Average Error: 5.4 → 2.1
Time: 25.6s
Precision: 64
\[\left(\left(\left(\left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4.0\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(j \cdot 27.0\right) \cdot k\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(\left(t \cdot \left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) - \left(a \cdot 4.0\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4.0\right) \cdot i = -\infty:\\ \;\;\;\;\mathsf{fma}\left(b, c, \left(\left(t \cdot x\right) \cdot \left(y \cdot 18.0\right)\right) \cdot z - \mathsf{fma}\left(4.0, \mathsf{fma}\left(t, a, x \cdot i\right), 27.0 \cdot \left(k \cdot j\right)\right)\right)\\ \mathbf{elif}\;\left(\left(t \cdot \left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) - \left(a \cdot 4.0\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4.0\right) \cdot i \le 2.403209667818115 \cdot 10^{+294}:\\ \;\;\;\;\left(\left(\left(t \cdot \left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) - \left(a \cdot 4.0\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4.0\right) \cdot i\right) - j \cdot \left(27.0 \cdot k\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(c \cdot b + \left(\left(y \cdot \left(\left(z \cdot x\right) \cdot t\right)\right) \cdot 18.0 - \left(a \cdot 4.0\right) \cdot t\right)\right) - \left(x \cdot 4.0\right) \cdot i\right) - \sqrt[3]{\left(k \cdot j\right) \cdot \left(27.0 \cdot \left(\left(27.0 \cdot \left(k \cdot j\right)\right) \cdot \left(27.0 \cdot \left(k \cdot j\right)\right)\right)\right)}\\ \end{array}\]
\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}\;\left(\left(t \cdot \left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) - \left(a \cdot 4.0\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4.0\right) \cdot i = -\infty:\\
\;\;\;\;\mathsf{fma}\left(b, c, \left(\left(t \cdot x\right) \cdot \left(y \cdot 18.0\right)\right) \cdot z - \mathsf{fma}\left(4.0, \mathsf{fma}\left(t, a, x \cdot i\right), 27.0 \cdot \left(k \cdot j\right)\right)\right)\\

\mathbf{elif}\;\left(\left(t \cdot \left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) - \left(a \cdot 4.0\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4.0\right) \cdot i \le 2.403209667818115 \cdot 10^{+294}:\\
\;\;\;\;\left(\left(\left(t \cdot \left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) - \left(a \cdot 4.0\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4.0\right) \cdot i\right) - j \cdot \left(27.0 \cdot k\right)\\

