Average Error: 5.9 → 5.1
Time: 12.7s
Precision: 64
\[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.67549533106865849 \cdot 10^{-122}:\\ \;\;\;\;\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot \left(z \cdot t\right) - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\\ \mathbf{elif}\;x \le 4.8142642785690852 \cdot 10^{-197}:\\ \;\;\;\;\mathsf{fma}\left(t, \left(\left(x \cdot 18\right) \cdot y\right) \cdot z - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, \left(\left(\left(\sqrt[3]{\sqrt[3]{j \cdot \left(27 \cdot k\right)}} \cdot \sqrt[3]{\sqrt[3]{j \cdot \left(27 \cdot k\right)}}\right) \cdot \sqrt[3]{\sqrt[3]{j \cdot \left(27 \cdot k\right)}}\right) \cdot \sqrt[3]{j \cdot \left(27 \cdot k\right)}\right) \cdot \sqrt[3]{j \cdot \left(27 \cdot k\right)}\right)\right)\\ \mathbf{elif}\;x \le 8.87126180649849532 \cdot 10^{-105}:\\ \;\;\;\;\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot \left(z \cdot t\right) - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, \left(x \cdot 18\right) \cdot \left(y \cdot z\right) - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, j \cdot \left(27 \cdot k\right)\right)\right)\\ \end{array}\]
\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.67549533106865849 \cdot 10^{-122}:\\
\;\;\;\;\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot \left(z \cdot t\right) - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\\

\mathbf{elif}\;x \le 4.8142642785690852 \cdot 10^{-197}:\\
\;\;\;\;\mathsf{fma}\left(t, \left(\left(x \cdot 18\right) \cdot y\right) \cdot z - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, \left(\left(\left(\sqrt[3]{\sqrt[3]{j \cdot \left(27 \cdot k\right)}} \cdot \sqrt[3]{\sqrt[3]{j \cdot \left(27 \cdot k\right)}}\right) \cdot \sqrt[3]{\sqrt[3]{j \cdot \left(27 \cdot k\right)}}\right) \cdot \sqrt[3]{j \cdot \left(27 \cdot k\right)}\right) \cdot \sqrt[3]{j \cdot \left(27 \cdot k\right)}\right)\right)\\

\mathbf{elif}\;x \le 8.87126180649849532 \cdot 10^{-105}:\\
\;\;\;\;\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot \left(z \cdot t\right) - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(t, \left(x \cdot 18\right) \cdot \left(y \cdot z\right) - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, j \cdot \left(27 \cdot k\right)\right)\right)\\

\end{array}
double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r107581 = x;
        double r107582 = 18.0;
        double r107583 = r107581 * r107582;
        double r107584 = y;
        double r107585 = r107583 * r107584;
        double r107586 = z;
        double r107587 = r107585 * r107586;
        double r107588 = t;
        double r107589 = r107587 * r107588;
        double r107590 = a;
        double r107591 = 4.0;
        double r107592 = r107590 * r107591;
        double r107593 = r107592 * r107588;
        double r107594 = r107589 - r107593;
        double r107595 = b;
        double r107596 = c;
        double r107597 = r107595 * r107596;
        double r107598 = r107594 + r107597;
        double r107599 = r107581 * r107591;
        double r107600 = i;
        double r107601 = r107599 * r107600;
        double r107602 = r107598 - r107601;
        double r107603 = j;
        double r107604 = 27.0;
        double r107605 = r107603 * r107604;
        double r107606 = k;
        double r107607 = r107605 * r107606;
        double r107608 = r107602 - r107607;
        return r107608;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r107609 = x;
        double r107610 = -1.6754953310686585e-122;
        bool r107611 = r107609 <= r107610;
        double r107612 = 18.0;
        double r107613 = r107609 * r107612;
        double r107614 = y;
        double r107615 = r107613 * r107614;
        double r107616 = z;
        double r107617 = t;
        double r107618 = r107616 * r107617;
        double r107619 = r107615 * r107618;
        double r107620 = a;
        double r107621 = 4.0;
        double r107622 = r107620 * r107621;
        double r107623 = r107622 * r107617;
        double r107624 = r107619 - r107623;
        double r107625 = b;
        double r107626 = c;
        double r107627 = r107625 * r107626;
        double r107628 = r107624 + r107627;
        double r107629 = r107609 * r107621;
        double r107630 = i;
        double r107631 = r107629 * r107630;
        double r107632 = r107628 - r107631;
        double r107633 = j;
        double r107634 = 27.0;
        double r107635 = r107633 * r107634;
        double r107636 = k;
        double r107637 = r107635 * r107636;
        double r107638 = r107632 - r107637;
        double r107639 = 4.814264278569085e-197;
        bool r107640 = r107609 <= r107639;
        double r107641 = r107615 * r107616;
        double r107642 = r107641 - r107622;
        double r107643 = r107621 * r107630;
        double r107644 = r107634 * r107636;
        double r107645 = r107633 * r107644;
        double r107646 = cbrt(r107645);
        double r107647 = cbrt(r107646);
        double r107648 = r107647 * r107647;
        double r107649 = r107648 * r107647;
        double r107650 = r107649 * r107646;
        double r107651 = r107650 * r107646;
        double r107652 = fma(r107609, r107643, r107651);
        double r107653 = r107627 - r107652;
        double r107654 = fma(r107617, r107642, r107653);
        double r107655 = 8.871261806498495e-105;
        bool r107656 = r107609 <= r107655;
        double r107657 = r107614 * r107616;
        double r107658 = r107613 * r107657;
        double r107659 = r107658 - r107622;
        double r107660 = fma(r107609, r107643, r107645);
        double r107661 = r107627 - r107660;
        double r107662 = fma(r107617, r107659, r107661);
        double r107663 = r107656 ? r107638 : r107662;
        double r107664 = r107640 ? r107654 : r107663;
        double r107665 = r107611 ? r107638 : r107664;
        return r107665;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus i

