Linear.Matrix:det33 from linear-1.19.1.3

Average Error: 12.6 → 12.1

Time: 23.8s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \le -1.218634213792080201745340966505075594605 \cdot 10^{-161}:\\ \;\;\;\;\left(t \cdot \left(j \cdot c\right) + \left(-i \cdot \left(j \cdot y\right)\right)\right) + \left(\left(\left(-\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \left(\sqrt[3]{t} \cdot \left(x \cdot a\right)\right)\right) + x \cdot \left(z \cdot y\right)\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right)\\ \mathbf{elif}\;b \le 4.019988014933805737651483930381726228251 \cdot 10^{-236}:\\ \;\;\;\;\left(x \cdot \left(z \cdot y\right) + \left(-a \cdot \left(x \cdot t\right)\right)\right) + \left(t \cdot \left(j \cdot c\right) + \left(-i \cdot \left(j \cdot y\right)\right)\right)\\ \mathbf{elif}\;b \le 4.890488450149844969276568701808490405517 \cdot 10^{-215}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot 0\right) + j \cdot \left(c \cdot t - i \cdot y\right)\\ \mathbf{elif}\;b \le 4.23029005514141909806817664724091704732 \cdot 10^{-90}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(z \cdot \left(b \cdot c\right) + b \cdot \left(-i \cdot a\right)\right)\right) + \left(t \cdot \left(j \cdot c\right) + \left(-i \cdot \left(j \cdot y\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(-t \cdot \left(x \cdot a\right)\right) + x \cdot \left(z \cdot y\right)\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(t \cdot \left(j \cdot c\right) + \left(-\left(i \cdot j\right) \cdot y\right)\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r393846 = x;
        double r393847 = y;
        double r393848 = z;
        double r393849 = r393847 * r393848;
        double r393850 = t;
        double r393851 = a;
        double r393852 = r393850 * r393851;
        double r393853 = r393849 - r393852;
        double r393854 = r393846 * r393853;
        double r393855 = b;
        double r393856 = c;
        double r393857 = r393856 * r393848;
        double r393858 = i;
        double r393859 = r393858 * r393851;
        double r393860 = r393857 - r393859;
        double r393861 = r393855 * r393860;
        double r393862 = r393854 - r393861;
        double r393863 = j;
        double r393864 = r393856 * r393850;
        double r393865 = r393858 * r393847;
        double r393866 = r393864 - r393865;
        double r393867 = r393863 * r393866;
        double r393868 = r393862 + r393867;
        return r393868;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r393869 = b;
        double r393870 = -1.2186342137920802e-161;
        bool r393871 = r393869 <= r393870;
        double r393872 = t;
        double r393873 = j;
        double r393874 = c;
        double r393875 = r393873 * r393874;
        double r393876 = r393872 * r393875;
        double r393877 = i;
        double r393878 = y;
        double r393879 = r393873 * r393878;
        double r393880 = r393877 * r393879;
        double r393881 = -r393880;
        double r393882 = r393876 + r393881;
        double r393883 = cbrt(r393872);
        double r393884 = r393883 * r393883;
        double r393885 = x;
        double r393886 = a;
        double r393887 = r393885 * r393886;
        double r393888 = r393883 * r393887;
        double r393889 = r393884 * r393888;
        double r393890 = -r393889;
        double r393891 = z;
        double r393892 = r393891 * r393878;
        double r393893 = r393885 * r393892;
        double r393894 = r393890 + r393893;
        double r393895 = r393874 * r393891;
        double r393896 = r393877 * r393886;
        double r393897 = r393895 - r393896;
        double r393898 = r393869 * r393897;
        double r393899 = r393894 - r393898;
        double r393900 = r393882 + r393899;
        double r393901 = 4.019988014933806e-236;
        bool r393902 = r393869 <= r393901;
        double r393903 = r393885 * r393872;
        double r393904 = r393886 * r393903;
        double r393905 = -r393904;
        double r393906 = r393893 + r393905;
        double r393907 = r393906 + r393882;
        double r393908 = 4.890488450149845e-215;
        bool r393909 = r393869 <= r393908;
        double r393910 = r393878 * r393891;
        double r393911 = r393872 * r393886;
        double r393912 = r393910 - r393911;
        double r393913 = r393885 * r393912;
        double r393914 = 0.0;
        double r393915 = r393869 * r393914;
        double r393916 = r393913 - r393915;
        double r393917 = r393874 * r393872;
        double r393918 = r393877 * r393878;
        double r393919 = r393917 - r393918;
        double r393920 = r393873 * r393919;
        double r393921 = r393916 + r393920;
        double r393922 = 4.230290055141419e-90;
        bool r393923 = r393869 <= r393922;
        double r393924 = r393869 * r393874;
        double r393925 = r393891 * r393924;
        double r393926 = -r393896;
        double r393927 = r393869 * r393926;
        double r393928 = r393925 + r393927;
        double r393929 = r393913 - r393928;
        double r393930 = r393929 + r393882;
        double r393931 = r393872 * r393887;
        double r393932 = -r393931;
        double r393933 = r393932 + r393893;
        double r393934 = r393933 - r393898;
        double r393935 = r393877 * r393873;
        double r393936 = r393935 * r393878;
        double r393937 = -r393936;
        double r393938 = r393876 + r393937;
        double r393939 = r393934 + r393938;
        double r393940 = r393923 ? r393930 : r393939;
        double r393941 = r393909 ? r393921 : r393940;
        double r393942 = r393902 ? r393907 : r393941;
        double r393943 = r393871 ? r393900 : r393942;
        return r393943;
}

