Linear.Matrix:det33 from linear-1.19.1.3

Average Error: 12.2 → 11.8

Time: 25.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;j \le -9.929536798422278605563067575704178711758 \cdot 10^{96}:\\ \;\;\;\;j \cdot \left(c \cdot t - i \cdot y\right) + \left(-b \cdot \left(c \cdot z - i \cdot a\right)\right)\\ \mathbf{elif}\;j \le -1.098607656333811782069347702956620830271 \cdot 10^{-146}:\\ \;\;\;\;\left(t \cdot \left(j \cdot c\right) + \left(-i \cdot \left(j \cdot y\right)\right)\right) + \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right)\\ \mathbf{elif}\;j \le -6.651508483195953646948775651706394971033 \cdot 10^{-258}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(z \cdot \left(b \cdot c\right) + \left(-i \cdot a\right) \cdot b\right)\right) + \left(c \cdot \left(t \cdot j\right) + \left(-i \cdot \left(j \cdot y\right)\right)\right)\\ \mathbf{elif}\;j \le 2.649066615389731199752472489106042554466 \cdot 10^{253}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(b \cdot \left(\sqrt[3]{c \cdot z - i \cdot a} \cdot \sqrt[3]{c \cdot z - i \cdot a}\right)\right) \cdot \sqrt[3]{c \cdot z - i \cdot a}\right) + \left(c \cdot \left(t \cdot j\right) + \left(-i \cdot \left(j \cdot y\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;j \cdot \left(c \cdot t - i \cdot y\right) + \left(-b \cdot \left(c \cdot z - i \cdot a\right)\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r80822 = x;
        double r80823 = y;
        double r80824 = z;
        double r80825 = r80823 * r80824;
        double r80826 = t;
        double r80827 = a;
        double r80828 = r80826 * r80827;
        double r80829 = r80825 - r80828;
        double r80830 = r80822 * r80829;
        double r80831 = b;
        double r80832 = c;
        double r80833 = r80832 * r80824;
        double r80834 = i;
        double r80835 = r80834 * r80827;
        double r80836 = r80833 - r80835;
        double r80837 = r80831 * r80836;
        double r80838 = r80830 - r80837;
        double r80839 = j;
        double r80840 = r80832 * r80826;
        double r80841 = r80834 * r80823;
        double r80842 = r80840 - r80841;
        double r80843 = r80839 * r80842;
        double r80844 = r80838 + r80843;
        return r80844;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r80845 = j;
        double r80846 = -9.929536798422279e+96;
        bool r80847 = r80845 <= r80846;
        double r80848 = c;
        double r80849 = t;
        double r80850 = r80848 * r80849;
        double r80851 = i;
        double r80852 = y;
        double r80853 = r80851 * r80852;
        double r80854 = r80850 - r80853;
        double r80855 = r80845 * r80854;
        double r80856 = b;
        double r80857 = z;
        double r80858 = r80848 * r80857;
        double r80859 = a;
        double r80860 = r80851 * r80859;
        double r80861 = r80858 - r80860;
        double r80862 = r80856 * r80861;
        double r80863 = -r80862;
        double r80864 = r80855 + r80863;
        double r80865 = -1.0986076563338118e-146;
        bool r80866 = r80845 <= r80865;
        double r80867 = r80845 * r80848;
        double r80868 = r80849 * r80867;
        double r80869 = r80845 * r80852;
        double r80870 = r80851 * r80869;
        double r80871 = -r80870;
        double r80872 = r80868 + r80871;
        double r80873 = x;
        double r80874 = r80852 * r80857;
        double r80875 = r80849 * r80859;
        double r80876 = r80874 - r80875;
        double r80877 = r80873 * r80876;
        double r80878 = r80877 - r80862;
        double r80879 = r80872 + r80878;
        double r80880 = -6.651508483195954e-258;
        bool r80881 = r80845 <= r80880;
        double r80882 = r80856 * r80848;
        double r80883 = r80857 * r80882;
        double r80884 = -r80860;
        double r80885 = r80884 * r80856;
        double r80886 = r80883 + r80885;
        double r80887 = r80877 - r80886;
        double r80888 = r80849 * r80845;
        double r80889 = r80848 * r80888;
        double r80890 = r80889 + r80871;
        double r80891 = r80887 + r80890;
        double r80892 = 2.6490666153897312e+253;
        bool r80893 = r80845 <= r80892;
        double r80894 = cbrt(r80861);
        double r80895 = r80894 * r80894;
        double r80896 = r80856 * r80895;
        double r80897 = r80896 * r80894;
        double r80898 = r80877 - r80897;
        double r80899 = r80898 + r80890;
        double r80900 = r80893 ? r80899 : r80864;
        double r80901 = r80881 ? r80891 : r80900;
        double r80902 = r80866 ? r80879 : r80901;
        double r80903 = r80847 ? r80864 : r80902;
        return r80903;
}

