Linear.Matrix:det33 from linear-1.19.1.3

Average Error: 11.8 → 10.3

Time: 1.1m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;i \le -1453916164036345344:\\ \;\;\;\;j \cdot \left(c \cdot t - y \cdot i\right) + \left(\sqrt[3]{y \cdot z - t \cdot a} \cdot \left(\left(\sqrt[3]{y \cdot z - t \cdot a} \cdot \sqrt[3]{y \cdot z - t \cdot a}\right) \cdot x\right) - \left(\left(a \cdot b\right) \cdot \left(-i\right) + c \cdot \left(b \cdot z\right)\right)\right)\\ \mathbf{elif}\;i \le -5.217128050956565151940192447921636214319 \cdot 10^{-116}:\\ \;\;\;\;\left(\left(x \cdot \left(y \cdot z\right) - \left(x \cdot t\right) \cdot a\right) - \left(\sqrt[3]{b} \cdot \sqrt[3]{b}\right) \cdot \left(\left(c \cdot z - i \cdot a\right) \cdot \sqrt[3]{b}\right)\right) + j \cdot \left(c \cdot t - y \cdot i\right)\\ \mathbf{elif}\;i \le -7.642484941746692331031071219799992809581 \cdot 10^{-197}:\\ \;\;\;\;\left(\left(y \cdot z - t \cdot a\right) \cdot x - \left(\left(-b\right) \cdot \left(i \cdot a\right) + c \cdot \left(b \cdot z\right)\right)\right) + \left(\sqrt[3]{j} \cdot \sqrt[3]{j}\right) \cdot \left(\left(c \cdot t - y \cdot i\right) \cdot \sqrt[3]{j}\right)\\ \mathbf{elif}\;i \le 1.392000762622429387159763147815204755346 \cdot 10^{-21}:\\ \;\;\;\;\left(\left(y \cdot z - t \cdot a\right) \cdot x - \left(\sqrt[3]{b} \cdot \sqrt[3]{b}\right) \cdot \left(\sqrt[3]{\left(c \cdot z - i \cdot a\right) \cdot \sqrt[3]{b}} \cdot \left(\sqrt[3]{\left(c \cdot z - i \cdot a\right) \cdot \sqrt[3]{b}} \cdot \sqrt[3]{\left(c \cdot z - i \cdot a\right) \cdot \sqrt[3]{b}}\right)\right)\right) + j \cdot \left(c \cdot t - y \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;j \cdot \left(c \cdot t - y \cdot i\right) + \left(\sqrt[3]{y \cdot z - t \cdot a} \cdot \left(\left(\sqrt[3]{y \cdot z - t \cdot a} \cdot \sqrt[3]{y \cdot z - t \cdot a}\right) \cdot x\right) - \left(\left(a \cdot b\right) \cdot \left(-i\right) + c \cdot \left(b \cdot z\right)\right)\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r27723797 = x;
        double r27723798 = y;
        double r27723799 = z;
        double r27723800 = r27723798 * r27723799;
        double r27723801 = t;
        double r27723802 = a;
        double r27723803 = r27723801 * r27723802;
        double r27723804 = r27723800 - r27723803;
        double r27723805 = r27723797 * r27723804;
        double r27723806 = b;
        double r27723807 = c;
        double r27723808 = r27723807 * r27723799;
        double r27723809 = i;
        double r27723810 = r27723809 * r27723802;
        double r27723811 = r27723808 - r27723810;
        double r27723812 = r27723806 * r27723811;
        double r27723813 = r27723805 - r27723812;
        double r27723814 = j;
        double r27723815 = r27723807 * r27723801;
        double r27723816 = r27723809 * r27723798;
        double r27723817 = r27723815 - r27723816;
        double r27723818 = r27723814 * r27723817;
        double r27723819 = r27723813 + r27723818;
        return r27723819;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r27723820 = i;
        double r27723821 = -1.4539161640363453e+18;
        bool r27723822 = r27723820 <= r27723821;
