Average Error: 16.1 → 6.0
Time: 6.1s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 25713141841.6247902:\\ \;\;\;\;\frac{\left(\left(\sqrt[3]{\frac{\beta}{\left(\alpha + \beta\right) + 2}} \cdot \sqrt[3]{\frac{\beta}{\left(\alpha + \beta\right) + 2}}\right) \cdot \log \left(e^{\sqrt[3]{\frac{\beta}{\left(\alpha + \beta\right) + 2}}}\right) - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)\right) + \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right) \cdot \left(\left(-1\right) + 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \mathsf{fma}\left(4, \frac{1}{{\alpha}^{2}}, -\mathsf{fma}\left(2, \frac{1}{\alpha}, 8 \cdot \frac{1}{{\alpha}^{3}}\right)\right)}{2}\\ \end{array}\]
\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \le 25713141841.6247902:\\
\;\;\;\;\frac{\left(\left(\sqrt[3]{\frac{\beta}{\left(\alpha + \beta\right) + 2}} \cdot \sqrt[3]{\frac{\beta}{\left(\alpha + \beta\right) + 2}}\right) \cdot \log \left(e^{\sqrt[3]{\frac{\beta}{\left(\alpha + \beta\right) + 2}}}\right) - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)\right) + \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right) \cdot \left(\left(-1\right) + 1\right)}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \mathsf{fma}\left(4, \frac{1}{{\alpha}^{2}}, -\mathsf{fma}\left(2, \frac{1}{\alpha}, 8 \cdot \frac{1}{{\alpha}^{3}}\right)\right)}{2}\\

\end{array}
double f(double alpha, double beta) {
        double r103468 = beta;
        double r103469 = alpha;
        double r103470 = r103468 - r103469;
        double r103471 = r103469 + r103468;
        double r103472 = 2.0;
        double r103473 = r103471 + r103472;
        double r103474 = r103470 / r103473;
        double r103475 = 1.0;
        double r103476 = r103474 + r103475;
        double r103477 = r103476 / r103472;
        return r103477;
}
↓
double f(double alpha, double beta) {
        double r103478 = alpha;
        double r103479 = 25713141841.62479;
        bool r103480 = r103478 <= r103479;
        double r103481 = beta;
        double r103482 = r103478 + r103481;
        double r103483 = 2.0;
        double r103484 = r103482 + r103483;
        double r103485 = r103481 / r103484;
        double r103486 = cbrt(r103485);
        double r103487 = r103486 * r103486;
        double r103488 = exp(r103486);
        double r103489 = log(r103488);
        double r103490 = r103487 * r103489;
        double r103491 = r103478 / r103484;
        double r103492 = 1.0;
        double r103493 = r103491 - r103492;
        double r103494 = r103490 - r103493;
        double r103495 = 1.0;
        double r103496 = -r103495;
        double r103497 = r103496 + r103495;
        double r103498 = r103493 * r103497;
        double r103499 = r103494 + r103498;
        double r103500 = r103499 / r103483;
        double r103501 = 4.0;
        double r103502 = 2.0;
        double r103503 = pow(r103478, r103502);
        double r103504 = r103495 / r103503;
        double r103505 = r103495 / r103478;
        double r103506 = 8.0;
        double r103507 = 3.0;
        double r103508 = pow(r103478, r103507);
        double r103509 = r103495 / r103508;
        double r103510 = r103506 * r103509;
        double r103511 = fma(r103483, r103505, r103510);
        double r103512 = -r103511;
        double r103513 = fma(r103501, r103504, r103512);
        double r103514 = r103485 - r103513;
        double r103515 = r103514 / r103483;
        double r103516 = r103480 ? r103500 : r103515;
        return r103516;
}

Error

Bits error versus alpha

Bits error versus beta

Derivation

  1. Split input into 2 regimes

  2. if alpha < 25713141841.62479

    1. Initial program 0.2

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

    3. Applied div-sub0.2

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

    4. Applied associate-+l-0.2

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

    6. Applied add-cube-cbrt0.2

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

    7. Applied add-cube-cbrt0.2

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

    8. Applied prod-diff0.2

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

    9. Simplified0.2

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

    10. Simplified0.2

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

    12. Applied add-log-exp0.2

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

    if 25713141841.62479 < alpha

    1. Initial program 49.7

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

    3. Applied div-sub49.7

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

    4. Applied associate-+l-48.1

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

    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

      \[\leadsto \frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \color{blue}{\mathsf{fma}\left(4, \frac{1}{{\alpha}^{2}}, -\mathsf{fma}\left(2, \frac{1}{\alpha}, 8 \cdot \frac{1}{{\alpha}^{3}}\right)\right)}}{2}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification6.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 25713141841.6247902:\\ \;\;\;\;\frac{\left(\left(\sqrt[3]{\frac{\beta}{\left(\alpha + \beta\right) + 2}} \cdot \sqrt[3]{\frac{\beta}{\left(\alpha + \beta\right) + 2}}\right) \cdot \log \left(e^{\sqrt[3]{\frac{\beta}{\left(\alpha + \beta\right) + 2}}}\right) - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)\right) + \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right) \cdot \left(\left(-1\right) + 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \mathsf{fma}\left(4, \frac{1}{{\alpha}^{2}}, -\mathsf{fma}\left(2, \frac{1}{\alpha}, 8 \cdot \frac{1}{{\alpha}^{3}}\right)\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020027 +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))