Linear.Matrix:det33 from linear-1.19.1.3

Average Error: 12.3 → 9.8

Time: 24.6s

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 -5.40768492793686555941761981699772927749 \cdot 10^{-39}:\\ \;\;\;\;\left(j \cdot \left(t \cdot c - y \cdot i\right) + \left(y \cdot \left(x \cdot z\right) - t \cdot \left(x \cdot a\right)\right)\right) + \left(a \cdot i - c \cdot z\right) \cdot b\\ \mathbf{elif}\;b \le -1.310708033188256852286497632350011018004 \cdot 10^{-167}:\\ \;\;\;\;\left(\left(j \cdot \left(-y \cdot i\right) + \left(j \cdot c\right) \cdot t\right) + \left(y \cdot \left(x \cdot z\right) - \left(t \cdot x\right) \cdot a\right)\right) + \left(\left(i \cdot b\right) \cdot a - \left(b \cdot z\right) \cdot c\right)\\ \mathbf{elif}\;b \le -5.939141620508838471044724084871683968029 \cdot 10^{-276}:\\ \;\;\;\;\left(\left(x \cdot \left(y \cdot z\right) - \left(t \cdot x\right) \cdot a\right) + j \cdot \left(t \cdot c - y \cdot i\right)\right) + \left(\sqrt[3]{\left(i \cdot b\right) \cdot a} \cdot \left(\sqrt[3]{\left(i \cdot b\right) \cdot a} \cdot \sqrt[3]{\left(i \cdot b\right) \cdot a}\right) - \left(b \cdot z\right) \cdot c\right)\\ \mathbf{elif}\;b \le 2.248851513589900759233480267753595172809 \cdot 10^{-207}:\\ \;\;\;\;\left(\left(j \cdot \left(-y \cdot i\right) + \left(j \cdot c\right) \cdot t\right) + \left(y \cdot \left(x \cdot z\right) - \left(t \cdot x\right) \cdot a\right)\right) + \left(\left(i \cdot b\right) \cdot a - \left(b \cdot z\right) \cdot c\right)\\ \mathbf{elif}\;b \le 1.659927328976305946610149570336434936816 \cdot 10^{118}:\\ \;\;\;\;\left(j \cdot \left(t \cdot c - y \cdot i\right) + \left(x \cdot \left(y \cdot z\right) - \sqrt[3]{a} \cdot \left(t \cdot \left(\left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right) \cdot x\right)\right)\right)\right) + \left(\left(i \cdot b\right) \cdot a - \left(b \cdot z\right) \cdot c\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot i - c \cdot z\right) \cdot b + \left(\left(\left(j \cdot c\right) \cdot t + y \cdot \left(\left(-i\right) \cdot j\right)\right) + \left(y \cdot z - t \cdot a\right) \cdot x\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 r93734 = x;
        double r93735 = y;
        double r93736 = z;
        double r93737 = r93735 * r93736;
        double r93738 = t;
        double r93739 = a;
        double r93740 = r93738 * r93739;
        double r93741 = r93737 - r93740;
        double r93742 = r93734 * r93741;
        double r93743 = b;
        double r93744 = c;
        double r93745 = r93744 * r93736;
        double r93746 = i;
        double r93747 = r93746 * r93739;
        double r93748 = r93745 - r93747;
        double r93749 = r93743 * r93748;
        double r93750 = r93742 - r93749;
        double r93751 = j;
        double r93752 = r93744 * r93738;
        double r93753 = r93746 * r93735;
        double r93754 = r93752 - r93753;
        double r93755 = r93751 * r93754;
        double r93756 = r93750 + r93755;
        return r93756;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r93757 = b;
        double r93758 = -5.4076849279368656e-39;
        bool r93759 = r93757 <= r93758;
