}{2.0}\\ \end{array}\]

Reproduce

herbie shell --seed 2019133 
(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))