Average Error: 16.5 → 6.1
Time: 15.1s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 8092481.162986398674547672271728515625:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \sqrt[3]{{\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)}^{3}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\left(\frac{4}{{\alpha}^{2}} - \frac{8}{{\alpha}^{3}}\right) - \frac{2}{\alpha}\right)}{2}\\ \end{array}\]
\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \le 8092481.162986398674547672271728515625:\\
\;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \sqrt[3]{{\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)}^{3}}}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\left(\frac{4}{{\alpha}^{2}} - \frac{8}{{\alpha}^{3}}\right) - \frac{2}{\alpha}\right)}{2}\\

\end{array}
double f(double alpha, double beta) {
        double r62242 = beta;
        double r62243 = alpha;
        double r62244 = r62242 - r62243;
        double r62245 = r62243 + r62242;
        double r62246 = 2.0;
        double r62247 = r62245 + r62246;
        double r62248 = r62244 / r62247;
        double r62249 = 1.0;
        double r62250 = r62248 + r62249;
        double r62251 = r62250 / r62246;
        return r62251;
}
↓
double f(double alpha, double beta) {
        double r62252 = alpha;
        double r62253 = 8092481.162986399;
        bool r62254 = r62252 <= r62253;
        double r62255 = beta;
        double r62256 = r62252 + r62255;
        double r62257 = 2.0;
        double r62258 = r62256 + r62257;
        double r62259 = r62255 / r62258;
        double r62260 = r62252 / r62258;
        double r62261 = 1.0;
        double r62262 = r62260 - r62261;
        double r62263 = 3.0;
        double r62264 = pow(r62262, r62263);
        double r62265 = cbrt(r62264);
        double r62266 = r62259 - r62265;
        double r62267 = r62266 / r62257;
        double r62268 = 4.0;
        double r62269 = 2.0;
        double r62270 = pow(r62252, r62269);
        double r62271 = r62268 / r62270;
        double r62272 = 8.0;
        double r62273 = pow(r62252, r62263);
        double r62274 = r62272 / r62273;
        double r62275 = r62271 - r62274;
        double r62276 = r62257 / r62252;
        double r62277 = r62275 - r62276;
        double r62278 = r62259 - r62277;
        double r62279 = r62278 / r62257;
        double r62280 = r62254 ? r62267 : r62279;
        return r62280;
}

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 < 8092481.162986399

    1. Initial program 0.1

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

    3. Applied div-sub0.1

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

    4. Applied associate-+l-0.1

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

    6. Applied add-cbrt-cube0.1

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

    7. Simplified0.1

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

    if 8092481.162986399 < alpha

    1. Initial program 50.0

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

    3. Applied div-sub49.9

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

    4. Applied associate-+l-48.3

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

    6. Applied add-cbrt-cube48.3

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

    7. Simplified48.3

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

    8. Taylor expanded around inf 18.4

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

    9. Simplified18.4

      \[\leadsto \frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \color{blue}{\left(\left(\frac{4}{{\alpha}^{2}} - \frac{8}{{\alpha}^{3}}\right) - \frac{2}{\alpha}\right)}}{2}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification6.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 8092481.162986398674547672271728515625:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \sqrt[3]{{\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)}^{3}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\left(\frac{4}{{\alpha}^{2}} - \frac{8}{{\alpha}^{3}}\right) - \frac{2}{\alpha}\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/1"
  :precision binary64
  :pre (and (> alpha -1) (> beta -1))
  (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2)) 1) 2))