Average Error: 15.9 → 5.9
Time: 25.1s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 536060982.7574347:\\ \;\;\;\;\frac{e^{\log \left(\frac{\beta}{2.0 + \left(\beta + \alpha\right)} - \left(\frac{\alpha}{2.0 + \left(\beta + \alpha\right)} - 1.0\right)\right)}}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{2.0 + \left(\beta + \alpha\right)} - \left(\left(\frac{4.0}{\alpha \cdot \alpha} - \frac{8.0}{\alpha \cdot \left(\alpha \cdot \alpha\right)}\right) - \frac{2.0}{\alpha}\right)}{2.0}\\ \end{array}\]
\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \le 536060982.7574347:\\
\;\;\;\;\frac{e^{\log \left(\frac{\beta}{2.0 + \left(\beta + \alpha\right)} - \left(\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 r2989578 = beta;
        double r2989579 = alpha;
        double r2989580 = r2989578 - r2989579;
        double r2989581 = r2989579 + r2989578;
        double r2989582 = 2.0;
        double r2989583 = r2989581 + r2989582;
        double r2989584 = r2989580 / r2989583;
        double r2989585 = 1.0;
        double r2989586 = r2989584 + r2989585;
        double r2989587 = r2989586 / r2989582;
        return r2989587;
}
↓
double f(double alpha, double beta) {
        double r2989588 = alpha;
        double r2989589 = 536060982.7574347;
        bool r2989590 = r2989588 <= r2989589;
        double r2989591 = beta;
        double r2989592 = 2.0;
        double r2989593 = r2989591 + r2989588;
        double r2989594 = r2989592 + r2989593;
        double r2989595 = r2989591 / r2989594;
        double r2989596 = r2989588 / r2989594;
        double r2989597 = 1.0;
        double r2989598 = r2989596 - r2989597;
        double r2989599 = r2989595 - r2989598;
        double r2989600 = log(r2989599);
        double r2989601 = exp(r2989600);
        double r2989602 = r2989601 / r2989592;
        double r2989603 = 4.0;
        double r2989604 = r2989588 * r2989588;
        double r2989605 = r2989603 / r2989604;
        double r2989606 = 8.0;
        double r2989607 = r2989588 * r2989604;
        double r2989608 = r2989606 / r2989607;
        double r2989609 = r2989605 - r2989608;
        double r2989610 = r2989592 / r2989588;
        double r2989611 = r2989609 - r2989610;
        double r2989612 = r2989595 - r2989611;
        double r2989613 = r2989612 / r2989592;
        double r2989614 = r2989590 ? r2989602 : r2989613;
        return r2989614;
}

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 < 536060982.7574347

    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-exp-log0.1

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

    if 536060982.7574347 < alpha

    1. Initial program 49.6

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

    3. Applied div-sub49.6

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

    4. Applied associate-+l-48.1

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

    5. Taylor expanded around inf 18.3

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

    6. Simplified18.3

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

  4. Final simplification5.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 536060982.7574347:\\ \;\;\;\;\frac{e^{\log \left(\frac{\beta}{2.0 + \left(\beta + \alpha\right)} - \left(\frac{\alpha}{2.0 + \left(\beta + \alpha\right)} - 1.0\right)\right)}}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{2.0 + \left(\beta + \alpha\right)} - \left(\left(\frac{4.0}{\alpha \cdot \alpha} - \frac{8.0}{\alpha \cdot \left(\alpha \cdot \alpha\right)}\right) - \frac{2.0}{\alpha}\right)}{2.0}\\ \end{array}\]

Reproduce

herbie shell --seed 2019151 
(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))