        double r27723823 = j;
        double r27723824 = c;
        double r27723825 = t;
        double r27723826 = r27723824 * r27723825;
        double r27723827 = y;
        double r27723828 = r27723827 * r27723820;
        double r27723829 = r27723826 - r27723828;
        double r27723830 = r27723823 * r27723829;
        double r27723831 = z;
        double r27723832 = r27723827 * r27723831;
        double r27723833 = a;
        double r27723834 = r27723825 * r27723833;
        double r27723835 = r27723832 - r27723834;
        double r27723836 = cbrt(r27723835);
        double r27723837 = r27723836 * r27723836;
        double r27723838 = x;
        double r27723839 = r27723837 * r27723838;
        double r27723840 = r27723836 * r27723839;
        double r27723841 = b;
        double r27723842 = r27723833 * r27723841;
        double r27723843 = -r27723820;
        double r27723844 = r27723842 * r27723843;
        double r27723845 = r27723841 * r27723831;
        double r27723846 = r27723824 * r27723845;
        double r27723847 = r27723844 + r27723846;
        double r27723848 = r27723840 - r27723847;
        double r27723849 = r27723830 + r27723848;
        double r27723850 = -5.217128050956565e-116;
        bool r27723851 = r27723820 <= r27723850;
        double r27723852 = r27723838 * r27723832;
        double r27723853 = r27723838 * r27723825;
        double r27723854 = r27723853 * r27723833;
        double r27723855 = r27723852 - r27723854;
        double r27723856 = cbrt(r27723841);
        double r27723857 = r27723856 * r27723856;
        double r27723858 = r27723824 * r27723831;
        double r27723859 = r27723820 * r27723833;
        double r27723860 = r27723858 - r27723859;
        double r27723861 = r27723860 * r27723856;
        double r27723862 = r27723857 * r27723861;
        double r27723863 = r27723855 - r27723862;
        double r27723864 = r27723863 + r27723830;
        double r27723865 = -7.642484941746692e-197;
        bool r27723866 = r27723820 <= r27723865;
        double r27723867 = r27723835 * r27723838;
        double r27723868 = -r27723841;
        double r27723869 = r27723868 * r27723859;
        double r27723870 = r27723869 + r27723846;
        double r27723871 = r27723867 - r27723870;
        double r27723872 = cbrt(r27723823);
        double r27723873 = r27723872 * r27723872;
        double r27723874 = r27723829 * r27723872;
        double r27723875 = r27723873 * r27723874;
        double r27723876 = r27723871 + r27723875;
        double r27723877 = 1.3920007626224294e-21;
        bool r27723878 = r27723820 <= r27723877;
        double r27723879 = cbrt(r27723861);
        double r27723880 = r27723879 * r27723879;
        double r27723881 = r27723879 * r27723880;
        double r27723882 = r27723857 * r27723881;
        double r27723883 = r27723867 - r27723882;
        double r27723884 = r27723883 + r27723830;
        double r27723885 = r27723878 ? r27723884 : r27723849;
        double r27723886 = r27723866 ? r27723876 : r27723885;
        double r27723887 = r27723851 ? r27723864 : r27723886;
        double r27723888 = r27723822 ? r27723849 : r27723887;
        return r27723888;
}

