Average Error: 16.2 → 6.4
Time: 25.7s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 2.1159100932126255 \cdot 10^{+26}:\\ \;\;\;\;e^{\log \left(\frac{\frac{\beta}{\left(\beta + \alpha\right) + 2.0} - \left(\frac{\alpha}{\left(\beta + \alpha\right) + 2.0} - 1.0\right)}{2.0}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\sqrt[3]{\beta} \cdot \sqrt[3]{\beta}}{\sqrt[3]{\left(\beta + \alpha\right) + 2.0} \cdot \sqrt[3]{\left(\beta + \alpha\right) + 2.0}}, \frac{\sqrt[3]{\beta}}{\sqrt[3]{\left(\beta + \alpha\right) + 2.0}}, -\left(\frac{4.0}{\alpha \cdot \alpha} - \left(\frac{2.0}{\alpha} + \frac{\frac{8.0}{\alpha \cdot \alpha}}{\alpha}\right)\right)\right)}{2.0}\\ \end{array}\]
\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \le 2.1159100932126255 \cdot 10^{+26}:\\
\;\;\;\;e^{\log \left(\frac{\frac{\beta}{\left(\beta + \alpha\right) + 2.0} - \left(\frac{\alpha}{\left(\beta + \alpha\right) + 2.0} - 1.0\right)}{2.0}\right)}\\

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

\end{array}
double f(double alpha, double beta) {
        double r1428448 = beta;
        double r1428449 = alpha;
        double r1428450 = r1428448 - r1428449;
        double r1428451 = r1428449 + r1428448;
        double r1428452 = 2.0;
        double r1428453 = r1428451 + r1428452;
        double r1428454 = r1428450 / r1428453;
        double r1428455 = 1.0;
        double r1428456 = r1428454 + r1428455;
        double r1428457 = r1428456 / r1428452;
        return r1428457;
}
↓
double f(double alpha, double beta) {
        double r1428458 = alpha;
        double r1428459 = 2.1159100932126255e+26;
        bool r1428460 = r1428458 <= r1428459;
        double r1428461 = beta;
        double r1428462 = r1428461 + r1428458;
        double r1428463 = 2.0;
        double r1428464 = r1428462 + r1428463;
        double r1428465 = r1428461 / r1428464;
        double r1428466 = r1428458 / r1428464;
        double r1428467 = 1.0;
        double r1428468 = r1428466 - r1428467;
        double r1428469 = r1428465 - r1428468;
        double r1428470 = r1428469 / r1428463;
        double r1428471 = log(r1428470);
        double r1428472 = exp(r1428471);
        double r1428473 = cbrt(r1428461);
        double r1428474 = r1428473 * r1428473;
        double r1428475 = cbrt(r1428464);
        double r1428476 = r1428475 * r1428475;
        double r1428477 = r1428474 / r1428476;
        double r1428478 = r1428473 / r1428475;
        double r1428479 = 4.0;
        double r1428480 = r1428458 * r1428458;
        double r1428481 = r1428479 / r1428480;
        double r1428482 = r1428463 / r1428458;
        double r1428483 = 8.0;
        double r1428484 = r1428483 / r1428480;
        double r1428485 = r1428484 / r1428458;
        double r1428486 = r1428482 + r1428485;
        double r1428487 = r1428481 - r1428486;
        double r1428488 = -r1428487;
        double r1428489 = fma(r1428477, r1428478, r1428488);
        double r1428490 = r1428489 / r1428463;
        double r1428491 = r1428460 ? r1428472 : r1428490;
        return r1428491;
}

Error

Bits error versus alpha

Bits error versus beta

Derivation

  1. Split input into 2 regimes

  2. if alpha < 2.1159100932126255e+26

    1. Initial program 1.2

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

    3. Applied div-sub1.2

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

    4. Applied associate-+l-1.2

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

    6. Applied add-exp-log1.2

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

    7. Applied add-exp-log1.2

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

    8. Applied div-exp1.2

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

    9. Simplified1.2

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

    if 2.1159100932126255e+26 < alpha

    1. Initial program 50.4

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

    3. Applied div-sub50.4

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

    4. Applied associate-+l-48.8

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

    6. Applied add-cube-cbrt48.9

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

    7. Applied add-cube-cbrt48.8

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

    8. Applied times-frac48.8

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

    9. Applied fma-neg48.8

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

    10. Taylor expanded around inf 18.3

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

    11. Simplified18.3

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

  4. Final simplification6.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 2.1159100932126255 \cdot 10^{+26}:\\ \;\;\;\;e^{\log \left(\frac{\frac{\beta}{\left(\beta + \alpha\right) + 2.0} - \left(\frac{\alpha}{\left(\beta + \alpha\right) + 2.0} - 1.0\right)}{2.0}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\sqrt[3]{\beta} \cdot \sqrt[3]{\beta}}{\sqrt[3]{\left(\beta + \alpha\right) + 2.0} \cdot \sqrt[3]{\left(\beta + \alpha\right) + 2.0}}, \frac{\sqrt[3]{\beta}}{\sqrt[3]{\left(\beta + \alpha\right) + 2.0}}, -\left(\frac{4.0}{\alpha \cdot \alpha} - \left(\frac{2.0}{\alpha} + \frac{\frac{8.0}{\alpha \cdot \alpha}}{\alpha}\right)\right)\right)}{2.0}\\ \end{array}\]

Reproduce

herbie shell --seed 2019152 +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))