Average Error: 5.3 → 4.3
Time: 19.0s
Precision: 64
\[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -7.802449997962363245079599405702539273998 \cdot 10^{-129} \lor \neg \left(t \le 7.690769560303890058907581900892530132644 \cdot 10^{-134}\right):\\ \;\;\;\;\mathsf{fma}\left(t, \left(\left(y \cdot 18\right) \cdot x\right) \cdot z - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, \left(j \cdot 27\right) \cdot k\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, 0 - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, j \cdot \left(27 \cdot k\right)\right)\right)\\ \end{array}\]
\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}\;t \le -7.802449997962363245079599405702539273998 \cdot 10^{-129} \lor \neg \left(t \le 7.690769560303890058907581900892530132644 \cdot 10^{-134}\right):\\
\;\;\;\;\mathsf{fma}\left(t, \left(\left(y \cdot 18\right) \cdot x\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)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(t, 0 - 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 r149724 = x;
        double r149725 = 18.0;
        double r149726 = r149724 * r149725;
        double r149727 = y;
        double r149728 = r149726 * r149727;
        double r149729 = z;
        double r149730 = r149728 * r149729;
        double r149731 = t;
        double r149732 = r149730 * r149731;
        double r149733 = a;
        double r149734 = 4.0;
        double r149735 = r149733 * r149734;
        double r149736 = r149735 * r149731;
        double r149737 = r149732 - r149736;
        double r149738 = b;
        double r149739 = c;
        double r149740 = r149738 * r149739;
        double r149741 = r149737 + r149740;
        double r149742 = r149724 * r149734;
        double r149743 = i;
        double r149744 = r149742 * r149743;
        double r149745 = r149741 - r149744;
        double r149746 = j;
        double r149747 = 27.0;
        double r149748 = r149746 * r149747;
        double r149749 = k;
        double r149750 = r149748 * r149749;
        double r149751 = r149745 - r149750;
        return r149751;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r149752 = t;
        double r149753 = -7.802449997962363e-129;
        bool r149754 = r149752 <= r149753;
        double r149755 = 7.69076956030389e-134;
        bool r149756 = r149752 <= r149755;
        double r149757 = !r149756;
        bool r149758 = r149754 || r149757;
        double r149759 = y;
        double r149760 = 18.0;
        double r149761 = r149759 * r149760;
        double r149762 = x;
        double r149763 = r149761 * r149762;
        double r149764 = z;
        double r149765 = r149763 * r149764;
        double r149766 = a;
        double r149767 = 4.0;
        double r149768 = r149766 * r149767;
        double r149769 = r149765 - r149768;
        double r149770 = b;
        double r149771 = c;
        double r149772 = r149770 * r149771;
        double r149773 = i;
        double r149774 = r149767 * r149773;
        double r149775 = j;
        double r149776 = 27.0;
        double r149777 = r149775 * r149776;
        double r149778 = k;
        double r149779 = r149777 * r149778;
        double r149780 = fma(r149762, r149774, r149779);
        double r149781 = r149772 - r149780;
        double r149782 = fma(r149752, r149769, r149781);
        double r149783 = 0.0;
        double r149784 = r149783 - r149768;
        double r149785 = r149776 * r149778;
        double r149786 = r149775 * r149785;
        double r149787 = fma(r149762, r149774, r149786);
        double r149788 = r149772 - r149787;
        double r149789 = fma(r149752, r149784, r149788);
        double r149790 = r149758 ? r149782 : r149789;
        return r149790;
}

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 2 regimes

  2. if t < -7.802449997962363e-129 or 7.69076956030389e-134 < t

    1. Initial program 3.0

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

    2. Simplified3.1

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

    4. Applied add-cube-cbrt3.2

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

    5. Applied associate-*r*3.2

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

    7. Applied *-un-lft-identity3.2

      \[\leadsto \mathsf{fma}\left(t, \left(\left(\left(x \cdot 18\right) \cdot \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right)\right) \cdot \sqrt[3]{y}\right) \cdot \color{blue}{\left(1 \cdot z\right)} - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, \left(j \cdot 27\right) \cdot k\right)\right)\]

    8. Applied associate-*r*3.2

      \[\leadsto \mathsf{fma}\left(t, \color{blue}{\left(\left(\left(\left(x \cdot 18\right) \cdot \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right)\right) \cdot \sqrt[3]{y}\right) \cdot 1\right) \cdot z} - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, \left(j \cdot 27\right) \cdot k\right)\right)\]

    9. Simplified3.1

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

    if -7.802449997962363e-129 < t < 7.69076956030389e-134

    1. Initial program 9.1

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

    2. Simplified9.1

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

    4. Applied associate-*l*9.1

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

    5. Taylor expanded around 0 6.3

      \[\leadsto \mathsf{fma}\left(t, \color{blue}{0} - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, j \cdot \left(27 \cdot k\right)\right)\right)\]
  3. Recombined 2 regimes into one program.

  4. Final simplification4.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -7.802449997962363245079599405702539273998 \cdot 10^{-129} \lor \neg \left(t \le 7.690769560303890058907581900892530132644 \cdot 10^{-134}\right):\\ \;\;\;\;\mathsf{fma}\left(t, \left(\left(y \cdot 18\right) \cdot x\right) \cdot z - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, \left(j \cdot 27\right) \cdot k\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, 0 - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, j \cdot \left(27 \cdot k\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019362 +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)))