Average Error: 15.9 → 6.0
Time: 23.6s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 644102.615074100787751376628875732421875:\\ \;\;\;\;\frac{\log \left(e^{\frac{\beta}{2 + \left(\beta + \alpha\right)}}\right) - \left(\frac{\alpha}{2 + \left(\beta + \alpha\right)} - 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{\frac{\beta}{2 + \left(\beta + \alpha\right)}} \cdot \left(\sqrt[3]{\frac{\beta}{2 + \left(\beta + \alpha\right)}} \cdot \sqrt[3]{\frac{\beta}{2 + \left(\beta + \alpha\right)}}\right) - \left(\left(\frac{\frac{4}{\alpha}}{\alpha} - \frac{8}{\alpha \cdot \left(\alpha \cdot \alpha\right)}\right) - \frac{2}{\alpha}\right)}{2}\\ \end{array}\]
\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \le 644102.615074100787751376628875732421875:\\
\;\;\;\;\frac{\log \left(e^{\frac{\beta}{2 + \left(\beta + \alpha\right)}}\right) - \left(\frac{\alpha}{2 + \left(\beta + \alpha\right)} - 1\right)}{2}\\

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

\end{array}
double f(double alpha, double beta) {
        double r3994378 = beta;
        double r3994379 = alpha;
        double r3994380 = r3994378 - r3994379;
        double r3994381 = r3994379 + r3994378;
        double r3994382 = 2.0;
        double r3994383 = r3994381 + r3994382;
        double r3994384 = r3994380 / r3994383;
        double r3994385 = 1.0;
        double r3994386 = r3994384 + r3994385;
        double r3994387 = r3994386 / r3994382;
        return r3994387;
}
↓
double f(double alpha, double beta) {
        double r3994388 = alpha;
        double r3994389 = 644102.6150741008;
        bool r3994390 = r3994388 <= r3994389;
        double r3994391 = beta;
        double r3994392 = 2.0;
        double r3994393 = r3994391 + r3994388;
        double r3994394 = r3994392 + r3994393;
        double r3994395 = r3994391 / r3994394;
        double r3994396 = exp(r3994395);
        double r3994397 = log(r3994396);
        double r3994398 = r3994388 / r3994394;
        double r3994399 = 1.0;
        double r3994400 = r3994398 - r3994399;
        double r3994401 = r3994397 - r3994400;
        double r3994402 = r3994401 / r3994392;
        double r3994403 = cbrt(r3994395);
        double r3994404 = r3994403 * r3994403;
        double r3994405 = r3994403 * r3994404;
        double r3994406 = 4.0;
        double r3994407 = r3994406 / r3994388;
        double r3994408 = r3994407 / r3994388;
        double r3994409 = 8.0;
        double r3994410 = r3994388 * r3994388;
        double r3994411 = r3994388 * r3994410;
        double r3994412 = r3994409 / r3994411;
        double r3994413 = r3994408 - r3994412;
        double r3994414 = r3994392 / r3994388;
        double r3994415 = r3994413 - r3994414;
        double r3994416 = r3994405 - r3994415;
        double r3994417 = r3994416 / r3994392;
        double r3994418 = r3994390 ? r3994402 : r3994417;
        return r3994418;
}

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 < 644102.6150741008

    1. Initial program 0.0

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

    3. Applied div-sub0.0

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

    4. Applied associate-+l-0.0

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

    6. Applied add-log-exp0.1

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

    if 644102.6150741008 < alpha

    1. Initial program 49.6

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

    3. Applied div-sub49.5

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

    4. Applied associate-+l-47.9

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

    6. Applied add-sqr-sqrt49.1

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

    7. Applied associate-/r*49.2

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

    9. Applied add-cube-cbrt49.2

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

    10. Taylor expanded around inf 18.6

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

    11. Simplified18.6

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

  4. Final simplification6.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 644102.615074100787751376628875732421875:\\ \;\;\;\;\frac{\log \left(e^{\frac{\beta}{2 + \left(\beta + \alpha\right)}}\right) - \left(\frac{\alpha}{2 + \left(\beta + \alpha\right)} - 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{\frac{\beta}{2 + \left(\beta + \alpha\right)}} \cdot \left(\sqrt[3]{\frac{\beta}{2 + \left(\beta + \alpha\right)}} \cdot \sqrt[3]{\frac{\beta}{2 + \left(\beta + \alpha\right)}}\right) - \left(\left(\frac{\frac{4}{\alpha}}{\alpha} - \frac{8}{\alpha \cdot \left(\alpha \cdot \alpha\right)}\right) - \frac{2}{\alpha}\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019170 
(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))