Average Error: 16.3 → 3.2
Time: 23.4s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} \le -0.9999999551912412:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \left(\left(\frac{4.0}{\alpha \cdot \alpha} - \frac{2.0}{\alpha}\right) - \frac{8.0}{\left(\alpha \cdot \alpha\right) \cdot \alpha}\right)}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(e^{1.0 + \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0}}\right)}{2.0}\\ \end{array}\]
\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 -0.9999999551912412:\\
\;\;\;\;\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 r4562442 = beta;
        double r4562443 = alpha;
        double r4562444 = r4562442 - r4562443;
        double r4562445 = r4562443 + r4562442;
        double r4562446 = 2.0;
        double r4562447 = r4562445 + r4562446;
        double r4562448 = r4562444 / r4562447;
        double r4562449 = 1.0;
        double r4562450 = r4562448 + r4562449;
        double r4562451 = r4562450 / r4562446;
        return r4562451;
}
↓
double f(double alpha, double beta) {
        double r4562452 = beta;
        double r4562453 = alpha;
        double r4562454 = r4562452 - r4562453;
        double r4562455 = r4562453 + r4562452;
        double r4562456 = 2.0;
        double r4562457 = r4562455 + r4562456;
        double r4562458 = r4562454 / r4562457;
        double r4562459 = -0.9999999551912412;
        bool r4562460 = r4562458 <= r4562459;
        double r4562461 = r4562452 / r4562457;
        double r4562462 = 4.0;
        double r4562463 = r4562453 * r4562453;
        double r4562464 = r4562462 / r4562463;
        double r4562465 = r4562456 / r4562453;
        double r4562466 = r4562464 - r4562465;
        double r4562467 = 8.0;
        double r4562468 = r4562463 * r4562453;
        double r4562469 = r4562467 / r4562468;
        double r4562470 = r4562466 - r4562469;
        double r4562471 = r4562461 - r4562470;
        double r4562472 = r4562471 / r4562456;
        double r4562473 = 1.0;
        double r4562474 = r4562473 + r4562458;
        double r4562475 = exp(r4562474);
        double r4562476 = log(r4562475);
        double r4562477 = r4562476 / r4562456;
        double r4562478 = r4562460 ? r4562472 : r4562477;
        return r4562478;
}

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)) < -0.9999999551912412

    1. Initial program 59.7

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

    3. Applied div-sub59.7

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

    4. Applied associate-+l-57.8

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

    5. Taylor expanded around inf 11.5

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

    6. Simplified11.5

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

    if -0.9999999551912412 < (/ (- beta alpha) (+ (+ alpha beta) 2.0))

    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 add-log-exp0.1

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

  4. Final simplification3.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} \le -0.9999999551912412:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \left(\left(\frac{4.0}{\alpha \cdot \alpha} - \frac{2.0}{\alpha}\right) - \frac{8.0}{\left(\alpha \cdot \alpha\right) \cdot \alpha}\right)}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(e^{1.0 + \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0}}\right)}{2.0}\\ \end{array}\]

Reproduce

herbie shell --seed 2019158 +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))