        double r93760 = j;
        double r93761 = t;
        double r93762 = c;
        double r93763 = r93761 * r93762;
        double r93764 = y;
        double r93765 = i;
        double r93766 = r93764 * r93765;
        double r93767 = r93763 - r93766;
        double r93768 = r93760 * r93767;
        double r93769 = x;
        double r93770 = z;
        double r93771 = r93769 * r93770;
        double r93772 = r93764 * r93771;
        double r93773 = a;
        double r93774 = r93769 * r93773;
        double r93775 = r93761 * r93774;
        double r93776 = r93772 - r93775;
        double r93777 = r93768 + r93776;
        double r93778 = r93773 * r93765;
        double r93779 = r93762 * r93770;
        double r93780 = r93778 - r93779;
        double r93781 = r93780 * r93757;
        double r93782 = r93777 + r93781;
        double r93783 = -1.3107080331882569e-167;
        bool r93784 = r93757 <= r93783;
        double r93785 = -r93766;
        double r93786 = r93760 * r93785;
        double r93787 = r93760 * r93762;
        double r93788 = r93787 * r93761;
        double r93789 = r93786 + r93788;
        double r93790 = r93761 * r93769;
        double r93791 = r93790 * r93773;
        double r93792 = r93772 - r93791;
        double r93793 = r93789 + r93792;
        double r93794 = r93765 * r93757;
        double r93795 = r93794 * r93773;
        double r93796 = r93757 * r93770;
        double r93797 = r93796 * r93762;
        double r93798 = r93795 - r93797;
        double r93799 = r93793 + r93798;
        double r93800 = -5.9391416205088385e-276;
        bool r93801 = r93757 <= r93800;
        double r93802 = r93764 * r93770;
        double r93803 = r93769 * r93802;
        double r93804 = r93803 - r93791;
        double r93805 = r93804 + r93768;
        double r93806 = cbrt(r93795);
        double r93807 = r93806 * r93806;
        double r93808 = r93806 * r93807;
        double r93809 = r93808 - r93797;
        double r93810 = r93805 + r93809;
        double r93811 = 2.2488515135899008e-207;
        bool r93812 = r93757 <= r93811;
        double r93813 = 1.659927328976306e+118;
        bool r93814 = r93757 <= r93813;
        double r93815 = cbrt(r93773);
        double r93816 = r93815 * r93815;
        double r93817 = r93816 * r93769;
        double r93818 = r93761 * r93817;
        double r93819 = r93815 * r93818;
        double r93820 = r93803 - r93819;
        double r93821 = r93768 + r93820;
        double r93822 = r93821 + r93798;
        double r93823 = -r93765;
        double r93824 = r93823 * r93760;
        double r93825 = r93764 * r93824;
        double r93826 = r93788 + r93825;
        double r93827 = r93761 * r93773;
        double r93828 = r93802 - r93827;
        double r93829 = r93828 * r93769;
        double r93830 = r93826 + r93829;
        double r93831 = r93781 + r93830;
        double r93832 = r93814 ? r93822 : r93831;
        double r93833 = r93812 ? r93799 : r93832;
        double r93834 = r93801 ? r93810 : r93833;
        double r93835 = r93784 ? r93799 : r93834;
        double r93836 = r93759 ? r93782 : r93835;
        return r93836;
}