\mathbf{else}:\\
\;\;\;\;\left(\left(c \cdot b + \left(\left(y \cdot \left(\left(z \cdot x\right) \cdot t\right)\right) \cdot 18.0 - \left(a \cdot 4.0\right) \cdot t\right)\right) - \left(x \cdot 4.0\right) \cdot i\right) - \sqrt[3]{\left(k \cdot j\right) \cdot \left(27.0 \cdot \left(\left(27.0 \cdot \left(k \cdot j\right)\right) \cdot \left(27.0 \cdot \left(k \cdot j\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 r35392756 = x;
        double r35392757 = 18.0;
        double r35392758 = r35392756 * r35392757;
        double r35392759 = y;
        double r35392760 = r35392758 * r35392759;
        double r35392761 = z;
        double r35392762 = r35392760 * r35392761;
        double r35392763 = t;
        double r35392764 = r35392762 * r35392763;
        double r35392765 = a;
        double r35392766 = 4.0;
        double r35392767 = r35392765 * r35392766;
        double r35392768 = r35392767 * r35392763;
        double r35392769 = r35392764 - r35392768;
        double r35392770 = b;
        double r35392771 = c;
        double r35392772 = r35392770 * r35392771;
        double r35392773 = r35392769 + r35392772;
        double r35392774 = r35392756 * r35392766;
        double r35392775 = i;
        double r35392776 = r35392774 * r35392775;
        double r35392777 = r35392773 - r35392776;
        double r35392778 = j;
        double r35392779 = 27.0;
        double r35392780 = r35392778 * r35392779;
        double r35392781 = k;
        double r35392782 = r35392780 * r35392781;
        double r35392783 = r35392777 - r35392782;
        return r35392783;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r35392784 = t;
        double r35392785 = x;
        double r35392786 = 18.0;
        double r35392787 = r35392785 * r35392786;
        double r35392788 = y;
        double r35392789 = r35392787 * r35392788;
        double r35392790 = z;
        double r35392791 = r35392789 * r35392790;
        double r35392792 = r35392784 * r35392791;
        double r35392793 = a;
        double r35392794 = 4.0;
        double r35392795 = r35392793 * r35392794;
        double r35392796 = r35392795 * r35392784;
        double r35392797 = r35392792 - r35392796;
        double r35392798 = c;
        double r35392799 = b;
        double r35392800 = r35392798 * r35392799;
        double r35392801 = r35392797 + r35392800;
        double r35392802 = r35392785 * r35392794;
        double r35392803 = i;
        double r35392804 = r35392802 * r35392803;
        double r35392805 = r35392801 - r35392804;
        double r35392806 = -inf.0;
        bool r35392807 = r35392805 <= r35392806;
        double r35392808 = r35392784 * r35392785;
        double r35392809 = r35392788 * r35392786;
        double r35392810 = r35392808 * r35392809;
        double r35392811 = r35392810 * r35392790;
        double r35392812 = r35392785 * r35392803;
        double r35392813 = fma(r35392784, r35392793, r35392812);
        double r35392814 = 27.0;
        double r35392815 = k;
        double r35392816 = j;
        double r35392817 = r35392815 * r35392816;
        double r35392818 = r35392814 * r35392817;
        double r35392819 = fma(r35392794, r35392813, r35392818);
        double r35392820 = r35392811 - r35392819;
        double r35392821 = fma(r35392799, r35392798, r35392820);
        double r35392822 = 2.403209667818115e+294;
        bool r35392823 = r35392805 <= r35392822;
        double r35392824 = r35392814 * r35392815;
        double r35392825 = r35392816 * r35392824;
        double r35392826 = r35392805 - r35392825;
        double r35392827 = r35392790 * r35392785;
        double r35392828 = r35392827 * r35392784;
        double r35392829 = r35392788 * r35392828;
        double r35392830 = r35392829 * r35392786;
        double r35392831 = r35392830 - r35392796;
        double r35392832 = r35392800 + r35392831;
        double r35392833 = r35392832 - r35392804;
        double r35392834 = r35392818 * r35392818;
        double r35392835 = r35392814 * r35392834;
        double r35392836 = r35392817 * r35392835;
        double r35392837 = cbrt(r35392836);
        double r35392838 = r35392833 - r35392837;
        double r35392839 = r35392823 ? r35392826 : r35392838;
        double r35392840 = r35392807 ? r35392821 : r35392839;
        return r35392840;
}

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

Target

Original5.4
Target1.4
Herbie2.1
\[\begin{array}{l} \mathbf{if}\;t \lt -1.6210815397541398 \cdot 10^{-69}:\\ \;\;\;\;\left(\left(18.0 \cdot t\right) \cdot \left(\left(x \cdot y\right) \cdot z\right) - \left(a \cdot t + i \cdot x\right) \cdot 4.0\right) - \left(\left(k \cdot j\right) \cdot 27.0 - c \cdot b\right)\\ \mathbf{elif}\;t \lt 165.68027943805222:\\ \;\;\;\;\left(\left(18.0 \cdot y\right) \cdot \left(x \cdot \left(z \cdot t\right)\right) - \left(a \cdot t + i \cdot x\right) \cdot 4.0\right) + \left(c \cdot b - 27.0 \cdot \left(k \cdot j\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(18.0 \cdot t\right) \cdot \left(\left(x \cdot y\right) \cdot z\right) - \left(a \cdot t + i \cdot x\right) \cdot 4.0\right) - \left(\left(k \cdot j\right) \cdot 27.0 - c \cdot b\right)\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if (- (+ (- (* (* (* (* x 18.0) y) z) t) (* (* a 4.0) t)) (* b c)) (* (* x 4.0) i)) < -inf.0

