Average Error: 16.1 → 6.1
Time: 20.4s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 7715362331151764:\\ \;\;\;\;\frac{\frac{\beta}{2 + \left(\beta + \alpha\right)} - \frac{\sqrt[3]{\left(\left(\frac{\alpha}{2 + \left(\beta + \alpha\right)} \cdot \frac{\alpha}{2 + \left(\beta + \alpha\right)} - 1 \cdot 1\right) \cdot \left(\frac{\alpha}{2 + \left(\beta + \alpha\right)} \cdot \frac{\alpha}{2 + \left(\beta + \alpha\right)} - 1 \cdot 1\right)\right) \cdot \left(\frac{\alpha}{2 + \left(\beta + \alpha\right)} \cdot \frac{\alpha}{2 + \left(\beta + \alpha\right)} - 1 \cdot 1\right)}}{1 + \frac{\alpha}{2 + \left(\beta + \alpha\right)}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{2 + \left(\beta + \alpha\right)} - \left(\left(\frac{4}{\alpha \cdot \alpha} - \frac{2}{\alpha}\right) - \frac{\frac{8}{\alpha}}{\alpha \cdot \alpha}\right)}{2}\\ \end{array}\]
\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \le 7715362331151764:\\
\;\;\;\;\frac{\frac{\beta}{2 + \left(\beta + \alpha\right)} - \frac{\sqrt[3]{\left(\left(\frac{\alpha}{2 + \left(\beta + \alpha\right)} \cdot \frac{\alpha}{2 + \left(\beta + \alpha\right)} - 1 \cdot 1\right) \cdot \left(\frac{\alpha}{2 + \left(\beta + \alpha\right)} \cdot \frac{\alpha}{2 + \left(\beta + \alpha\right)} - 1 \cdot 1\right)\right) \cdot \left(\frac{\alpha}{2 + \left(\beta + \alpha\right)} \cdot \frac{\alpha}{2 + \left(\beta + \alpha\right)} - 1 \cdot 1\right)}}{1 + \frac{\alpha}{2 + \left(\beta + \alpha\right)}}}{2}\\

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

\end{array}
double f(double alpha, double beta) {
        double r3965276 = beta;
        double r3965277 = alpha;
        double r3965278 = r3965276 - r3965277;
        double r3965279 = r3965277 + r3965276;
        double r3965280 = 2.0;
        double r3965281 = r3965279 + r3965280;
        double r3965282 = r3965278 / r3965281;
        double r3965283 = 1.0;
        double r3965284 = r3965282 + r3965283;
        double r3965285 = r3965284 / r3965280;
        return r3965285;
}
↓
double f(double alpha, double beta) {
        double r3965286 = alpha;
        double r3965287 = 7715362331151764.0;
        bool r3965288 = r3965286 <= r3965287;
        double r3965289 = beta;
        double r3965290 = 2.0;
        double r3965291 = r3965289 + r3965286;
        double r3965292 = r3965290 + r3965291;
        double r3965293 = r3965289 / r3965292;
        double r3965294 = r3965286 / r3965292;
        double r3965295 = r3965294 * r3965294;
        double r3965296 = 1.0;
        double r3965297 = r3965296 * r3965296;
        double r3965298 = r3965295 - r3965297;
        double r3965299 = r3965298 * r3965298;
        double r3965300 = r3965299 * r3965298;
        double r3965301 = cbrt(r3965300);
        double r3965302 = r3965296 + r3965294;
        double r3965303 = r3965301 / r3965302;
        double r3965304 = r3965293 - r3965303;
        double r3965305 = r3965304 / r3965290;
        double r3965306 = 4.0;
        double r3965307 = r3965286 * r3965286;
        double r3965308 = r3965306 / r3965307;
        double r3965309 = r3965290 / r3965286;
        double r3965310 = r3965308 - r3965309;
        double r3965311 = 8.0;
        double r3965312 = r3965311 / r3965286;
        double r3965313 = r3965312 / r3965307;
        double r3965314 = r3965310 - r3965313;
        double r3965315 = r3965293 - r3965314;
        double r3965316 = r3965315 / r3965290;
        double r3965317 = r3965288 ? r3965305 : r3965316;
        return r3965317;
}

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 < 7715362331151764.0

    1. Initial program 0.4

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

    3. Applied div-sub0.4

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

    4. Applied associate-+l-0.4

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

    6. Applied flip--0.5

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

    8. Applied add-cbrt-cube0.5

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

    if 7715362331151764.0 < alpha

    1. Initial program 49.9

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
    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. Taylor expanded around inf 18.4

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

    6. Simplified18.4

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

  4. Final simplification6.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 7715362331151764:\\ \;\;\;\;\frac{\frac{\beta}{2 + \left(\beta + \alpha\right)} - \frac{\sqrt[3]{\left(\left(\frac{\alpha}{2 + \left(\beta + \alpha\right)} \cdot \frac{\alpha}{2 + \left(\beta + \alpha\right)} - 1 \cdot 1\right) \cdot \left(\frac{\alpha}{2 + \left(\beta + \alpha\right)} \cdot \frac{\alpha}{2 + \left(\beta + \alpha\right)} - 1 \cdot 1\right)\right) \cdot \left(\frac{\alpha}{2 + \left(\beta + \alpha\right)} \cdot \frac{\alpha}{2 + \left(\beta + \alpha\right)} - 1 \cdot 1\right)}}{1 + \frac{\alpha}{2 + \left(\beta + \alpha\right)}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{2 + \left(\beta + \alpha\right)} - \left(\left(\frac{4}{\alpha \cdot \alpha} - \frac{2}{\alpha}\right) - \frac{\frac{8}{\alpha}}{\alpha \cdot \alpha}\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/1"
  :pre (and (> alpha -1.0) (> beta -1.0))
  (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0))