Average Error: 16.4 → 6.4
Time: 19.8s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 1485119.434541507:\\ \;\;\;\;\frac{e^{\log \left(\frac{\beta}{2.0 + \left(\beta + \alpha\right)} - \left(\frac{\alpha}{2.0 + \left(\beta + \alpha\right)} - 1.0\right)\right)}}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt[3]{\beta} \cdot \sqrt[3]{\beta}}{\sqrt[3]{2.0 + \left(\beta + \alpha\right)} \cdot \sqrt[3]{2.0 + \left(\beta + \alpha\right)}} \cdot \frac{\sqrt[3]{\beta}}{\sqrt[3]{2.0 + \left(\beta + \alpha\right)}} - \left(\left(4.0 - \frac{8.0}{\alpha}\right) \cdot \frac{1}{\alpha \cdot \alpha} - \frac{2.0}{\alpha}\right)}{2.0}\\ \end{array}\]
\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \le 1485119.434541507:\\
\;\;\;\;\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{\sqrt[3]{\beta} \cdot \sqrt[3]{\beta}}{\sqrt[3]{2.0 + \left(\beta + \alpha\right)} \cdot \sqrt[3]{2.0 + \left(\beta + \alpha\right)}} \cdot \frac{\sqrt[3]{\beta}}{\sqrt[3]{2.0 + \left(\beta + \alpha\right)}} - \left(\left(4.0 - \frac{8.0}{\alpha}\right) \cdot \frac{1}{\alpha \cdot \alpha} - \frac{2.0}{\alpha}\right)}{2.0}\\

\end{array}
double f(double alpha, double beta) {
        double r3423220 = beta;
        double r3423221 = alpha;
        double r3423222 = r3423220 - r3423221;
        double r3423223 = r3423221 + r3423220;
        double r3423224 = 2.0;
        double r3423225 = r3423223 + r3423224;
        double r3423226 = r3423222 / r3423225;
        double r3423227 = 1.0;
        double r3423228 = r3423226 + r3423227;
        double r3423229 = r3423228 / r3423224;
        return r3423229;
}
↓
double f(double alpha, double beta) {
        double r3423230 = alpha;
        double r3423231 = 1485119.434541507;
        bool r3423232 = r3423230 <= r3423231;
        double r3423233 = beta;
        double r3423234 = 2.0;
        double r3423235 = r3423233 + r3423230;
        double r3423236 = r3423234 + r3423235;
        double r3423237 = r3423233 / r3423236;
        double r3423238 = r3423230 / r3423236;
        double r3423239 = 1.0;
        double r3423240 = r3423238 - r3423239;
        double r3423241 = r3423237 - r3423240;
        double r3423242 = log(r3423241);
        double r3423243 = exp(r3423242);
        double r3423244 = r3423243 / r3423234;
        double r3423245 = cbrt(r3423233);
        double r3423246 = r3423245 * r3423245;
        double r3423247 = cbrt(r3423236);
        double r3423248 = r3423247 * r3423247;
        double r3423249 = r3423246 / r3423248;
        double r3423250 = r3423245 / r3423247;
        double r3423251 = r3423249 * r3423250;
        double r3423252 = 4.0;
        double r3423253 = 8.0;
        double r3423254 = r3423253 / r3423230;
        double r3423255 = r3423252 - r3423254;
        double r3423256 = 1.0;
        double r3423257 = r3423230 * r3423230;
        double r3423258 = r3423256 / r3423257;
        double r3423259 = r3423255 * r3423258;
        double r3423260 = r3423234 / r3423230;
        double r3423261 = r3423259 - r3423260;
        double r3423262 = r3423251 - r3423261;
        double r3423263 = r3423262 / r3423234;
        double r3423264 = r3423232 ? r3423244 : r3423263;
        return r3423264;
}

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 < 1485119.434541507

    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 1485119.434541507 < alpha

    1. Initial program 49.0

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

    3. Applied div-sub49.0

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

    4. Applied associate-+l-47.4

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

    6. Applied add-cube-cbrt47.5

      \[\leadsto \frac{\frac{\beta}{\color{blue}{\left(\sqrt[3]{\left(\alpha + \beta\right) + 2.0} \cdot \sqrt[3]{\left(\alpha + \beta\right) + 2.0}\right) \cdot \sqrt[3]{\left(\alpha + \beta\right) + 2.0}}} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0\right)}{2.0}\]

    7. Applied add-cube-cbrt47.4

      \[\leadsto \frac{\frac{\color{blue}{\left(\sqrt[3]{\beta} \cdot \sqrt[3]{\beta}\right) \cdot \sqrt[3]{\beta}}}{\left(\sqrt[3]{\left(\alpha + \beta\right) + 2.0} \cdot \sqrt[3]{\left(\alpha + \beta\right) + 2.0}\right) \cdot \sqrt[3]{\left(\alpha + \beta\right) + 2.0}} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0\right)}{2.0}\]

    8. Applied times-frac47.4

      \[\leadsto \frac{\color{blue}{\frac{\sqrt[3]{\beta} \cdot \sqrt[3]{\beta}}{\sqrt[3]{\left(\alpha + \beta\right) + 2.0} \cdot \sqrt[3]{\left(\alpha + \beta\right) + 2.0}} \cdot \frac{\sqrt[3]{\beta}}{\sqrt[3]{\left(\alpha + \beta\right) + 2.0}}} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0\right)}{2.0}\]

    9. Taylor expanded around -inf 18.9

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

    10. Simplified18.9

      \[\leadsto \frac{\frac{\sqrt[3]{\beta} \cdot \sqrt[3]{\beta}}{\sqrt[3]{\left(\alpha + \beta\right) + 2.0} \cdot \sqrt[3]{\left(\alpha + \beta\right) + 2.0}} \cdot \frac{\sqrt[3]{\beta}}{\sqrt[3]{\left(\alpha + \beta\right) + 2.0}} - \color{blue}{\left(\frac{1}{\alpha \cdot \alpha} \cdot \left(4.0 - \frac{8.0}{\alpha}\right) - \frac{2.0}{\alpha}\right)}}{2.0}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification6.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 1485119.434541507:\\ \;\;\;\;\frac{e^{\log \left(\frac{\beta}{2.0 + \left(\beta + \alpha\right)} - \left(\frac{\alpha}{2.0 + \left(\beta + \alpha\right)} - 1.0\right)\right)}}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt[3]{\beta} \cdot \sqrt[3]{\beta}}{\sqrt[3]{2.0 + \left(\beta + \alpha\right)} \cdot \sqrt[3]{2.0 + \left(\beta + \alpha\right)}} \cdot \frac{\sqrt[3]{\beta}}{\sqrt[3]{2.0 + \left(\beta + \alpha\right)}} - \left(\left(4.0 - \frac{8.0}{\alpha}\right) \cdot \frac{1}{\alpha \cdot \alpha} - \frac{2.0}{\alpha}\right)}{2.0}\\ \end{array}\]

Reproduce

herbie shell --seed 2019130 
(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))