Average Error: 15.9 → 6.5
Time: 47.9s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 686668042177335661463411015811072:\\ \;\;\;\;\frac{\frac{\beta}{2 + \left(\beta + \alpha\right)} - \mathsf{fma}\left(\frac{1}{\sqrt[3]{2 + \left(\beta + \alpha\right)} \cdot \sqrt[3]{2 + \left(\beta + \alpha\right)}}, \frac{\alpha}{\sqrt[3]{2 + \left(\beta + \alpha\right)}}, -1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{2 + \left(\beta + \alpha\right)} - \left(\frac{4}{\alpha \cdot \alpha} - \left(\frac{2}{\alpha} + \frac{8}{\alpha \cdot \left(\alpha \cdot \alpha\right)}\right)\right)}{2}\\ \end{array}\]
\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \le 686668042177335661463411015811072:\\
\;\;\;\;\frac{\frac{\beta}{2 + \left(\beta + \alpha\right)} - \mathsf{fma}\left(\frac{1}{\sqrt[3]{2 + \left(\beta + \alpha\right)} \cdot \sqrt[3]{2 + \left(\beta + \alpha\right)}}, \frac{\alpha}{\sqrt[3]{2 + \left(\beta + \alpha\right)}}, -1\right)}{2}\\

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

\end{array}
double f(double alpha, double beta) {
        double r3525424 = beta;
        double r3525425 = alpha;
        double r3525426 = r3525424 - r3525425;
        double r3525427 = r3525425 + r3525424;
        double r3525428 = 2.0;
        double r3525429 = r3525427 + r3525428;
        double r3525430 = r3525426 / r3525429;
        double r3525431 = 1.0;
        double r3525432 = r3525430 + r3525431;
        double r3525433 = r3525432 / r3525428;
        return r3525433;
}
↓
double f(double alpha, double beta) {
        double r3525434 = alpha;
        double r3525435 = 6.866680421773357e+32;
        bool r3525436 = r3525434 <= r3525435;
        double r3525437 = beta;
        double r3525438 = 2.0;
        double r3525439 = r3525437 + r3525434;
        double r3525440 = r3525438 + r3525439;
        double r3525441 = r3525437 / r3525440;
        double r3525442 = 1.0;
        double r3525443 = cbrt(r3525440);
        double r3525444 = r3525443 * r3525443;
        double r3525445 = r3525442 / r3525444;
        double r3525446 = r3525434 / r3525443;
        double r3525447 = 1.0;
        double r3525448 = -r3525447;
        double r3525449 = fma(r3525445, r3525446, r3525448);
        double r3525450 = r3525441 - r3525449;
        double r3525451 = r3525450 / r3525438;
        double r3525452 = 4.0;
        double r3525453 = r3525434 * r3525434;
        double r3525454 = r3525452 / r3525453;
        double r3525455 = r3525438 / r3525434;
        double r3525456 = 8.0;
        double r3525457 = r3525434 * r3525453;
        double r3525458 = r3525456 / r3525457;
        double r3525459 = r3525455 + r3525458;
        double r3525460 = r3525454 - r3525459;
        double r3525461 = r3525441 - r3525460;
        double r3525462 = r3525461 / r3525438;
        double r3525463 = r3525436 ? r3525451 : r3525462;
        return r3525463;
}

Error

Bits error versus alpha

Bits error versus beta

Derivation

  1. Split input into 2 regimes

  2. if alpha < 6.866680421773357e+32

    1. Initial program 1.6

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

    3. Applied div-sub1.6

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

    4. Applied associate-+l-1.6

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

    6. Applied add-cube-cbrt1.6

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

    7. Applied *-un-lft-identity1.6

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

    8. Applied times-frac1.6

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

    9. Applied fma-neg1.6

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

    if 6.866680421773357e+32 < alpha

    1. Initial program 50.6

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

    3. Applied div-sub50.6

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

    4. Applied associate-+l-49.0

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

    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

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

  4. Final simplification6.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 686668042177335661463411015811072:\\ \;\;\;\;\frac{\frac{\beta}{2 + \left(\beta + \alpha\right)} - \mathsf{fma}\left(\frac{1}{\sqrt[3]{2 + \left(\beta + \alpha\right)} \cdot \sqrt[3]{2 + \left(\beta + \alpha\right)}}, \frac{\alpha}{\sqrt[3]{2 + \left(\beta + \alpha\right)}}, -1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{2 + \left(\beta + \alpha\right)} - \left(\frac{4}{\alpha \cdot \alpha} - \left(\frac{2}{\alpha} + \frac{8}{\alpha \cdot \left(\alpha \cdot \alpha\right)}\right)\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019168 +o rules:numerics
(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))