Average Error: 16.6 → 3.3
Time: 5.0s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} \le -1:\\ \;\;\;\;\frac{\frac{1}{\frac{\left(\alpha + \beta\right) + 2}{\beta}} - \left(4 \cdot \frac{1}{{\alpha}^{2}} - \left(2 \cdot \frac{1}{\alpha} + 8 \cdot \frac{1}{{\alpha}^{3}}\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(e^{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2}}\right) + 1}{2}\\ \end{array}\]
\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}
↓
\begin{array}{l}
\mathbf{if}\;\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} \le -1:\\
\;\;\;\;\frac{\frac{1}{\frac{\left(\alpha + \beta\right) + 2}{\beta}} - \left(4 \cdot \frac{1}{{\alpha}^{2}} - \left(2 \cdot \frac{1}{\alpha} + 8 \cdot \frac{1}{{\alpha}^{3}}\right)\right)}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\log \left(e^{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2}}\right) + 1}{2}\\

\end{array}
double f(double alpha, double beta) {
        double r96505 = beta;
        double r96506 = alpha;
        double r96507 = r96505 - r96506;
        double r96508 = r96506 + r96505;
        double r96509 = 2.0;
        double r96510 = r96508 + r96509;
        double r96511 = r96507 / r96510;
        double r96512 = 1.0;
        double r96513 = r96511 + r96512;
        double r96514 = r96513 / r96509;
        return r96514;
}
↓
double f(double alpha, double beta) {
        double r96515 = beta;
        double r96516 = alpha;
        double r96517 = r96515 - r96516;
        double r96518 = r96516 + r96515;
        double r96519 = 2.0;
        double r96520 = r96518 + r96519;
        double r96521 = r96517 / r96520;
        double r96522 = -1.0;
        bool r96523 = r96521 <= r96522;
        double r96524 = 1.0;
        double r96525 = r96520 / r96515;
        double r96526 = r96524 / r96525;
        double r96527 = 4.0;
        double r96528 = 2.0;
        double r96529 = pow(r96516, r96528);
        double r96530 = r96524 / r96529;
        double r96531 = r96527 * r96530;
        double r96532 = r96524 / r96516;
        double r96533 = r96519 * r96532;
        double r96534 = 8.0;
        double r96535 = 3.0;
        double r96536 = pow(r96516, r96535);
        double r96537 = r96524 / r96536;
        double r96538 = r96534 * r96537;
        double r96539 = r96533 + r96538;
        double r96540 = r96531 - r96539;
        double r96541 = r96526 - r96540;
        double r96542 = r96541 / r96519;
        double r96543 = exp(r96521);
        double r96544 = log(r96543);
        double r96545 = 1.0;
        double r96546 = r96544 + r96545;
        double r96547 = r96546 / r96519;
        double r96548 = r96523 ? r96542 : r96547;
        return r96548;
}

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)) < -1.0

    1. Initial program 60.6

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
    2. Using strategy rm

    3. Applied div-sub60.6

      \[\leadsto \frac{\color{blue}{\left(\frac{\beta}{\left(\alpha + \beta\right) + 2} - \frac{\alpha}{\left(\alpha + \beta\right) + 2}\right)} + 1}{2}\]

    4. Applied associate-+l-58.7

      \[\leadsto \frac{\color{blue}{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)}}{2}\]
    5. Using strategy rm

    6. Applied clear-num58.7

      \[\leadsto \frac{\color{blue}{\frac{1}{\frac{\left(\alpha + \beta\right) + 2}{\beta}}} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)}{2}\]

    7. Taylor expanded around inf 10.6

      \[\leadsto \frac{\frac{1}{\frac{\left(\alpha + \beta\right) + 2}{\beta}} - \color{blue}{\left(4 \cdot \frac{1}{{\alpha}^{2}} - \left(2 \cdot \frac{1}{\alpha} + 8 \cdot \frac{1}{{\alpha}^{3}}\right)\right)}}{2}\]

    if -1.0 < (/ (- beta alpha) (+ (+ alpha beta) 2.0))

    1. Initial program 0.6

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
    2. Using strategy rm

    3. Applied add-log-exp0.6

      \[\leadsto \frac{\color{blue}{\log \left(e^{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2}}\right)} + 1}{2}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification3.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} \le -1:\\ \;\;\;\;\frac{\frac{1}{\frac{\left(\alpha + \beta\right) + 2}{\beta}} - \left(4 \cdot \frac{1}{{\alpha}^{2}} - \left(2 \cdot \frac{1}{\alpha} + 8 \cdot \frac{1}{{\alpha}^{3}}\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(e^{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2}}\right) + 1}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020089 
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/1"
  :precision binary64
  :pre (and (> alpha -1) (> beta -1))
  (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2)) 1) 2))