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

Target

Original	12.6
Target	16.4
Herbie	12.1

\[\begin{array}{l} \mathbf{if}\;t \lt -8.12097891919591218149793027759825150959 \cdot 10^{-33}:\\ \;\;\;\;x \cdot \left(z \cdot y - a \cdot t\right) - \left(b \cdot \left(z \cdot c - a \cdot i\right) - \left(c \cdot t - y \cdot i\right) \cdot j\right)\\ \mathbf{elif}\;t \lt -4.712553818218485141757938537793350881052 \cdot 10^{-169}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \frac{j \cdot \left({\left(c \cdot t\right)}^{2} - {\left(i \cdot y\right)}^{2}\right)}{c \cdot t + i \cdot y}\\ \mathbf{elif}\;t \lt -7.633533346031583686060259351057142920433 \cdot 10^{-308}:\\ \;\;\;\;x \cdot \left(z \cdot y - a \cdot t\right) - \left(b \cdot \left(z \cdot c - a \cdot i\right) - \left(c \cdot t - y \cdot i\right) \cdot j\right)\\ \mathbf{elif}\;t \lt 1.053588855745548710002760210539645467715 \cdot 10^{-139}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \frac{j \cdot \left({\left(c \cdot t\right)}^{2} - {\left(i \cdot y\right)}^{2}\right)}{c \cdot t + i \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(z \cdot y - a \cdot t\right) - \left(b \cdot \left(z \cdot c - a \cdot i\right) - \left(c \cdot t - y \cdot i\right) \cdot j\right)\\ \end{array}\]

Derivation

1. Split input into 5 regimes

2. if b < -1.2186342137920802e-161

   1. Initial program 10.8

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

   3. Applied sub-neg 10.8

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

   4. Applied distribute-lft-in 10.8

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

   5. Simplified 10.9

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

   6. Simplified 10.5

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

   8. Applied sub-neg 10.5

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

   9. Applied distribute-lft-in 10.5

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

   10. Simplified 10.5

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

   11. Simplified 11.3

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

   13. Applied *-un-lft-identity 11.3

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

   14. Applied associate-*l* 11.3

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

   15. Simplified 10.6

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

   17. Applied add-cube-cbrt 10.7

       \[\leadsto \left(\left(x \cdot \left(z \cdot y\right) + \left(-1 \cdot \left(\color{blue}{\left(\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \sqrt[3]{t}\right)} \cdot \left(x \cdot a\right)\right)\right)\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(t \cdot \left(j \cdot c\right) + \left(-i \cdot \left(j \cdot y\right)\right)\right)\]

   18. Applied associate-*l* 10.7

       \[\leadsto \left(\left(x \cdot \left(z \cdot y\right) + \left(-1 \cdot \color{blue}{\left(\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \left(\sqrt[3]{t} \cdot \left(x \cdot a\right)\right)\right)}\right)\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(t \cdot \left(j \cdot c\right) + \left(-i \cdot \left(j \cdot y\right)\right)\right)\]

