Average Error: 16.4 → 5.9
Time: 23.5s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 460518623.3564523:\\ \;\;\;\;e^{\log \left(\frac{\frac{1}{\frac{\left(\alpha + \beta\right) + 2.0}{\beta}} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0\right)}{2.0}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \left(\frac{4.0}{\alpha \cdot \alpha} - \left(\frac{2.0}{\alpha} + \frac{\frac{8.0}{\alpha}}{\alpha \cdot \alpha}\right)\right)}{2.0}\\ \end{array}\]
\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \le 460518623.3564523:\\
\;\;\;\;e^{\log \left(\frac{\frac{1}{\frac{\left(\alpha + \beta\right) + 2.0}{\beta}} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0\right)}{2.0}\right)}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \left(\frac{4.0}{\alpha \cdot \alpha} - \left(\frac{2.0}{\alpha} + \frac{\frac{8.0}{\alpha}}{\alpha \cdot \alpha}\right)\right)}{2.0}\\

\end{array}
double f(double alpha, double beta) {
        double r2987273 = beta;
        double r2987274 = alpha;
        double r2987275 = r2987273 - r2987274;
        double r2987276 = r2987274 + r2987273;
        double r2987277 = 2.0;
        double r2987278 = r2987276 + r2987277;
        double r2987279 = r2987275 / r2987278;
        double r2987280 = 1.0;
        double r2987281 = r2987279 + r2987280;
        double r2987282 = r2987281 / r2987277;
        return r2987282;
}
↓
double f(double alpha, double beta) {
        double r2987283 = alpha;
        double r2987284 = 460518623.3564523;
        bool r2987285 = r2987283 <= r2987284;
        double r2987286 = 1.0;
        double r2987287 = beta;
        double r2987288 = r2987283 + r2987287;
        double r2987289 = 2.0;
        double r2987290 = r2987288 + r2987289;
        double r2987291 = r2987290 / r2987287;
        double r2987292 = r2987286 / r2987291;
        double r2987293 = r2987283 / r2987290;
        double r2987294 = 1.0;
        double r2987295 = r2987293 - r2987294;
        double r2987296 = r2987292 - r2987295;
        double r2987297 = r2987296 / r2987289;
        double r2987298 = log(r2987297);
        double r2987299 = exp(r2987298);
        double r2987300 = r2987287 / r2987290;
        double r2987301 = 4.0;
        double r2987302 = r2987283 * r2987283;
        double r2987303 = r2987301 / r2987302;
        double r2987304 = r2987289 / r2987283;
        double r2987305 = 8.0;
        double r2987306 = r2987305 / r2987283;
        double r2987307 = r2987306 / r2987302;
        double r2987308 = r2987304 + r2987307;
        double r2987309 = r2987303 - r2987308;
        double r2987310 = r2987300 - r2987309;
        double r2987311 = r2987310 / r2987289;
        double r2987312 = r2987285 ? r2987299 : r2987311;
        return r2987312;
}

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 < 460518623.3564523

    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

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

    8. Applied clear-num0.1

      \[\leadsto e^{\log \left(\frac{\color{blue}{\frac{1}{\frac{\left(\alpha + \beta\right) + 2.0}{\beta}}} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0\right)}{2.0}\right)}\]

    if 460518623.3564523 < alpha

    1. Initial program 50.0

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

    3. Applied div-sub50.0

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

    4. Applied associate-+l-48.5

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

    5. Taylor expanded around inf 17.9

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

    6. Simplified17.9

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

  4. Final simplification5.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 460518623.3564523:\\ \;\;\;\;e^{\log \left(\frac{\frac{1}{\frac{\left(\alpha + \beta\right) + 2.0}{\beta}} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0\right)}{2.0}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \left(\frac{4.0}{\alpha \cdot \alpha} - \left(\frac{2.0}{\alpha} + \frac{\frac{8.0}{\alpha}}{\alpha \cdot \alpha}\right)\right)}{2.0}\\ \end{array}\]

Reproduce

herbie shell --seed 2019162 +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))