Average Error: 15.8 → 3.0
Time: 20.0s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} \le -0.9999999999999994:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \left(\left(\frac{4.0}{\alpha \cdot \alpha} - \frac{2.0}{\alpha}\right) - \frac{8.0}{\left(\alpha \cdot \alpha\right) \cdot \alpha}\right)}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(e^{1.0 + \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0}}\right)}{2.0}\\ \end{array}\]
\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}
↓
\begin{array}{l}
\mathbf{if}\;\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} \le -0.9999999999999994:\\
\;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \left(\left(\frac{4.0}{\alpha \cdot \alpha} - \frac{2.0}{\alpha}\right) - \frac{8.0}{\left(\alpha \cdot \alpha\right) \cdot \alpha}\right)}{2.0}\\

\mathbf{else}:\\
\;\;\;\;\frac{\log \left(e^{1.0 + \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0}}\right)}{2.0}\\

\end{array}
double f(double alpha, double beta) {
        double r3409499 = beta;
        double r3409500 = alpha;
        double r3409501 = r3409499 - r3409500;
        double r3409502 = r3409500 + r3409499;
        double r3409503 = 2.0;
        double r3409504 = r3409502 + r3409503;
        double r3409505 = r3409501 / r3409504;
        double r3409506 = 1.0;
        double r3409507 = r3409505 + r3409506;
        double r3409508 = r3409507 / r3409503;
        return r3409508;
}
↓
double f(double alpha, double beta) {
        double r3409509 = beta;
        double r3409510 = alpha;
        double r3409511 = r3409509 - r3409510;
        double r3409512 = r3409510 + r3409509;
        double r3409513 = 2.0;
        double r3409514 = r3409512 + r3409513;
        double r3409515 = r3409511 / r3409514;
        double r3409516 = -0.9999999999999994;
        bool r3409517 = r3409515 <= r3409516;
        double r3409518 = r3409509 / r3409514;
        double r3409519 = 4.0;
        double r3409520 = r3409510 * r3409510;
        double r3409521 = r3409519 / r3409520;
        double r3409522 = r3409513 / r3409510;
        double r3409523 = r3409521 - r3409522;
        double r3409524 = 8.0;
        double r3409525 = r3409520 * r3409510;
        double r3409526 = r3409524 / r3409525;
        double r3409527 = r3409523 - r3409526;
        double r3409528 = r3409518 - r3409527;
        double r3409529 = r3409528 / r3409513;
        double r3409530 = 1.0;
        double r3409531 = r3409530 + r3409515;
        double r3409532 = exp(r3409531);
        double r3409533 = log(r3409532);
        double r3409534 = r3409533 / r3409513;
        double r3409535 = r3409517 ? r3409529 : r3409534;
        return r3409535;
}

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 (/ (- beta alpha) (+ (+ alpha beta) 2.0)) < -0.9999999999999994

    1. Initial program 60.5

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

    3. Applied div-sub60.5

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

    4. Applied associate-+l-58.8

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

    5. Taylor expanded around inf 10.4

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

    6. Simplified10.4

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

    if -0.9999999999999994 < (/ (- beta alpha) (+ (+ alpha beta) 2.0))

    1. Initial program 0.5

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

    3. Applied add-log-exp0.5

      \[\leadsto \frac{\color{blue}{\log \left(e^{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}\right)}}{2.0}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification3.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} \le -0.9999999999999994:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \left(\left(\frac{4.0}{\alpha \cdot \alpha} - \frac{2.0}{\alpha}\right) - \frac{8.0}{\left(\alpha \cdot \alpha\right) \cdot \alpha}\right)}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(e^{1.0 + \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0}}\right)}{2.0}\\ \end{array}\]

Reproduce

herbie shell --seed 2019143 
(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))