   if -1.2186342137920802e-161 < b < 4.019988014933806e-236

   1. Initial program 18.3

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

   3. Applied sub-neg 18.3

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

   4. Applied distribute-lft-in 18.3

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

   5. Simplified 18.4

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

   6. Simplified 18.1

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

   8. Applied sub-neg 18.1

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

   9. Applied distribute-lft-in 18.1

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

   10. Simplified 18.1

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

   11. Simplified 18.5

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

   12. Taylor expanded around 0 17.4

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

   if 4.019988014933806e-236 < b < 4.890488450149845e-215

   1. Initial program 17.7

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

   2. Taylor expanded around 0 17.9

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

   if 4.890488450149845e-215 < b < 4.230290055141419e-90

   1. Initial program 16.9

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

   3. Applied sub-neg 16.9

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

   4. Applied distribute-lft-in 16.9

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

   5. Simplified 17.1

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

   6. Simplified 17.0

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

   8. Applied sub-neg 17.0

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

   9. Applied distribute-lft-in 17.0

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

   10. Simplified 14.4

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

   if 4.230290055141419e-90 < b

   1. Initial program 7.6

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

   3. Applied sub-neg 7.6

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

   4. Applied distribute-lft-in 7.6

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

   5. Simplified 7.7

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

   6. Simplified 7.9

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

   8. Applied sub-neg 7.9

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

   9. Applied distribute-lft-in 7.9

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

   10. Simplified 7.9

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

   11. Simplified 8.8

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

   13. Applied *-un-lft-identity 8.8

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

   14. Applied associate-*l* 8.8

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

   15. Simplified 8.3

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

   17. Applied associate-*r* 7.9

       \[\leadsto \left(\left(x \cdot \left(z \cdot y\right) + \left(-1 \cdot \left(t \cdot \left(x \cdot a\right)\right)\right)\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(t \cdot \left(j \cdot c\right) + \left(-\color{blue}{\left(i \cdot j\right) \cdot y}\right)\right)\]
3. Recombined 5 regimes into one program.

4. Final simplification 12.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.218634213792080201745340966505075594605 \cdot 10^{-161}:\\ \;\;\;\;\left(t \cdot \left(j \cdot c\right) + \left(-i \cdot \left(j \cdot y\right)\right)\right) + \left(\left(\left(-\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \left(\sqrt[3]{t} \cdot \left(x \cdot a\right)\right)\right) + x \cdot \left(z \cdot y\right)\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right)\\ \mathbf{elif}\;b \le 4.019988014933805737651483930381726228251 \cdot 10^{-236}:\\ \;\;\;\;\left(x \cdot \left(z \cdot y\right) + \left(-a \cdot \left(x \cdot t\right)\right)\right) + \left(t \cdot \left(j \cdot c\right) + \left(-i \cdot \left(j \cdot y\right)\right)\right)\\ \mathbf{elif}\;b \le 4.890488450149844969276568701808490405517 \cdot 10^{-215}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot 0\right) + j \cdot \left(c \cdot t - i \cdot y\right)\\ \mathbf{elif}\;b \le 4.23029005514141909806817664724091704732 \cdot 10^{-90}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(z \cdot \left(b \cdot c\right) + b \cdot \left(-i \cdot a\right)\right)\right) + \left(t \cdot \left(j \cdot c\right) + \left(-i \cdot \left(j \cdot y\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(-t \cdot \left(x \cdot a\right)\right) + x \cdot \left(z \cdot y\right)\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(t \cdot \left(j \cdot c\right) + \left(-\left(i \cdot j\right) \cdot y\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019347 +o rules:numerics
(FPCore (x y z t a b c i j)
  :name "Linear.Matrix:det33 from linear-1.19.1.3"
  :precision binary64

  :herbie-target
  (if (< t -8.120978919195912e-33) (- (* x (- (* z y) (* a t))) (- (* b (- (* z c) (* a i))) (* (- (* c t) (* y i)) j))) (if (< t -4.712553818218485e-169) (+ (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* i a)))) (/ (* j (- (pow (* c t) 2) (pow (* i y) 2))) (+ (* c t) (* i y)))) (if (< t -7.633533346031584e-308) (- (* x (- (* z y) (* a t))) (- (* b (- (* z c) (* a i))) (* (- (* c t) (* y i)) j))) (if (< t 1.0535888557455487e-139) (+ (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* i a)))) (/ (* j (- (pow (* c t) 2) (pow (* i y) 2))) (+ (* c t) (* i y)))) (- (* x (- (* z y) (* a t))) (- (* b (- (* z c) (* a i))) (* (- (* c t) (* y i)) j)))))))

  (+ (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* i a)))) (* j (- (* c t) (* i y)))))