Average Error: 5.4 → 4.5
Time: 41.4s
Precision: 64
\[\left(\left(\left(\left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4.0\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(j \cdot 27.0\right) \cdot k\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -3.88366167002168 \cdot 10^{-170}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\left(\sqrt[3]{\left(x \cdot 18.0\right) \cdot y} \cdot \sqrt[3]{\left(x \cdot 18.0\right) \cdot y}\right) \cdot \sqrt[3]{\left(x \cdot 18.0\right) \cdot y}\right) \cdot z - 4.0 \cdot a\right), t, \left(b \cdot c - \mathsf{fma}\left(k, \left(27.0 \cdot j\right), \left(\left(4.0 \cdot x\right) \cdot i\right)\right)\right)\right)\\ \mathbf{elif}\;t \le 4.6889661927618364 \cdot 10^{-163}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(-4.0\right) \cdot a\right), t, \left(b \cdot c - \mathsf{fma}\left(k, \left(27.0 \cdot j\right), \left(\left(4.0 \cdot x\right) \cdot i\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\left(\sqrt[3]{\left(x \cdot 18.0\right) \cdot y} \cdot \sqrt[3]{\left(x \cdot 18.0\right) \cdot y}\right) \cdot \sqrt[3]{\left(x \cdot 18.0\right) \cdot y}\right) \cdot z - 4.0 \cdot a\right), t, \left(b \cdot c - \mathsf{fma}\left(k, \left(27.0 \cdot j\right), \left(\left(4.0 \cdot x\right) \cdot i\right)\right)\right)\right)\\ \end{array}\]
\left(\left(\left(\left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4.0\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(j \cdot 27.0\right) \cdot k
↓
\begin{array}{l}
\mathbf{if}\;t \le -3.88366167002168 \cdot 10^{-170}:\\
\;\;\;\;\mathsf{fma}\left(\left(\left(\left(\sqrt[3]{\left(x \cdot 18.0\right) \cdot y} \cdot \sqrt[3]{\left(x \cdot 18.0\right) \cdot y}\right) \cdot \sqrt[3]{\left(x \cdot 18.0\right) \cdot y}\right) \cdot z - 4.0 \cdot a\right), t, \left(b \cdot c - \mathsf{fma}\left(k, \left(27.0 \cdot j\right), \left(\left(4.0 \cdot x\right) \cdot i\right)\right)\right)\right)\\

\mathbf{elif}\;t \le 4.6889661927618364 \cdot 10^{-163}:\\
\;\;\;\;\mathsf{fma}\left(\left(\left(-4.0\right) \cdot a\right), t, \left(b \cdot c - \mathsf{fma}\left(k, \left(27.0 \cdot j\right), \left(\left(4.0 \cdot x\right) \cdot i\right)\right)\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(\left(\left(\left(\sqrt[3]{\left(x \cdot 18.0\right) \cdot y} \cdot \sqrt[3]{\left(x \cdot 18.0\right) \cdot y}\right) \cdot \sqrt[3]{\left(x \cdot 18.0\right) \cdot y}\right) \cdot z - 4.0 \cdot a\right), t, \left(b \cdot c - \mathsf{fma}\left(k, \left(27.0 \cdot j\right), \left(\left(4.0 \cdot x\right) \cdot i\right)\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 r3760662 = x;
        double r3760663 = 18.0;
        double r3760664 = r3760662 * r3760663;
        double r3760665 = y;
        double r3760666 = r3760664 * r3760665;
        double r3760667 = z;
        double r3760668 = r3760666 * r3760667;
        double r3760669 = t;
        double r3760670 = r3760668 * r3760669;
        double r3760671 = a;
        double r3760672 = 4.0;
        double r3760673 = r3760671 * r3760672;
        double r3760674 = r3760673 * r3760669;
        double r3760675 = r3760670 - r3760674;
        double r3760676 = b;
        double r3760677 = c;
        double r3760678 = r3760676 * r3760677;
        double r3760679 = r3760675 + r3760678;
        double r3760680 = r3760662 * r3760672;
        double r3760681 = i;
        double r3760682 = r3760680 * r3760681;
        double r3760683 = r3760679 - r3760682;
        double r3760684 = j;
        double r3760685 = 27.0;
        double r3760686 = r3760684 * r3760685;
        double r3760687 = k;
        double r3760688 = r3760686 * r3760687;
        double r3760689 = r3760683 - r3760688;
        return r3760689;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r3760690 = t;
        double r3760691 = -3.88366167002168e-170;
        bool r3760692 = r3760690 <= r3760691;
        double r3760693 = x;
        double r3760694 = 18.0;
        double r3760695 = r3760693 * r3760694;
        double r3760696 = y;
        double r3760697 = r3760695 * r3760696;
        double r3760698 = cbrt(r3760697);
        double r3760699 = r3760698 * r3760698;
        double r3760700 = r3760699 * r3760698;
        double r3760701 = z;
        double r3760702 = r3760700 * r3760701;
        double r3760703 = 4.0;
        double r3760704 = a;
        double r3760705 = r3760703 * r3760704;
        double r3760706 = r3760702 - r3760705;
        double r3760707 = b;
        double r3760708 = c;
        double r3760709 = r3760707 * r3760708;
        double r3760710 = k;
        double r3760711 = 27.0;
        double r3760712 = j;
        double r3760713 = r3760711 * r3760712;
        double r3760714 = r3760703 * r3760693;
        double r3760715 = i;
        double r3760716 = r3760714 * r3760715;
        double r3760717 = fma(r3760710, r3760713, r3760716);
        double r3760718 = r3760709 - r3760717;
        double r3760719 = fma(r3760706, r3760690, r3760718);
        double r3760720 = 4.6889661927618364e-163;
        bool r3760721 = r3760690 <= r3760720;
        double r3760722 = -r3760703;
        double r3760723 = r3760722 * r3760704;
        double r3760724 = fma(r3760723, r3760690, r3760718);
        double r3760725 = r3760721 ? r3760724 : r3760719;
        double r3760726 = r3760692 ? r3760719 : r3760725;
        return r3760726;
}

