Average Error: 5.1 → 4.1
Time: 14.9s
Precision: 64
\[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -2.05110049757782854 \cdot 10^{65}:\\ \;\;\;\;\mathsf{fma}\left(t, x \cdot \left(\left(18 \cdot y\right) \cdot z\right) - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, \left(j \cdot 27\right) \cdot k\right)\right)\\ \mathbf{elif}\;t \le 9.3308809483877645 \cdot 10^{-261}:\\ \;\;\;\;\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot \left(z \cdot t\right) - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - a \cdot \left(4 \cdot t\right)\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\\ \end{array}\]
\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 -2.05110049757782854 \cdot 10^{65}:\\
\;\;\;\;\mathsf{fma}\left(t, x \cdot \left(\left(18 \cdot y\right) \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)\\

\mathbf{elif}\;t \le 9.3308809483877645 \cdot 10^{-261}:\\
\;\;\;\;\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}:\\
\;\;\;\;\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - a \cdot \left(4 \cdot t\right)\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\\

\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 r173799 = x;
        double r173800 = 18.0;
        double r173801 = r173799 * r173800;
        double r173802 = y;
        double r173803 = r173801 * r173802;
        double r173804 = z;
        double r173805 = r173803 * r173804;
        double r173806 = t;
        double r173807 = r173805 * r173806;
        double r173808 = a;
        double r173809 = 4.0;
        double r173810 = r173808 * r173809;
        double r173811 = r173810 * r173806;
        double r173812 = r173807 - r173811;
        double r173813 = b;
        double r173814 = c;
        double r173815 = r173813 * r173814;
        double r173816 = r173812 + r173815;
        double r173817 = r173799 * r173809;
        double r173818 = i;
        double r173819 = r173817 * r173818;
        double r173820 = r173816 - r173819;
        double r173821 = j;
        double r173822 = 27.0;
        double r173823 = r173821 * r173822;
        double r173824 = k;
        double r173825 = r173823 * r173824;
        double r173826 = r173820 - r173825;
        return r173826;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r173827 = t;
        double r173828 = -2.0511004975778285e+65;
        bool r173829 = r173827 <= r173828;
        double r173830 = x;
        double r173831 = 18.0;
        double r173832 = y;
        double r173833 = r173831 * r173832;
        double r173834 = z;
        double r173835 = r173833 * r173834;
        double r173836 = r173830 * r173835;
        double r173837 = a;
        double r173838 = 4.0;
        double r173839 = r173837 * r173838;
        double r173840 = r173836 - r173839;
        double r173841 = b;
        double r173842 = c;
        double r173843 = r173841 * r173842;
        double r173844 = i;
        double r173845 = r173838 * r173844;
        double r173846 = j;
        double r173847 = 27.0;
        double r173848 = r173846 * r173847;
        double r173849 = k;
        double r173850 = r173848 * r173849;
        double r173851 = fma(r173830, r173845, r173850);
        double r173852 = r173843 - r173851;
        double r173853 = fma(r173827, r173840, r173852);
        double r173854 = 9.330880948387764e-261;
        bool r173855 = r173827 <= r173854;
        double r173856 = r173830 * r173831;
        double r173857 = r173856 * r173832;
        double r173858 = r173834 * r173827;
        double r173859 = r173857 * r173858;
        double r173860 = r173839 * r173827;
        double r173861 = r173859 - r173860;
        double r173862 = r173861 + r173843;
        double r173863 = r173830 * r173838;
        double r173864 = r173863 * r173844;
        double r173865 = r173862 - r173864;
        double r173866 = r173865 - r173850;
        double r173867 = r173857 * r173834;
        double r173868 = r173867 * r173827;
        double r173869 = r173838 * r173827;
        double r173870 = r173837 * r173869;
        double r173871 = r173868 - r173870;
        double r173872 = r173871 + r173843;
        double r173873 = r173872 - r173864;
        double r173874 = r173873 - r173850;
        double r173875 = r173855 ? r173866 : r173874;
        double r173876 = r173829 ? r173853 : r173875;
        return r173876;
}

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 t < -2.0511004975778285e+65

    1. Initial program 1.2

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

    2. Simplified1.2

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

    4. Applied associate-*l*1.3

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

    6. Applied associate-*l*1.9

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

    if -2.0511004975778285e+65 < t < 9.330880948387764e-261

    1. Initial program 6.8

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

    3. Applied associate-*l*4.2

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

    if 9.330880948387764e-261 < t

    1. Initial program 4.5

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

    3. Applied associate-*l*4.6

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

  4. Final simplification4.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -2.05110049757782854 \cdot 10^{65}:\\ \;\;\;\;\mathsf{fma}\left(t, x \cdot \left(\left(18 \cdot y\right) \cdot z\right) - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, \left(j \cdot 27\right) \cdot k\right)\right)\\ \mathbf{elif}\;t \le 9.3308809483877645 \cdot 10^{-261}:\\ \;\;\;\;\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot \left(z \cdot t\right) - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - a \cdot \left(4 \cdot t\right)\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\\ \end{array}\]

Reproduce

herbie shell --seed 2020060 +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)))