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	11.8
Target	15.9
Herbie	10.3

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

Derivation

1. Split input into 4 regimes

2. if i < -1.4539161640363453e+18 or 1.3920007626224294e-21 < i

   1. Initial program 15.8

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

   3. Applied add-cube-cbrt 16.1

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

   4. Applied associate-*l* 16.1

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

   6. Applied sub-neg 16.1

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

   7. Applied distribute-lft-in 16.1

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

   8. Applied distribute-lft-in 16.1

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

   9. Simplified 16.0

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

   10. Simplified 15.8

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

   11. Taylor expanded around inf 15.9

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

   12. Simplified 11.9

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

   14. Applied add-cube-cbrt 12.1

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

   15. Applied associate-*r* 12.1

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

   if -1.4539161640363453e+18 < i < -5.217128050956565e-116

   1. Initial program 8.8

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

   3. Applied add-cube-cbrt 9.1

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

   4. Applied associate-*l* 9.2

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

   5. Taylor expanded around inf 8.2

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

   if -5.217128050956565e-116 < i < -7.642484941746692e-197

   1. Initial program 9.3

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

   3. Applied add-cube-cbrt 9.6

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

   4. Applied associate-*l* 9.6

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

   6. Applied sub-neg 9.6

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

   7. Applied distribute-lft-in 9.6

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

   8. Applied distribute-lft-in 9.6

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

   9. Simplified 8.6

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

   10. Simplified 8.5

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

   12. Applied add-cube-cbrt 8.9

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

   13. Applied associate-*l* 8.9

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

   if -7.642484941746692e-197 < i < 1.3920007626224294e-21

   1. Initial program 9.3

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

   3. Applied add-cube-cbrt 9.5

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

   4. Applied associate-*l* 9.5

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

   6. Applied add-cube-cbrt 9.6

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

4. Final simplification 10.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -1453916164036345344:\\ \;\;\;\;j \cdot \left(c \cdot t - y \cdot i\right) + \left(\sqrt[3]{y \cdot z - t \cdot a} \cdot \left(\left(\sqrt[3]{y \cdot z - t \cdot a} \cdot \sqrt[3]{y \cdot z - t \cdot a}\right) \cdot x\right) - \left(\left(a \cdot b\right) \cdot \left(-i\right) + c \cdot \left(b \cdot z\right)\right)\right)\\ \mathbf{elif}\;i \le -5.217128050956565151940192447921636214319 \cdot 10^{-116}:\\ \;\;\;\;\left(\left(x \cdot \left(y \cdot z\right) - \left(x \cdot t\right) \cdot a\right) - \left(\sqrt[3]{b} \cdot \sqrt[3]{b}\right) \cdot \left(\left(c \cdot z - i \cdot a\right) \cdot \sqrt[3]{b}\right)\right) + j \cdot \left(c \cdot t - y \cdot i\right)\\ \mathbf{elif}\;i \le -7.642484941746692331031071219799992809581 \cdot 10^{-197}:\\ \;\;\;\;\left(\left(y \cdot z - t \cdot a\right) \cdot x - \left(\left(-b\right) \cdot \left(i \cdot a\right) + c \cdot \left(b \cdot z\right)\right)\right) + \left(\sqrt[3]{j} \cdot \sqrt[3]{j}\right) \cdot \left(\left(c \cdot t - y \cdot i\right) \cdot \sqrt[3]{j}\right)\\ \mathbf{elif}\;i \le 1.392000762622429387159763147815204755346 \cdot 10^{-21}:\\ \;\;\;\;\left(\left(y \cdot z - t \cdot a\right) \cdot x - \left(\sqrt[3]{b} \cdot \sqrt[3]{b}\right) \cdot \left(\sqrt[3]{\left(c \cdot z - i \cdot a\right) \cdot \sqrt[3]{b}} \cdot \left(\sqrt[3]{\left(c \cdot z - i \cdot a\right) \cdot \sqrt[3]{b}} \cdot \sqrt[3]{\left(c \cdot z - i \cdot a\right) \cdot \sqrt[3]{b}}\right)\right)\right) + j \cdot \left(c \cdot t - y \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;j \cdot \left(c \cdot t - y \cdot i\right) + \left(\sqrt[3]{y \cdot z - t \cdot a} \cdot \left(\left(\sqrt[3]{y \cdot z - t \cdot a} \cdot \sqrt[3]{y \cdot z - t \cdot a}\right) \cdot x\right) - \left(\left(a \cdot b\right) \cdot \left(-i\right) + c \cdot \left(b \cdot z\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019168 
(FPCore (x y z t a b c i j)
  :name "Linear.Matrix:det33 from linear-1.19.1.3"

  :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) 2.0) (pow (* i y) 2.0))) (+ (* 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) 2.0) (pow (* i y) 2.0))) (+ (* 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)))))