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 5 regimes

2. if b < -5.4076849279368656e-39

   1. Initial program 8.0

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

   2. Simplified 8.0

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

   3. Taylor expanded around inf 9.4

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

   4. Simplified 8.2

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

   if -5.4076849279368656e-39 < b < -1.3107080331882569e-167 or -5.9391416205088385e-276 < b < 2.2488515135899008e-207

   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. Simplified 15.8

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

   3. Taylor expanded around inf 10.5

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

   4. Simplified 10.4

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

   5. Taylor expanded around inf 10.2

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

   6. Simplified 10.2

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

   8. Applied sub-neg 10.2

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

   9. Applied distribute-lft-in 10.2

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

   10. Simplified 10.9

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

   11. Simplified 10.9

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

   13. Applied associate-*r* 10.5

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

   if -1.3107080331882569e-167 < b < -5.9391416205088385e-276

   1. Initial program 17.3

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

   2. Simplified 17.3

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

   3. Taylor expanded around inf 10.9

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

   4. Simplified 11.0

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

   5. Taylor expanded around inf 10.0

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

   6. Simplified 10.0

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

   8. Applied add-cube-cbrt 10.1

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

   9. Simplified 10.1

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

   10. Simplified 10.1

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

   if 2.2488515135899008e-207 < b < 1.659927328976306e+118

   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. Simplified 12.9

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

   3. Taylor expanded around inf 11.2

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

   4. Simplified 11.3

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

   5. Taylor expanded around inf 11.2

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

   6. Simplified 11.2

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

   8. Applied add-cube-cbrt 11.4

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

   9. Applied associate-*r* 11.4

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

   10. Simplified 11.3

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

   if 1.659927328976306e+118 < b

   1. Initial program 6.0

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

   2. Simplified 6.0

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

   4. Applied sub-neg 6.0

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

   5. Applied distribute-lft-in 6.0

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

   6. Simplified 6.5

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

   7. Simplified 6.6

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

4. Final simplification 9.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -5.40768492793686555941761981699772927749 \cdot 10^{-39}:\\ \;\;\;\;\left(j \cdot \left(t \cdot c - y \cdot i\right) + \left(y \cdot \left(x \cdot z\right) - t \cdot \left(x \cdot a\right)\right)\right) + \left(a \cdot i - c \cdot z\right) \cdot b\\ \mathbf{elif}\;b \le -1.310708033188256852286497632350011018004 \cdot 10^{-167}:\\ \;\;\;\;\left(\left(j \cdot \left(-y \cdot i\right) + \left(j \cdot c\right) \cdot t\right) + \left(y \cdot \left(x \cdot z\right) - \left(t \cdot x\right) \cdot a\right)\right) + \left(\left(i \cdot b\right) \cdot a - \left(b \cdot z\right) \cdot c\right)\\ \mathbf{elif}\;b \le -5.939141620508838471044724084871683968029 \cdot 10^{-276}:\\ \;\;\;\;\left(\left(x \cdot \left(y \cdot z\right) - \left(t \cdot x\right) \cdot a\right) + j \cdot \left(t \cdot c - y \cdot i\right)\right) + \left(\sqrt[3]{\left(i \cdot b\right) \cdot a} \cdot \left(\sqrt[3]{\left(i \cdot b\right) \cdot a} \cdot \sqrt[3]{\left(i \cdot b\right) \cdot a}\right) - \left(b \cdot z\right) \cdot c\right)\\ \mathbf{elif}\;b \le 2.248851513589900759233480267753595172809 \cdot 10^{-207}:\\ \;\;\;\;\left(\left(j \cdot \left(-y \cdot i\right) + \left(j \cdot c\right) \cdot t\right) + \left(y \cdot \left(x \cdot z\right) - \left(t \cdot x\right) \cdot a\right)\right) + \left(\left(i \cdot b\right) \cdot a - \left(b \cdot z\right) \cdot c\right)\\ \mathbf{elif}\;b \le 1.659927328976305946610149570336434936816 \cdot 10^{118}:\\ \;\;\;\;\left(j \cdot \left(t \cdot c - y \cdot i\right) + \left(x \cdot \left(y \cdot z\right) - \sqrt[3]{a} \cdot \left(t \cdot \left(\left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right) \cdot x\right)\right)\right)\right) + \left(\left(i \cdot b\right) \cdot a - \left(b \cdot z\right) \cdot c\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot i - c \cdot z\right) \cdot b + \left(\left(\left(j \cdot c\right) \cdot t + y \cdot \left(\left(-i\right) \cdot j\right)\right) + \left(y \cdot z - t \cdot a\right) \cdot x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019194 
(FPCore (x y z t a b c i j)
  :name "Linear.Matrix:det33 from linear-1.19.1.3"
  (+ (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* i a)))) (* j (- (* c t) (* i y)))))