Average Error: 15.7 → 3.0
Time: 15.7s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} \le -0.9999999999999637:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \left(\frac{4.0}{\alpha \cdot \alpha} - \left(\frac{8.0}{\left(\alpha \cdot \alpha\right) \cdot \alpha} + \frac{2.0}{\alpha}\right)\right)}{2.0}\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0\right)}{2.0}\right)}\\ \end{array}\]
\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 -0.9999999999999637:\\
\;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \left(\frac{4.0}{\alpha \cdot \alpha} - \left(\frac{8.0}{\left(\alpha \cdot \alpha\right) \cdot \alpha} + \frac{2.0}{\alpha}\right)\right)}{2.0}\\

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

\end{array}
double f(double alpha, double beta) {
        double r2004415 = beta;
        double r2004416 = alpha;
        double r2004417 = r2004415 - r2004416;
        double r2004418 = r2004416 + r2004415;
        double r2004419 = 2.0;
        double r2004420 = r2004418 + r2004419;
        double r2004421 = r2004417 / r2004420;
        double r2004422 = 1.0;
        double r2004423 = r2004421 + r2004422;
        double r2004424 = r2004423 / r2004419;
        return r2004424;
}
↓
double f(double alpha, double beta) {
        double r2004425 = beta;
        double r2004426 = alpha;
        double r2004427 = r2004425 - r2004426;
        double r2004428 = r2004426 + r2004425;
        double r2004429 = 2.0;
        double r2004430 = r2004428 + r2004429;
        double r2004431 = r2004427 / r2004430;
        double r2004432 = -0.9999999999999637;
        bool r2004433 = r2004431 <= r2004432;
        double r2004434 = r2004425 / r2004430;
        double r2004435 = 4.0;
        double r2004436 = r2004426 * r2004426;
        double r2004437 = r2004435 / r2004436;
        double r2004438 = 8.0;
        double r2004439 = r2004436 * r2004426;
        double r2004440 = r2004438 / r2004439;
        double r2004441 = r2004429 / r2004426;
        double r2004442 = r2004440 + r2004441;
        double r2004443 = r2004437 - r2004442;
        double r2004444 = r2004434 - r2004443;
        double r2004445 = r2004444 / r2004429;
        double r2004446 = r2004426 / r2004430;
        double r2004447 = 1.0;
        double r2004448 = r2004446 - r2004447;
        double r2004449 = r2004434 - r2004448;
        double r2004450 = r2004449 / r2004429;
        double r2004451 = log(r2004450);
        double r2004452 = exp(r2004451);
        double r2004453 = r2004433 ? r2004445 : r2004452;
        return r2004453;
}

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)) < -0.9999999999999637

    1. Initial program 60.4

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

    3. Applied div-sub60.4

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

    4. Applied associate-+l-58.7

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

    6. Applied add-log-exp58.7

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

    7. Taylor expanded around inf 10.6

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

    8. Simplified10.6

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

    if -0.9999999999999637 < (/ (- beta alpha) (+ (+ alpha beta) 2.0))

    1. Initial program 0.4

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

    3. Applied div-sub0.4

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

    4. Applied associate-+l-0.4

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

    6. Applied add-exp-log0.4

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

  4. Final simplification3.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} \le -0.9999999999999637:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \left(\frac{4.0}{\alpha \cdot \alpha} - \left(\frac{8.0}{\left(\alpha \cdot \alpha\right) \cdot \alpha} + \frac{2.0}{\alpha}\right)\right)}{2.0}\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0\right)}{2.0}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019153 
(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))