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

Derivation

1. Split input into 4 regimes

2. if j < -9.929536798422279e+96 or 2.6490666153897312e+253 < j

   1. Initial program 7.3

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

   2. Taylor expanded around 0 15.8

      \[\leadsto \left(\color{blue}{0} - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]

   if -9.929536798422279e+96 < j < -1.0986076563338118e-146

   1. Initial program 11.3

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

   3. Applied sub-neg 11.3

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

   4. Applied distribute-lft-in 11.3

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

   5. Simplified 11.3

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

   6. Simplified 10.4

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

   8. Applied associate-*l* 9.2

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

   10. Applied *-un-lft-identity 9.2

       \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(\color{blue}{\left(1 \cdot c\right)} \cdot \left(t \cdot j\right) + \left(-i \cdot \left(j \cdot y\right)\right)\right)\]

   11. Applied associate-*l* 9.2

       \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(\color{blue}{1 \cdot \left(c \cdot \left(t \cdot j\right)\right)} + \left(-i \cdot \left(j \cdot y\right)\right)\right)\]

   12. Simplified 9.1

       \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(1 \cdot \color{blue}{\left(t \cdot \left(j \cdot c\right)\right)} + \left(-i \cdot \left(j \cdot y\right)\right)\right)\]

   if -1.0986076563338118e-146 < j < -6.651508483195954e-258

   1. Initial program 17.1

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

   3. Applied sub-neg 17.1

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

   4. Applied distribute-lft-in 17.1

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

   5. Simplified 17.1

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

   6. Simplified 14.0

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

   8. Applied associate-*l* 10.9

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

   10. Applied sub-neg 10.9

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

   11. Applied distribute-lft-in 10.9

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

   12. Simplified 11.5

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

   13. Simplified 11.5

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

   if -6.651508483195954e-258 < j < 2.6490666153897312e+253

   1. Initial program 12.9

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

   3. Applied sub-neg 12.9

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

   4. Applied distribute-lft-in 12.9

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

   5. Simplified 12.9

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

   6. Simplified 12.4

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

   8. Applied associate-*l* 11.6

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

   10. Applied add-cube-cbrt 11.9

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

   11. Applied associate-*r* 11.9

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

4. Final simplification 11.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;j \le -9.929536798422278605563067575704178711758 \cdot 10^{96}:\\ \;\;\;\;j \cdot \left(c \cdot t - i \cdot y\right) + \left(-b \cdot \left(c \cdot z - i \cdot a\right)\right)\\ \mathbf{elif}\;j \le -1.098607656333811782069347702956620830271 \cdot 10^{-146}:\\ \;\;\;\;\left(t \cdot \left(j \cdot c\right) + \left(-i \cdot \left(j \cdot y\right)\right)\right) + \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right)\\ \mathbf{elif}\;j \le -6.651508483195953646948775651706394971033 \cdot 10^{-258}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(z \cdot \left(b \cdot c\right) + \left(-i \cdot a\right) \cdot b\right)\right) + \left(c \cdot \left(t \cdot j\right) + \left(-i \cdot \left(j \cdot y\right)\right)\right)\\ \mathbf{elif}\;j \le 2.649066615389731199752472489106042554466 \cdot 10^{253}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(b \cdot \left(\sqrt[3]{c \cdot z - i \cdot a} \cdot \sqrt[3]{c \cdot z - i \cdot a}\right)\right) \cdot \sqrt[3]{c \cdot z - i \cdot a}\right) + \left(c \cdot \left(t \cdot j\right) + \left(-i \cdot \left(j \cdot y\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;j \cdot \left(c \cdot t - i \cdot y\right) + \left(-b \cdot \left(c \cdot z - i \cdot a\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019325 
(FPCore (x y z t a b c i j)
  :name "Linear.Matrix:det33 from linear-1.19.1.3"
  :precision binary64
  (+ (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* i a)))) (* j (- (* c t) (* i y)))))