Bits error versus j

Bits error versus k

Derivation

  1. Split input into 3 regimes

  2. if x < -1.6754953310686585e-122 or 4.814264278569085e-197 < x < 8.871261806498495e-105

    1. Initial program 7.1

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\]
    2. Using strategy rm

    3. Applied associate-*l*6.8

      \[\leadsto \left(\left(\left(\color{blue}{\left(\left(x \cdot 18\right) \cdot y\right) \cdot \left(z \cdot t\right)} - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\]

    if -1.6754953310686585e-122 < x < 4.814264278569085e-197

    1. Initial program 0.9

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\]

    2. Simplified0.9

      \[\leadsto \color{blue}{\mathsf{fma}\left(t, \left(\left(x \cdot 18\right) \cdot y\right) \cdot z - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, \left(j \cdot 27\right) \cdot k\right)\right)}\]
    3. Using strategy rm

    4. Applied associate-*l*0.8

      \[\leadsto \mathsf{fma}\left(t, \left(\left(x \cdot 18\right) \cdot y\right) \cdot z - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, \color{blue}{j \cdot \left(27 \cdot k\right)}\right)\right)\]
    5. Using strategy rm

    6. Applied add-cube-cbrt1.1

      \[\leadsto \mathsf{fma}\left(t, \left(\left(x \cdot 18\right) \cdot y\right) \cdot z - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, \color{blue}{\left(\sqrt[3]{j \cdot \left(27 \cdot k\right)} \cdot \sqrt[3]{j \cdot \left(27 \cdot k\right)}\right) \cdot \sqrt[3]{j \cdot \left(27 \cdot k\right)}}\right)\right)\]
    7. Using strategy rm

    8. Applied add-cube-cbrt1.2

      \[\leadsto \mathsf{fma}\left(t, \left(\left(x \cdot 18\right) \cdot y\right) \cdot z - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, \left(\color{blue}{\left(\left(\sqrt[3]{\sqrt[3]{j \cdot \left(27 \cdot k\right)}} \cdot \sqrt[3]{\sqrt[3]{j \cdot \left(27 \cdot k\right)}}\right) \cdot \sqrt[3]{\sqrt[3]{j \cdot \left(27 \cdot k\right)}}\right)} \cdot \sqrt[3]{j \cdot \left(27 \cdot k\right)}\right) \cdot \sqrt[3]{j \cdot \left(27 \cdot k\right)}\right)\right)\]

    if 8.871261806498495e-105 < x

    1. Initial program 9.7

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\]

    2. Simplified9.7

      \[\leadsto \color{blue}{\mathsf{fma}\left(t, \left(\left(x \cdot 18\right) \cdot y\right) \cdot z - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, \left(j \cdot 27\right) \cdot k\right)\right)}\]
    3. Using strategy rm

    4. Applied associate-*l*9.7

      \[\leadsto \mathsf{fma}\left(t, \left(\left(x \cdot 18\right) \cdot y\right) \cdot z - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, \color{blue}{j \cdot \left(27 \cdot k\right)}\right)\right)\]
    5. Using strategy rm

    6. Applied associate-*l*6.9

      \[\leadsto \mathsf{fma}\left(t, \color{blue}{\left(x \cdot 18\right) \cdot \left(y \cdot z\right)} - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, j \cdot \left(27 \cdot k\right)\right)\right)\]
  3. Recombined 3 regimes into one program.

  4. Final simplification5.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.67549533106865849 \cdot 10^{-122}:\\ \;\;\;\;\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot \left(z \cdot t\right) - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\\ \mathbf{elif}\;x \le 4.8142642785690852 \cdot 10^{-197}:\\ \;\;\;\;\mathsf{fma}\left(t, \left(\left(x \cdot 18\right) \cdot y\right) \cdot z - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, \left(\left(\left(\sqrt[3]{\sqrt[3]{j \cdot \left(27 \cdot k\right)}} \cdot \sqrt[3]{\sqrt[3]{j \cdot \left(27 \cdot k\right)}}\right) \cdot \sqrt[3]{\sqrt[3]{j \cdot \left(27 \cdot k\right)}}\right) \cdot \sqrt[3]{j \cdot \left(27 \cdot k\right)}\right) \cdot \sqrt[3]{j \cdot \left(27 \cdot k\right)}\right)\right)\\ \mathbf{elif}\;x \le 8.87126180649849532 \cdot 10^{-105}:\\ \;\;\;\;\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot \left(z \cdot t\right) - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, \left(x \cdot 18\right) \cdot \left(y \cdot z\right) - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, j \cdot \left(27 \cdot k\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020027 +o rules:numerics
(FPCore (x y z t a b c i j k)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1"
  :precision binary64
  (- (- (+ (- (* (* (* (* x 18) y) z) t) (* (* a 4) t)) (* b c)) (* (* x 4) i)) (* (* j 27) k)))