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 < -3.88366167002168e-170 or 4.6889661927618364e-163 < t

    1. Initial program 3.7

      \[\left(\left(\left(\left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4.0\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(j \cdot 27.0\right) \cdot k\]

    2. Simplified3.7

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z - a \cdot 4.0\right), t, \left(c \cdot b - \mathsf{fma}\left(k, \left(27.0 \cdot j\right), \left(\left(x \cdot 4.0\right) \cdot i\right)\right)\right)\right)}\]
    3. Using strategy rm

    4. Applied add-cube-cbrt3.9

      \[\leadsto \mathsf{fma}\left(\left(\color{blue}{\left(\left(\sqrt[3]{\left(x \cdot 18.0\right) \cdot y} \cdot \sqrt[3]{\left(x \cdot 18.0\right) \cdot y}\right) \cdot \sqrt[3]{\left(x \cdot 18.0\right) \cdot y}\right)} \cdot z - a \cdot 4.0\right), t, \left(c \cdot b - \mathsf{fma}\left(k, \left(27.0 \cdot j\right), \left(\left(x \cdot 4.0\right) \cdot i\right)\right)\right)\right)\]

    if -3.88366167002168e-170 < t < 4.6889661927618364e-163

    1. Initial program 9.3

      \[\left(\left(\left(\left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4.0\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(j \cdot 27.0\right) \cdot k\]

    2. Simplified9.3

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z - a \cdot 4.0\right), t, \left(c \cdot b - \mathsf{fma}\left(k, \left(27.0 \cdot j\right), \left(\left(x \cdot 4.0\right) \cdot i\right)\right)\right)\right)}\]

    3. Taylor expanded around 0 5.9

      \[\leadsto \mathsf{fma}\left(\left(\color{blue}{0} - a \cdot 4.0\right), t, \left(c \cdot b - \mathsf{fma}\left(k, \left(27.0 \cdot j\right), \left(\left(x \cdot 4.0\right) \cdot i\right)\right)\right)\right)\]
  3. Recombined 2 regimes into one program.

  4. Final simplification4.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -3.88366167002168 \cdot 10^{-170}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\left(\sqrt[3]{\left(x \cdot 18.0\right) \cdot y} \cdot \sqrt[3]{\left(x \cdot 18.0\right) \cdot y}\right) \cdot \sqrt[3]{\left(x \cdot 18.0\right) \cdot y}\right) \cdot z - 4.0 \cdot a\right), t, \left(b \cdot c - \mathsf{fma}\left(k, \left(27.0 \cdot j\right), \left(\left(4.0 \cdot x\right) \cdot i\right)\right)\right)\right)\\ \mathbf{elif}\;t \le 4.6889661927618364 \cdot 10^{-163}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(-4.0\right) \cdot a\right), t, \left(b \cdot c - \mathsf{fma}\left(k, \left(27.0 \cdot j\right), \left(\left(4.0 \cdot x\right) \cdot i\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\left(\sqrt[3]{\left(x \cdot 18.0\right) \cdot y} \cdot \sqrt[3]{\left(x \cdot 18.0\right) \cdot y}\right) \cdot \sqrt[3]{\left(x \cdot 18.0\right) \cdot y}\right) \cdot z - 4.0 \cdot a\right), t, \left(b \cdot c - \mathsf{fma}\left(k, \left(27.0 \cdot j\right), \left(\left(4.0 \cdot x\right) \cdot i\right)\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019133 +o rules:numerics
(FPCore (x y z t a b c i j k)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1"
  (- (- (+ (- (* (* (* (* x 18.0) y) z) t) (* (* a 4.0) t)) (* b c)) (* (* x 4.0) i)) (* (* j 27.0) k)))