Average Error: 16.1 → 6.0
Time: 18.7s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 1188863810682620.75:\\ \;\;\;\;\frac{\sqrt[3]{{\left(\frac{\beta}{\left(\alpha + \beta\right) + 2}\right)}^{3}} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\frac{\frac{4}{\alpha}}{\alpha} - \left(\frac{2}{\alpha} - \frac{-8}{{\alpha}^{3}}\right)\right)}{2}\\ \end{array}\]
\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \le 1188863810682620.75:\\
\;\;\;\;\frac{\sqrt[3]{{\left(\frac{\beta}{\left(\alpha + \beta\right) + 2}\right)}^{3}} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)}{2}\\

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

\end{array}
double f(double alpha, double beta) {
        double r363 = beta;
        double r364 = alpha;
        double r365 = r363 - r364;
        double r366 = r364 + r363;
        double r367 = 2.0;
        double r368 = r366 + r367;
        double r369 = r365 / r368;
        double r370 = 1.0;
        double r371 = r369 + r370;
        double r372 = r371 / r367;
        return r372;
}
↓
double f(double alpha, double beta) {
        double r373 = alpha;
        double r374 = 1188863810682620.8;
        bool r375 = r373 <= r374;
        double r376 = beta;
        double r377 = r373 + r376;
        double r378 = 2.0;
        double r379 = r377 + r378;
        double r380 = r376 / r379;
        double r381 = 3.0;
        double r382 = pow(r380, r381);
        double r383 = cbrt(r382);
        double r384 = r373 / r379;
        double r385 = 1.0;
        double r386 = r384 - r385;
        double r387 = r383 - r386;
        double r388 = r387 / r378;
        double r389 = 4.0;
        double r390 = r389 / r373;
        double r391 = r390 / r373;
        double r392 = r378 / r373;
        double r393 = 8.0;
        double r394 = -r393;
        double r395 = pow(r373, r381);
        double r396 = r394 / r395;
        double r397 = r392 - r396;
        double r398 = r391 - r397;
        double r399 = r380 - r398;
        double r400 = r399 / r378;
        double r401 = r375 ? r388 : r400;
        return r401;
}

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 < 1188863810682620.8

    1. Initial program 0.3

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

    3. Applied div-sub0.3

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

    4. Applied associate-+l-0.3

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

    6. Applied add-cbrt-cube11.2

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

    7. Applied add-cbrt-cube13.7

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

    8. Applied cbrt-undiv13.7

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

    9. Simplified0.3

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

    if 1188863810682620.8 < alpha

    1. Initial program 50.2

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

    3. Applied div-sub50.2

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

    4. Applied associate-+l-48.7

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

    5. Taylor expanded around inf 18.3

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

    6. Simplified18.3

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

  4. Final simplification6.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 1188863810682620.75:\\ \;\;\;\;\frac{\sqrt[3]{{\left(\frac{\beta}{\left(\alpha + \beta\right) + 2}\right)}^{3}} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\frac{\frac{4}{\alpha}}{\alpha} - \left(\frac{2}{\alpha} - \frac{-8}{{\alpha}^{3}}\right)\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020025 +o rules:numerics
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/1"
  :precision binary64
  :pre (and (> alpha -1) (> beta -1))
  (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2)) 1) 2))