    1. Initial program 60.5

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

    2. Simplified12.8

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

    if -inf.0 < (- (+ (- (* (* (* (* x 18.0) y) z) t) (* (* a 4.0) t)) (* b c)) (* (* x 4.0) i)) < 2.403209667818115e+294

    1. Initial program 0.3

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

    3. Applied associate-*l*0.3

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

    if 2.403209667818115e+294 < (- (+ (- (* (* (* (* x 18.0) y) z) t) (* (* a 4.0) t)) (* b c)) (* (* x 4.0) i))

    1. Initial program 42.0

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

    2. Taylor expanded around inf 27.1

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

    4. Applied associate-*r*27.2

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

    6. Applied associate-*r*10.2

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

    8. Applied add-cbrt-cube21.3

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

    9. Applied add-cbrt-cube21.3

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

    10. Applied add-cbrt-cube34.1

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

    11. Applied cbrt-unprod34.1

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

    12. Applied cbrt-unprod34.5

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

    13. Simplified21.9

      \[\leadsto \left(\left(\left(18.0 \cdot \left(\left(t \cdot \left(x \cdot z\right)\right) \cdot y\right) - \left(a \cdot 4.0\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4.0\right) \cdot i\right) - \sqrt[3]{\color{blue}{\left(\left(\left(27.0 \cdot \left(j \cdot k\right)\right) \cdot \left(27.0 \cdot \left(j \cdot k\right)\right)\right) \cdot 27.0\right) \cdot \left(j \cdot k\right)}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification2.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(t \cdot \left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) - \left(a \cdot 4.0\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4.0\right) \cdot i = -\infty:\\ \;\;\;\;\mathsf{fma}\left(b, c, \left(\left(t \cdot x\right) \cdot \left(y \cdot 18.0\right)\right) \cdot z - \mathsf{fma}\left(4.0, \mathsf{fma}\left(t, a, x \cdot i\right), 27.0 \cdot \left(k \cdot j\right)\right)\right)\\ \mathbf{elif}\;\left(\left(t \cdot \left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) - \left(a \cdot 4.0\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4.0\right) \cdot i \le 2.403209667818115 \cdot 10^{+294}:\\ \;\;\;\;\left(\left(\left(t \cdot \left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) - \left(a \cdot 4.0\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4.0\right) \cdot i\right) - j \cdot \left(27.0 \cdot k\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(c \cdot b + \left(\left(y \cdot \left(\left(z \cdot x\right) \cdot t\right)\right) \cdot 18.0 - \left(a \cdot 4.0\right) \cdot t\right)\right) - \left(x \cdot 4.0\right) \cdot i\right) - \sqrt[3]{\left(k \cdot j\right) \cdot \left(27.0 \cdot \left(\left(27.0 \cdot \left(k \cdot j\right)\right) \cdot \left(27.0 \cdot \left(k \cdot j\right)\right)\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019163 +o rules:numerics
(FPCore (x y z t a b c i j k)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, E"

  :herbie-target
  (if (< t -1.6210815397541398e-69) (- (- (* (* 18.0 t) (* (* x y) z)) (* (+ (* a t) (* i x)) 4.0)) (- (* (* k j) 27.0) (* c b))) (if (< t 165.68027943805222) (+ (- (* (* 18.0 y) (* x (* z t))) (* (+ (* a t) (* i x)) 4.0)) (- (* c b) (* 27.0 (* k j)))) (- (- (* (* 18.0 t) (* (* x y) z)) (* (+ (* a t) (* i x)) 4.0)) (- (* (* k j) 27.0) (* c b)))))

  (- (- (+ (- (* (* (* (* x 18.0) y) z) t) (* (* a 4.0) t)) (* b c)) (* (* x 4.0) i)) (* (* j 27.0) k)))