Average Error: 23.6 → 11.3
Time: 10.1s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 0.0\]
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 1.60680517543021152 \cdot 10^{177}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt[3]{{\left(\frac{\alpha + \beta}{\mathsf{fma}\left(2, i, \alpha + \beta\right) + 2}\right)}^{3}}, \frac{\beta}{\mathsf{fma}\left(2, i, \alpha + \beta\right)} - \frac{1}{\sqrt[3]{\mathsf{fma}\left(2, i, \alpha + \beta\right)} \cdot \sqrt[3]{\mathsf{fma}\left(2, i, \alpha + \beta\right)}} \cdot \frac{\alpha}{\sqrt[3]{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}, 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{8}{{\alpha}^{3}} - \frac{4}{\alpha \cdot \alpha}\right) + \frac{2}{\alpha}}{2}\\ \end{array}\]
\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \le 1.60680517543021152 \cdot 10^{177}:\\
\;\;\;\;\frac{\mathsf{fma}\left(\sqrt[3]{{\left(\frac{\alpha + \beta}{\mathsf{fma}\left(2, i, \alpha + \beta\right) + 2}\right)}^{3}}, \frac{\beta}{\mathsf{fma}\left(2, i, \alpha + \beta\right)} - \frac{1}{\sqrt[3]{\mathsf{fma}\left(2, i, \alpha + \beta\right)} \cdot \sqrt[3]{\mathsf{fma}\left(2, i, \alpha + \beta\right)}} \cdot \frac{\alpha}{\sqrt[3]{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}, 1\right)}{2}\\

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

\end{array}
double f(double alpha, double beta, double i) {
        double r112450 = alpha;
        double r112451 = beta;
        double r112452 = r112450 + r112451;
        double r112453 = r112451 - r112450;
        double r112454 = r112452 * r112453;
        double r112455 = 2.0;
        double r112456 = i;
        double r112457 = r112455 * r112456;
        double r112458 = r112452 + r112457;
        double r112459 = r112454 / r112458;
        double r112460 = r112458 + r112455;
        double r112461 = r112459 / r112460;
        double r112462 = 1.0;
        double r112463 = r112461 + r112462;
        double r112464 = r112463 / r112455;
        return r112464;
}
↓
double f(double alpha, double beta, double i) {
        double r112465 = alpha;
        double r112466 = 1.6068051754302115e+177;
        bool r112467 = r112465 <= r112466;
        double r112468 = beta;
        double r112469 = r112465 + r112468;
        double r112470 = 2.0;
        double r112471 = i;
        double r112472 = fma(r112470, r112471, r112469);
        double r112473 = r112472 + r112470;
        double r112474 = r112469 / r112473;
        double r112475 = 3.0;
        double r112476 = pow(r112474, r112475);
        double r112477 = cbrt(r112476);
        double r112478 = r112468 / r112472;
        double r112479 = 1.0;
        double r112480 = cbrt(r112472);
        double r112481 = r112480 * r112480;
        double r112482 = r112479 / r112481;
        double r112483 = r112465 / r112480;
        double r112484 = r112482 * r112483;
        double r112485 = r112478 - r112484;
        double r112486 = 1.0;
        double r112487 = fma(r112477, r112485, r112486);
        double r112488 = r112487 / r112470;
        double r112489 = 8.0;
        double r112490 = pow(r112465, r112475);
        double r112491 = r112489 / r112490;
        double r112492 = 4.0;
        double r112493 = r112465 * r112465;
        double r112494 = r112492 / r112493;
        double r112495 = r112491 - r112494;
        double r112496 = r112470 / r112465;
        double r112497 = r112495 + r112496;
        double r112498 = r112497 / r112470;
        double r112499 = r112467 ? r112488 : r112498;
        return r112499;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

  1. Split input into 2 regimes

  2. if alpha < 1.6068051754302115e+177

    1. Initial program 17.0

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2}\]

    2. Simplified6.3

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{\alpha + \beta}{\mathsf{fma}\left(2, i, \alpha + \beta\right) + 2}, \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}, 1\right)}{2}}\]
    3. Using strategy rm

    4. Applied add-cbrt-cube15.7

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

    5. Applied add-cbrt-cube21.9

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

    6. Applied cbrt-undiv21.9

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

    7. Simplified6.3

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt[3]{\color{blue}{{\left(\frac{\alpha + \beta}{\mathsf{fma}\left(2, i, \alpha + \beta\right) + 2}\right)}^{3}}}, \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}, 1\right)}{2}\]
    8. Using strategy rm

    9. Applied div-sub6.3

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt[3]{{\left(\frac{\alpha + \beta}{\mathsf{fma}\left(2, i, \alpha + \beta\right) + 2}\right)}^{3}}, \color{blue}{\frac{\beta}{\mathsf{fma}\left(2, i, \alpha + \beta\right)} - \frac{\alpha}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}, 1\right)}{2}\]
    10. Using strategy rm

    11. Applied add-cube-cbrt6.2

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

    12. Applied *-un-lft-identity6.2

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

    13. Applied times-frac6.2

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

    if 1.6068051754302115e+177 < alpha

    1. Initial program 64.0

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2}\]

    2. Simplified49.2

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{\alpha + \beta}{\mathsf{fma}\left(2, i, \alpha + \beta\right) + 2}, \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}, 1\right)}{2}}\]
    3. Using strategy rm

    4. Applied add-exp-log49.2

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

    5. Taylor expanded around inf 42.2

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

    6. Simplified42.2

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

  4. Final simplification11.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 1.60680517543021152 \cdot 10^{177}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt[3]{{\left(\frac{\alpha + \beta}{\mathsf{fma}\left(2, i, \alpha + \beta\right) + 2}\right)}^{3}}, \frac{\beta}{\mathsf{fma}\left(2, i, \alpha + \beta\right)} - \frac{1}{\sqrt[3]{\mathsf{fma}\left(2, i, \alpha + \beta\right)} \cdot \sqrt[3]{\mathsf{fma}\left(2, i, \alpha + \beta\right)}} \cdot \frac{\alpha}{\sqrt[3]{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}, 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{8}{{\alpha}^{3}} - \frac{4}{\alpha \cdot \alpha}\right) + \frac{2}{\alpha}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020045 +o rules:numerics
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/2"
  :precision binary64
  :pre (and (> alpha -1) (> beta -1) (> i 0.0))
  (/ (+ (/ (/ (* (+ alpha beta) (- beta alpha)) (+ (+ alpha beta) (* 2 i))) (+ (+ (+ alpha beta) (* 2 i)) 2)) 1) 2))