Average Error: 16.7 → 6.1
Time: 25.9s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 457427713.59383404:\\ \;\;\;\;\frac{e^{\log \left(\frac{1}{\left(\beta + \alpha\right) + 2.0} \cdot \beta - \left(\frac{\alpha}{\left(\beta + \alpha\right) + 2.0} - 1.0\right)\right)}}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt[3]{\beta} \cdot \sqrt[3]{\beta}}{\sqrt[3]{\left(\beta + \alpha\right) + 2.0} \cdot \sqrt[3]{\left(\beta + \alpha\right) + 2.0}} \cdot \frac{\sqrt[3]{\beta}}{\sqrt[3]{\left(\beta + \alpha\right) + 2.0}} - \left(\left(\frac{\frac{4.0}{\alpha}}{\alpha} - \frac{\frac{8.0}{\alpha}}{\alpha \cdot \alpha}\right) - \frac{2.0}{\alpha}\right)}{2.0}\\ \end{array}\]
\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \le 457427713.59383404:\\
\;\;\;\;\frac{e^{\log \left(\frac{1}{\left(\beta + \alpha\right) + 2.0} \cdot \beta - \left(\frac{\alpha}{\left(\beta + \alpha\right) + 2.0} - 1.0\right)\right)}}{2.0}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{\sqrt[3]{\beta} \cdot \sqrt[3]{\beta}}{\sqrt[3]{\left(\beta + \alpha\right) + 2.0} \cdot \sqrt[3]{\left(\beta + \alpha\right) + 2.0}} \cdot \frac{\sqrt[3]{\beta}}{\sqrt[3]{\left(\beta + \alpha\right) + 2.0}} - \left(\left(\frac{\frac{4.0}{\alpha}}{\alpha} - \frac{\frac{8.0}{\alpha}}{\alpha \cdot \alpha}\right) - \frac{2.0}{\alpha}\right)}{2.0}\\

\end{array}
double f(double alpha, double beta) {
        double r1169278 = beta;
        double r1169279 = alpha;
        double r1169280 = r1169278 - r1169279;
        double r1169281 = r1169279 + r1169278;
        double r1169282 = 2.0;
        double r1169283 = r1169281 + r1169282;
        double r1169284 = r1169280 / r1169283;
        double r1169285 = 1.0;
        double r1169286 = r1169284 + r1169285;
        double r1169287 = r1169286 / r1169282;
        return r1169287;
}
↓
double f(double alpha, double beta) {
        double r1169288 = alpha;
        double r1169289 = 457427713.59383404;
        bool r1169290 = r1169288 <= r1169289;
        double r1169291 = 1.0;
        double r1169292 = beta;
        double r1169293 = r1169292 + r1169288;
        double r1169294 = 2.0;
        double r1169295 = r1169293 + r1169294;
        double r1169296 = r1169291 / r1169295;
        double r1169297 = r1169296 * r1169292;
        double r1169298 = r1169288 / r1169295;
        double r1169299 = 1.0;
        double r1169300 = r1169298 - r1169299;
        double r1169301 = r1169297 - r1169300;
        double r1169302 = log(r1169301);
        double r1169303 = exp(r1169302);
        double r1169304 = r1169303 / r1169294;
        double r1169305 = cbrt(r1169292);
        double r1169306 = r1169305 * r1169305;
        double r1169307 = cbrt(r1169295);
        double r1169308 = r1169307 * r1169307;
        double r1169309 = r1169306 / r1169308;
        double r1169310 = r1169305 / r1169307;
        double r1169311 = r1169309 * r1169310;
        double r1169312 = 4.0;
        double r1169313 = r1169312 / r1169288;
        double r1169314 = r1169313 / r1169288;
        double r1169315 = 8.0;
        double r1169316 = r1169315 / r1169288;
        double r1169317 = r1169288 * r1169288;
        double r1169318 = r1169316 / r1169317;
        double r1169319 = r1169314 - r1169318;
        double r1169320 = r1169294 / r1169288;
        double r1169321 = r1169319 - r1169320;
        double r1169322 = r1169311 - r1169321;
        double r1169323 = r1169322 / r1169294;
        double r1169324 = r1169290 ? r1169304 : r1169323;
        return r1169324;
}

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 < 457427713.59383404

    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

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

    8. Applied div-inv0.1

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

    if 457427713.59383404 < alpha

    1. Initial program 49.9

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

    3. Applied div-sub49.9

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

    4. Applied associate-+l-48.3

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

    6. Applied add-cube-cbrt48.4

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

    7. Applied add-cube-cbrt48.3

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

    8. Applied times-frac48.3

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

    9. Taylor expanded around -inf 18.2

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

    10. Simplified18.2

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

  4. Final simplification6.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 457427713.59383404:\\ \;\;\;\;\frac{e^{\log \left(\frac{1}{\left(\beta + \alpha\right) + 2.0} \cdot \beta - \left(\frac{\alpha}{\left(\beta + \alpha\right) + 2.0} - 1.0\right)\right)}}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt[3]{\beta} \cdot \sqrt[3]{\beta}}{\sqrt[3]{\left(\beta + \alpha\right) + 2.0} \cdot \sqrt[3]{\left(\beta + \alpha\right) + 2.0}} \cdot \frac{\sqrt[3]{\beta}}{\sqrt[3]{\left(\beta + \alpha\right) + 2.0}} - \left(\left(\frac{\frac{4.0}{\alpha}}{\alpha} - \frac{\frac{8.0}{\alpha}}{\alpha \cdot \alpha}\right) - \frac{2.0}{\alpha}\right)}{2.0}\\ \end{array}\]

Reproduce

herbie shell --seed 2019142 +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))